Iterative Inference in a Chess-Playing Neural Network

TL;DR

This paper investigates how a chess neural network refines its representations across layers, revealing complex computational phases and late-layer reversals influenced by safety considerations.

Contribution

It extends the logit lens analysis to a chess engine, uncovering detailed insights into the layered computational processes and heuristic influences.

Findings

Capability improves across layers but in distinct phases

Move preferences are reevaluated throughout the network

Final layers prioritize safety over aggression

Abstract

Do neural networks build their representations through smooth, gradual refinement, or via more complex computational processes? We investigate this by extending the logit lens to analyze the policy network of Leela Chess Zero, a superhuman chess engine. Although playing strength and puzzle-solving ability improve consistently across layers, capability progression occurs in distinct computational phases with move preferences undergoing continuous reevaluation--move rankings remain poorly correlated with final outputs until late, and correct puzzle solutions found in middle layers are sometimes overridden. This late-layer reversal is accompanied by concept preference analyses showing final layers prioritize safety over aggression, suggesting a mechanism by which heuristic priors can override tactical solutions.

Tables

Table 1. Table 1 : Playing strength (Elo rating) across transformer layers and evaluation methods

Evaluation	Input	L0	L1	L2	L3	L4	L5	L6	L7	L8	L9	L10	L11	L12	L13	Full	Anchor
Internal Tournament ($\tau =0$)																	2292
Internal Tournament ($\tau =1$)																	–
Lichess Blitz	518	651	681	688	741	769	774	850	803	843	832	960	948	1252	1581	2274	–
Lichess Bullet	693	904	891	916	972	1009	926	984	1052	994	1061	1064	1129	1331	1659	2246	–
Lichess Rapid	558	816	709	697	915	717	939	1021	1018	1032	957	1107	1095	1290	1581	2253	–

Table 2. Table 2 : Playing strength evaluation through internal tournament ( τ = 0 \tau=0 )

Rank	Model	Elo	$+$	$-$	Games	Score (%)	Avg. Oppo.	Draws (%)
1	Lc0 Policy Net (Anchor)	2292	25	25	6400	97	1043	2
2	Full Model	2268	25	25	6400	96	1044	3
3	Logit Lens Layer 13	1670	22	21	6400	85	1081	2
4	Logit Lens Layer 12	1356	13	12	6400	74	1101	5
5	Logit Lens Layer 11	1117	10	10	6400	58	1116	7
6	Logit Lens Layer 10	1082	10	10	6400	55	1118	11
7	Logit Lens Layer 9	1055	10	10	6400	53	1120	12
8	Logit Lens Layer 8	1030	10	10	6400	51	1121	11
9	Logit Lens Layer 7	1028	10	10	6400	51	1122	12
10	Logit Lens Layer 6	997	10	10	6400	48	1124	12
11	Logit Lens Layer 5	993	9	9	6400	48	1124	17
12	Logit Lens Layer 4	908	9	9	6400	40	1129	18
13	Logit Lens Layer 3	832	9	9	6400	32	1134	16
14	Logit Lens Layer 2	730	10	10	6400	23	1140	16
15	Logit Lens Layer 1	655	11	10	6400	17	1145	17
16	Logit Lens Layer 0	609	11	12	6400	16	1148	8
17	Logit Lens Input	351	17	17	6400	4	1164	4

Table 3. Table 3 : Playing strength evaluation through internal tournament ( τ = 1 \tau=1 )

Rank	Model	Elo	$+$	$-$	Games	Score (%)	Avg. Oppo.	Draws (%)
1	Lc0 Policy Net (Anchor)	2292	–	–	32000	100	749	0
2	Full Model	1367	8	8	32000	88	807	1
3	Logit Lens Layer 13	1119	6	6	32000	76	822	2
4	Logit Lens Layer 12	1065	5	5	32000	73	826	3
5	Logit Lens Layer 11	876	5	5	32000	58	837	5
6	Logit Lens Layer 10	872	5	5	32000	57	838	5
7	Logit Lens Layer 9	851	5	5	32000	55	839	5
8	Logit Lens Layer 7	816	5	5	32000	52	841	6
9	Logit Lens Layer 8	813	5	5	32000	52	841	6
10	Logit Lens Layer 5	810	5	5	32000	52	842	6
11	Logit Lens Layer 6	799	5	5	32000	51	842	5
12	Logit Lens Layer 4	718	5	5	32000	43	847	7
13	Logit Lens Layer 3	585	5	5	32000	31	856	7
14	Logit Lens Layer 2	469	6	6	32000	22	863	7
15	Logit Lens Layer 1	391	6	6	32000	17	868	5
16	Logit Lens Layer 0	387	6	6	32000	17	868	4
17	Logit Lens Input	46	9	10	32000	4	889	0

Table 4. Table 4 : Performance of layer-wise policies on Lichess across different time controls

Bot	Rating			Total	Performance
	Bullet	Blitz	Rapid	Games	W	D	L
LLLBot-In	693 $\pm$ 54	518 $\pm$ 45	558 $\pm$ 45	437	2	150	285
LLLBot-0	904 $\pm$ 54	651 $\pm$ 45	816 $\pm$ 50	419	2	85	332
LLLBot-1	891 $\pm$ 54	681 $\pm$ 45	709 $\pm$ 46	431	5	92	334
LLLBot-2	916 $\pm$ 50	688 $\pm$ 45	697 $\pm$ 45	453	13	134	306
LLLBot-3	972 $\pm$ 55	741 $\pm$ 45	915 $\pm$ 50	463	9	80	374
LLLBot-4	1009 $\pm$ 55	769 $\pm$ 45	717 $\pm$ 45	461	12	136	313
LLLBot-5	926 $\pm$ 56	774 $\pm$ 45	939 $\pm$ 51	468	23	66	379
LLLBot-6	984 $\pm$ 64	850 $\pm$ 45	1021 $\pm$ 46	446	22	62	362
LLLBot-7	1052 $\pm$ 55	803 $\pm$ 45	1018 $\pm$ 49	452	23	50	379
LLLBot-8	994 $\pm$ 63	843 $\pm$ 45	1032 $\pm$ 47	450	23	58	369
LLLBot-9	1061 $\pm$ 58	832 $\pm$ 45	957 $\pm$ 54	445	32	47	366
LLLBot-10	1064 $\pm$ 59	960 $\pm$ 45	1107 $\pm$ 45	494	39	73	382
LLLBot-11	1129 $\pm$ 54	948 $\pm$ 45	1095 $\pm$ 45	436	34	46	356
LLLBot-12	1331 $\pm$ 49	1252 $\pm$ 45	1290 $\pm$ 45	415	74	63	278
LLLBot-13	1659 $\pm$ 48	1581 $\pm$ 45	1581 $\pm$ 45	368	124	39	205
LLLBot-Full	2246 $\pm$ 52	2274 $\pm$ 45	2253 $\pm$ 52	316	197	32	87

Table 5. Table 5 : Move probabilities by layer for puzzle 62FeU (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveNg3+	6.54%	0.09%	0.14%	1.79%	7.59%	0.41%	15.23%	26.71%
\wmoveRh6	0.01%	0.03%	0.04%	0.13%	0.12%	0.15%	0.02%	2.56%
\wmoveQg1+	9.16%	1.11%	4.36%	7.52%	6.52%	0.43%	1.30%	22.74%
\wmoveQxb2	2.80%	17.91%	19.80%	24.22%	24.28%	27.55%	13.09%	0.96%
\wmoveQxd2	6.94%	15.90%	19.44%	21.33%	26.00%	21.59%	9.50%	0.66%
\wmoveRxe5	0.07%	16.82%	21.35%	12.49%	5.78%	9.91%	24.47%	10.04%
\wmoveQf2	24.04%	0.17%	0.33%	0.12%	0.14%	0.26%	0.09%	0.05%
\wmoveQxe5	0.03%	23.35%	18.84%	1.76%	0.63%	0.52%	0.72%	0.26%
\wmoveQxf4	0.49%	6.44%	6.63%	13.56%	11.31%	13.92%	13.76%	20.34%
\wmoveQxc4	0.80%	9.96%	6.49%	9.72%	11.84%	11.95%	16.31%	10.29%
\wmoveh6	0.01%	1.04%	0.02%	0.01%	0.00%	0.02%	0.01%	0.00%
\wmoveQd3	12.24%	0.07%	0.12%	0.04%	0.09%	0.04%	0.08%	0.08%
\wmoveQe3	11.37%	0.05%	0.07%	0.58%	0.16%	0.31%	0.26%	0.56%
\wmoveQc3	10.54%	0.04%	0.09%	2.37%	1.38%	1.16%	0.97%	0.70%
\wmoveNe3	10.26%	0.09%	0.18%	1.02%	0.40%	0.44%	0.21%	0.07%
\wmoveNh6	0.13%	0.24%	0.03%	0.08%	0.01%	0.07%	0.01%	1.01%
\wmoveRc6	0.12%	1.51%	0.77%	0.66%	0.67%	4.08%	1.47%	0.87%
\wmoveh5	0.05%	2.57%	0.11%	0.04%	0.02%	0.03%	0.03%	0.02%
\wmoveKh8	0.15%	0.05%	0.02%	0.22%	0.81%	2.20%	0.34%	0.24%
\wmoveNd6	0.08%	0.01%	0.02%	0.07%	0.14%	0.08%	0.09%	0.10%
\wmoveQe4	1.62%	0.09%	0.13%	0.06%	0.06%	0.07%	0.10%	0.13%
\wmoveRb6	0.07%	0.01%	0.01%	0.08%	0.32%	1.36%	0.30%	0.18%
\wmoveRe8	0.06%	0.60%	0.22%	0.17%	0.10%	0.06%	0.02%	0.03%
\wmovef6	0.07%	0.06%	0.00%	0.01%	0.01%	0.02%	0.05%	0.05%
\wmoveg6	0.07%	0.11%	0.01%	0.07%	0.05%	0.04%	0.03%	0.02%
\wmovea5	0.04%	0.91%	0.13%	0.17%	0.06%	0.07%	0.09%	0.02%
\wmoveNe7	0.25%	0.01%	0.01%	0.05%	0.06%	0.10%	0.12%	0.09%
\wmoveQd8	0.05%	0.13%	0.04%	0.76%	0.24%	0.22%	0.12%	0.06%
\wmoveRe7	0.32%	0.01%	0.05%	0.05%	0.12%	0.68%	0.17%	0.09%
\wmoveKf8	0.06%	0.04%	0.02%	0.04%	0.25%	0.68%	0.17%	0.22%
\wmoveNh4	0.65%	0.04%	0.01%	0.04%	0.05%	0.03%	0.03%	0.05%
\wmoveg5	0.54%	0.38%	0.23%	0.07%	0.04%	0.03%	0.01%	0.01%
\wmoveQd5	0.02%	0.05%	0.06%	0.31%	0.22%	0.14%	0.52%	0.38%
\wmoveQd7	0.04%	0.04%	0.02%	0.09%	0.22%	0.41%	0.11%	0.16%
\wmoveRg6	0.07%	0.01%	0.02%	0.09%	0.16%	0.41%	0.04%	0.05%
\wmoveQd6	0.02%	0.02%	0.03%	0.08%	0.06%	0.19%	0.07%	0.07%
\wmoveRd6	0.10%	0.01%	0.03%	0.06%	0.06%	0.25%	0.07%	0.06%
\wmoveRf6	0.10%	0.01%	0.12%	0.07%	0.03%	0.11%	0.03%	0.03%

Table 6. Table 6 : Move probabilities by layer for puzzle 62FeU (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveNg3+	27.02%	20.06%	19.00%	25.38%	39.13%	28.03%	49.15%	87.55%
\wmoveRh6	8.66%	15.63%	9.35%	10.04%	28.26%	51.73%	9.61%	0.24%
\wmoveQg1+	18.86%	15.30%	49.02%	47.22%	22.42%	2.68%	2.08%	0.28%
\wmoveQxb2	0.27%	2.92%	0.42%	0.35%	0.20%	0.10%	0.86%	0.25%
\wmoveQxd2	0.19%	1.72%	0.34%	0.27%	0.13%	0.08%	0.63%	0.24%
\wmoveRxe5	9.74%	6.51%	5.83%	3.66%	2.71%	1.44%	0.71%	0.37%
\wmoveQf2	0.07%	0.21%	0.16%	0.55%	0.08%	0.04%	0.34%	0.27%
\wmoveQxe5	0.17%	0.25%	0.37%	0.13%	0.09%	0.06%	0.67%	0.37%
\wmoveQxf4	13.50%	18.19%	5.50%	6.82%	0.50%	0.17%	0.55%	0.27%
\wmoveQxc4	12.17%	13.07%	4.70%	1.31%	0.37%	0.20%	0.75%	0.29%
\wmoveh6	0.02%	0.14%	0.12%	0.61%	1.38%	3.87%	14.66%	0.25%
\wmoveQd3	0.07%	0.15%	0.22%	0.16%	0.33%	0.15%	0.59%	0.26%
\wmoveQe3	0.37%	0.15%	0.06%	0.03%	0.08%	0.04%	0.37%	0.25%
\wmoveQc3	1.42%	0.38%	0.22%	0.08%	0.12%	0.03%	0.50%	0.22%
\wmoveNe3	0.04%	0.04%	0.03%	0.02%	0.03%	0.03%	0.24%	0.21%
\wmoveNh6	4.29%	3.37%	2.22%	1.66%	2.45%	9.23%	6.39%	0.23%
\wmoveRc6	0.58%	0.22%	0.24%	0.06%	0.06%	0.05%	0.21%	0.34%
\wmoveh5	0.20%	0.07%	0.19%	0.32%	0.24%	0.14%	0.88%	0.25%
\wmoveKh8	0.33%	0.25%	0.48%	0.32%	0.34%	0.48%	0.27%	0.27%
\wmoveNd6	0.17%	0.12%	0.34%	0.15%	0.09%	0.12%	1.97%	0.82%
\wmoveQe4	0.11%	0.07%	0.07%	0.03%	0.04%	0.02%	0.31%	0.26%
\wmoveRb6	0.12%	0.05%	0.05%	0.02%	0.03%	0.05%	0.28%	0.28%
\wmoveRe8	0.10%	0.13%	0.12%	0.07%	0.06%	0.20%	0.70%	1.35%
\wmovef6	0.33%	0.13%	0.11%	0.14%	0.15%	0.27%	1.28%	0.29%
\wmoveg6	0.04%	0.03%	0.05%	0.08%	0.07%	0.12%	1.05%	0.31%
\wmovea5	0.04%	0.01%	0.02%	0.01%	0.01%	0.02%	0.18%	0.23%
\wmoveNe7	0.07%	0.05%	0.04%	0.02%	0.02%	0.01%	0.84%	0.39%
\wmoveQd8	0.08%	0.08%	0.07%	0.05%	0.06%	0.05%	0.48%	0.81%
\wmoveRe7	0.08%	0.04%	0.03%	0.01%	0.02%	0.03%	0.33%	0.26%
\wmoveKf8	0.21%	0.22%	0.21%	0.10%	0.05%	0.08%	0.22%	0.35%
\wmoveNh4	0.05%	0.04%	0.03%	0.04%	0.05%	0.07%	0.39%	0.20%
\wmoveg5	0.02%	0.02%	0.05%	0.04%	0.03%	0.04%	0.61%	0.29%
\wmoveQd5	0.35%	0.13%	0.14%	0.05%	0.07%	0.05%	0.41%	0.27%
\wmoveQd7	0.04%	0.09%	0.10%	0.07%	0.08%	0.06%	0.22%	0.35%
\wmoveRg6	0.05%	0.04%	0.05%	0.05%	0.10%	0.09%	0.23%	0.28%
\wmoveQd6	0.07%	0.04%	0.03%	0.03%	0.05%	0.06%	0.41%	0.28%
\wmoveRd6	0.06%	0.04%	0.02%	0.01%	0.04%	0.04%	0.34%	0.29%
\wmoveRf6	0.05%	0.05%	0.03%	0.02%	0.04%	0.06%	0.29%	0.28%

Table 7. Table 7 : Move probabilities by layer for puzzle 9JvtA (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveNd6+	32.13%	0.05%	3.60%	11.55%	0.14%	0.10%	0.75%	0.50%
\wmoveQxa5	0.20%	4.42%	6.22%	9.20%	26.54%	50.71%	47.82%	58.34%
\wmoveQxc6+	0.90%	26.71%	56.31%	19.67%	20.88%	16.16%	7.71%	5.78%
\wmoveb4	1.44%	46.77%	0.36%	0.88%	0.58%	0.01%	0.01%	0.16%
\wmoveQxd4	1.50%	7.14%	22.75%	39.84%	34.48%	23.90%	29.99%	23.33%
\wmoveNf6+	36.04%	0.01%	0.17%	2.27%	0.05%	0.13%	0.01%	0.01%
\wmoveNc5	17.13%	0.04%	0.27%	0.60%	0.17%	0.09%	0.13%	0.09%
\wmovea3	0.03%	7.50%	2.85%	6.10%	0.13%	0.57%	0.16%	0.20%
\wmoveNf3	0.07%	0.01%	0.03%	0.20%	4.38%	0.12%	4.86%	3.17%
\wmovef4	3.27%	0.27%	0.03%	0.04%	0.04%	0.15%	0.05%	0.11%
\wmoveBf3	0.29%	0.02%	0.51%	0.37%	1.85%	0.86%	3.02%	1.93%
\wmoveQb4	0.10%	0.44%	0.51%	0.92%	2.59%	0.20%	0.11%	0.27%
\wmoveNg5	2.51%	0.05%	0.02%	0.31%	0.17%	0.29%	0.02%	0.05%
\wmoveQb3	0.09%	0.10%	0.20%	1.28%	2.22%	0.42%	0.98%	1.08%
\wmoveQa3	0.04%	1.51%	0.93%	1.41%	1.87%	0.50%	0.50%	0.87%
\wmoveNc3	0.36%	0.02%	1.86%	1.02%	0.21%	0.19%	0.04%	0.20%
\wmoveg4	1.75%	0.30%	0.24%	0.71%	0.38%	0.25%	0.03%	0.02%
\wmoveRb1	0.16%	0.10%	0.09%	0.17%	0.21%	1.25%	1.09%	1.68%
\wmovef3	0.20%	0.00%	0.00%	0.00%	0.00%	0.11%	0.04%	0.08%
\wmoveh3	0.03%	1.44%	0.12%	0.41%	0.03%	0.12%	0.22%	0.08%
\wmoveBh3	0.02%	0.06%	0.11%	0.15%	0.98%	0.96%	0.43%	0.24%
\wmoveKf1	0.11%	0.37%	0.96%	0.25%	0.25%	0.19%	0.17%	0.09%
\wmoveb3	0.17%	0.21%	0.08%	0.92%	0.01%	0.03%	0.03%	0.01%
\wmoveBf1	0.01%	0.08%	0.87%	0.35%	0.51%	0.29%	0.17%	0.05%
\wmoveNh3	0.15%	0.83%	0.12%	0.21%	0.04%	0.13%	0.23%	0.22%
\wmoveKd1	0.12%	0.11%	0.05%	0.09%	0.41%	0.75%	0.40%	0.56%
\wmovee3	0.42%	0.49%	0.03%	0.23%	0.16%	0.75%	0.24%	0.25%
\wmoveQc4	0.25%	0.11%	0.03%	0.08%	0.16%	0.11%	0.18%	0.16%
\wmoveh4	0.09%	0.52%	0.56%	0.45%	0.15%	0.21%	0.16%	0.14%
\wmoveQd1	0.11%	0.09%	0.03%	0.08%	0.07%	0.17%	0.16%	0.08%
\wmoveQb5	0.12%	0.12%	0.06%	0.11%	0.30%	0.13%	0.16%	0.15%
\wmoveQc2	0.16%	0.10%	0.04%	0.12%	0.06%	0.16%	0.14%	0.10%

Table 8. Table 8 : Move probabilities by layer for puzzle 9JvtA (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveNd6+	1.22%	1.26%	2.22%	2.73%	1.72%	1.65%	24.37%	65.34%
\wmoveQxa5	58.41%	52.60%	50.88%	61.05%	56.84%	56.52%	46.15%	12.18%
\wmoveQxc6+	6.60%	14.67%	25.41%	22.02%	30.43%	28.79%	3.26%	0.61%
\wmoveb4	0.44%	1.59%	0.40%	0.70%	1.71%	3.37%	3.06%	0.29%
\wmoveQxd4	22.78%	19.53%	14.91%	8.88%	4.27%	4.65%	0.79%	0.29%
\wmoveNf6+	0.01%	0.02%	0.02%	0.02%	0.02%	0.02%	11.00%	13.67%
\wmoveNc5	0.08%	0.12%	0.20%	0.13%	0.14%	0.09%	0.69%	0.29%
\wmovea3	0.93%	0.78%	0.28%	0.55%	0.22%	0.19%	1.90%	0.28%
\wmoveNf3	1.91%	1.70%	0.89%	0.83%	0.41%	0.14%	0.73%	0.28%
\wmovef4	0.40%	0.48%	0.31%	0.57%	1.95%	1.26%	1.22%	0.26%
\wmoveBf3	0.72%	0.55%	0.35%	0.16%	0.11%	0.51%	0.38%	0.29%
\wmoveQb4	0.68%	0.60%	0.26%	0.12%	0.08%	0.06%	0.05%	0.27%
\wmoveNg5	0.22%	0.45%	0.03%	0.03%	0.02%	0.19%	0.34%	0.27%
\wmoveQb3	1.94%	2.00%	0.67%	0.28%	0.12%	0.32%	0.26%	0.29%
\wmoveQa3	0.78%	1.30%	0.72%	0.37%	0.21%	0.14%	0.12%	0.29%
\wmoveNc3	0.25%	0.13%	0.07%	0.11%	0.06%	0.17%	0.66%	0.29%
\wmoveg4	0.02%	0.02%	0.02%	0.04%	0.04%	0.04%	0.16%	0.29%
\wmoveRb1	0.55%	0.51%	0.77%	0.10%	0.10%	0.12%	0.38%	0.31%
\wmovef3	0.03%	0.06%	0.02%	0.02%	0.02%	0.13%	1.47%	0.26%
\wmoveh3	0.06%	0.04%	0.04%	0.03%	0.05%	0.04%	0.33%	0.30%
\wmoveBh3	0.20%	0.22%	0.21%	0.05%	0.07%	0.05%	0.16%	0.32%
\wmoveKf1	0.07%	0.05%	0.08%	0.07%	0.04%	0.02%	0.13%	0.33%
\wmoveb3	0.08%	0.05%	0.03%	0.04%	0.02%	0.01%	0.15%	0.25%
\wmoveBf1	0.04%	0.02%	0.03%	0.03%	0.05%	0.05%	0.15%	0.34%
\wmoveNh3	0.18%	0.17%	0.08%	0.03%	0.08%	0.19%	0.36%	0.29%
\wmoveKd1	0.54%	0.29%	0.37%	0.32%	0.40%	0.28%	0.21%	0.32%
\wmovee3	0.26%	0.25%	0.19%	0.18%	0.19%	0.10%	0.55%	0.28%
\wmoveQc4	0.14%	0.21%	0.19%	0.19%	0.25%	0.56%	0.32%	0.28%
\wmoveh4	0.17%	0.11%	0.06%	0.06%	0.08%	0.06%	0.20%	0.33%
\wmoveQd1	0.08%	0.04%	0.05%	0.04%	0.04%	0.04%	0.15%	0.32%
\wmoveQb5	0.10%	0.13%	0.17%	0.17%	0.11%	0.13%	0.10%	0.29%
\wmoveQc2	0.11%	0.06%	0.07%	0.09%	0.14%	0.13%	0.18%	0.29%

Table 9. Table 9 : Move probabilities by layer for puzzle 70v1B (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQxe8+	11.14%	50.54%	71.54%	80.98%	40.69%	64.80%	66.88%	68.29%
\wmoveQe7	56.93%	0.68%	0.16%	0.06%	0.08%	0.09%	0.06%	0.09%
\wmovebxc3	0.18%	20.37%	21.51%	13.64%	56.27%	30.75%	30.98%	27.02%
\wmoveQe6	20.58%	1.07%	1.00%	0.16%	0.12%	0.10%	0.10%	0.16%
\wmoveh4	0.08%	6.79%	0.81%	0.44%	0.37%	0.11%	0.07%	0.29%
\wmoveb4	0.72%	6.44%	1.49%	0.42%	0.06%	0.05%	0.02%	0.04%
\wmovea4	0.07%	4.65%	0.20%	0.08%	0.02%	0.02%	0.09%	0.14%
\wmoveQe5	0.69%	1.88%	0.49%	0.30%	0.22%	0.13%	0.06%	0.10%
\wmoveQf2	1.59%	0.15%	0.03%	0.12%	0.09%	0.03%	0.04%	0.10%
\wmoveQc4	0.62%	0.56%	1.50%	1.15%	0.20%	0.13%	0.12%	0.23%
\wmovef5	0.63%	1.44%	0.01%	0.01%	0.01%	0.01%	0.04%	0.14%
\wmoveRb1	0.11%	0.04%	0.02%	0.12%	0.19%	1.31%	0.06%	0.29%
\wmoveQg2	1.14%	0.10%	0.02%	0.07%	0.03%	0.04%	0.03%	0.08%
\wmovea3	0.02%	0.94%	0.05%	0.04%	0.04%	0.07%	0.06%	0.25%
\wmoveg4	0.82%	0.83%	0.02%	0.04%	0.09%	0.04%	0.05%	0.12%
\wmoveQb5	0.62%	0.10%	0.06%	0.09%	0.05%	0.07%	0.08%	0.17%
\wmoveQg4	0.61%	0.34%	0.09%	0.23%	0.16%	0.12%	0.10%	0.19%
\wmoveRd1	0.13%	0.17%	0.06%	0.05%	0.04%	0.55%	0.07%	0.09%
\wmoveQd1	0.08%	0.49%	0.16%	0.26%	0.09%	0.14%	0.07%	0.14%
\wmoveBe3	0.05%	0.01%	0.02%	0.13%	0.02%	0.05%	0.08%	0.23%
\wmoveQf3	0.48%	0.17%	0.02%	0.05%	0.04%	0.07%	0.07%	0.11%
\wmoveQe3	0.42%	0.11%	0.03%	0.07%	0.03%	0.03%	0.09%	0.10%
\wmoveb3	0.11%	0.34%	0.04%	0.06%	0.05%	0.07%	0.05%	0.30%
\wmoveQe4	0.40%	0.14%	0.15%	0.16%	0.11%	0.08%	0.06%	0.16%
\wmoveQa6	0.05%	0.31%	0.07%	0.06%	0.04%	0.05%	0.05%	0.08%
\wmoveQc2	0.37%	0.13%	0.04%	0.13%	0.06%	0.12%	0.07%	0.17%
\wmoveQh5	0.13%	0.30%	0.05%	0.11%	0.10%	0.09%	0.08%	0.18%
\wmoveh3	0.02%	0.29%	0.05%	0.07%	0.04%	0.01%	0.03%	0.04%
\wmoveKh1	0.26%	0.06%	0.04%	0.06%	0.04%	0.10%	0.06%	0.13%
\wmoveKf2	0.10%	0.12%	0.07%	0.30%	0.28%	0.10%	0.03%	0.08%
\wmoveQd3	0.22%	0.07%	0.04%	0.09%	0.05%	0.10%	0.09%	0.15%
\wmoveKg2	0.11%	0.03%	0.03%	0.16%	0.11%	0.14%	0.06%	0.12%
\wmoveQd2	0.07%	0.09%	0.03%	0.04%	0.04%	0.05%	0.03%	0.04%
\wmoveBd2	0.10%	0.01%	0.01%	0.02%	0.02%	0.06%	0.05%	0.07%
\wmoveKf1	0.10%	0.04%	0.03%	0.10%	0.07%	0.12%	0.04%	0.04%
\wmoveQf1	0.10%	0.16%	0.03%	0.08%	0.04%	0.05%	0.03%	0.05%
\wmoveRf1	0.16%	0.04%	0.02%	0.07%	0.03%	0.13%	0.03%	0.03%

Table 10. Table 10 : Move probabilities by layer for puzzle 70v1B (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQxe8+	42.88%	60.35%	74.76%	69.45%	78.19%	75.30%	53.67%	88.91%
\wmoveQe7	0.11%	0.11%	0.12%	0.30%	0.43%	0.21%	0.26%	0.33%
\wmovebxc3	52.88%	35.65%	20.83%	25.08%	17.09%	20.41%	35.91%	0.28%
\wmoveQe6	0.13%	0.07%	0.08%	0.08%	0.15%	0.12%	0.25%	0.41%
\wmoveh4	0.38%	0.25%	0.22%	0.86%	0.28%	0.14%	0.31%	0.34%
\wmoveb4	0.05%	0.11%	0.47%	0.56%	0.10%	0.22%	1.02%	0.33%
\wmovea4	0.14%	0.10%	0.04%	0.09%	0.10%	0.17%	0.97%	0.29%
\wmoveQe5	0.11%	0.09%	0.13%	0.14%	0.24%	0.23%	0.22%	0.37%
\wmoveQf2	0.05%	0.04%	0.04%	0.04%	0.04%	0.07%	0.15%	0.24%
\wmoveQc4	0.17%	0.09%	0.16%	0.14%	0.12%	0.10%	0.33%	0.45%
\wmovef5	0.29%	1.17%	0.44%	0.36%	0.45%	0.22%	0.56%	0.30%
\wmoveRb1	0.08%	0.05%	0.13%	0.19%	0.17%	0.12%	0.37%	0.37%
\wmoveQg2	0.09%	0.04%	0.04%	0.04%	0.06%	0.10%	0.16%	0.27%
\wmovea3	0.18%	0.11%	0.10%	0.12%	0.07%	0.07%	0.50%	0.32%
\wmoveg4	0.13%	0.14%	0.25%	0.44%	0.23%	0.22%	0.75%	0.30%
\wmoveQb5	0.13%	0.12%	0.10%	0.08%	0.08%	0.10%	0.21%	0.21%
\wmoveQg4	0.13%	0.08%	0.10%	0.11%	0.10%	0.06%	0.20%	0.35%
\wmoveRd1	0.09%	0.06%	0.10%	0.06%	0.06%	0.06%	0.26%	0.27%
\wmoveQd1	0.11%	0.05%	0.08%	0.12%	0.12%	0.06%	0.17%	0.29%
\wmoveBe3	0.15%	0.12%	0.20%	0.10%	0.06%	0.17%	0.49%	0.21%
\wmoveQf3	0.12%	0.05%	0.09%	0.09%	0.10%	0.07%	0.19%	0.34%
\wmoveQe3	0.07%	0.16%	0.12%	0.06%	0.12%	0.26%	0.23%	0.21%
\wmoveb3	0.33%	0.15%	0.33%	0.25%	0.19%	0.09%	0.39%	0.41%
\wmoveQe4	0.13%	0.08%	0.13%	0.18%	0.27%	0.21%	0.23%	0.40%
\wmoveQa6	0.07%	0.09%	0.09%	0.10%	0.11%	0.06%	0.15%	0.38%
\wmoveQc2	0.20%	0.09%	0.14%	0.16%	0.15%	0.10%	0.21%	0.34%
\wmoveQh5	0.14%	0.10%	0.10%	0.14%	0.12%	0.07%	0.23%	0.37%
\wmoveh3	0.10%	0.05%	0.06%	0.14%	0.07%	0.04%	0.27%	0.33%
\wmoveKh1	0.08%	0.06%	0.05%	0.03%	0.05%	0.07%	0.09%	0.33%
\wmoveKf2	0.04%	0.03%	0.04%	0.07%	0.09%	0.15%	0.10%	0.24%
\wmoveQd3	0.14%	0.12%	0.12%	0.08%	0.11%	0.11%	0.22%	0.28%
\wmoveKg2	0.07%	0.06%	0.05%	0.09%	0.19%	0.20%	0.11%	0.27%
\wmoveQd2	0.02%	0.03%	0.05%	0.02%	0.02%	0.04%	0.08%	0.27%
\wmoveBd2	0.10%	0.08%	0.16%	0.10%	0.07%	0.12%	0.27%	0.23%
\wmoveKf1	0.02%	0.02%	0.03%	0.05%	0.10%	0.11%	0.08%	0.26%
\wmoveQf1	0.05%	0.02%	0.03%	0.05%	0.10%	0.09%	0.17%	0.26%
\wmoveRf1	0.03%	0.01%	0.02%	0.02%	0.03%	0.05%	0.23%	0.25%

Table 11. Table 11 : Move probabilities by layer for puzzle 0KBPy (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveNg6+	9.76%	0.46%	0.88%	0.35%	6.34%	0.12%	1.38%	0.77%
\wmoveRc4	0.15%	0.32%	0.36%	1.20%	14.79%	69.39%	48.78%	41.74%
\wmoveQg8+	0.07%	40.00%	7.11%	18.17%	4.87%	0.36%	0.60%	0.77%
\wmoveNe6	19.23%	6.12%	30.11%	7.38%	8.51%	0.56%	10.91%	10.97%
\wmoveh4	0.09%	6.94%	0.16%	0.36%	0.28%	0.07%	0.23%	0.04%
\wmoveQa4	0.05%	2.77%	5.95%	10.38%	16.30%	2.73%	5.80%	3.49%
\wmoveQxb5	16.21%	8.15%	7.62%	6.20%	2.83%	3.07%	2.87%	1.41%
\wmoveRd7	15.49%	0.31%	0.19%	0.42%	1.07%	0.75%	0.35%	0.89%
\wmoveQc2	0.38%	0.71%	1.63%	13.84%	4.56%	0.78%	0.24%	0.88%
\wmoveQe6	1.77%	4.49%	1.31%	0.20%	0.70%	0.16%	1.50%	5.27%
\wmoveRa4	0.12%	0.51%	1.20%	1.83%	3.50%	2.12%	12.92%	6.79%
\wmoveQb4	0.19%	12.14%	11.15%	3.00%	2.59%	0.49%	0.36%	0.37%
\wmoveRd6	10.35%	0.26%	0.23%	0.53%	1.96%	1.40%	0.44%	0.74%
\wmoveQd3	0.18%	0.28%	0.59%	0.45%	0.44%	0.22%	0.19%	0.31%
\wmoveNd3	0.33%	0.18%	0.06%	0.62%	1.27%	0.14%	1.53%	1.31%
\wmoveRb4	0.09%	2.99%	9.44%	0.75%	0.50%	0.25%	0.04%	0.04%
\wmoveNd5	9.23%	0.10%	0.21%	2.13%	3.00%	0.24%	0.92%	0.73%
\wmoveQc4	0.31%	1.70%	5.11%	8.20%	4.18%	1.06%	2.18%	2.54%
\wmoveRe4	0.15%	0.29%	1.43%	2.45%	2.82%	0.97%	1.38%	7.30%
\wmovea4	0.08%	3.03%	0.08%	0.24%	0.26%	0.11%	0.22%	0.07%
\wmoveKh2	0.20%	0.93%	0.20%	0.38%	0.18%	0.19%	0.66%	2.47%
\wmoveRd8	2.55%	0.12%	0.06%	0.31%	0.67%	1.80%	0.30%	3.34%
\wmoveRd2	0.18%	0.20%	0.24%	0.73%	2.22%	4.69%	1.23%	1.35%
\wmoveQf7	1.78%	0.73%	3.87%	4.31%	2.14%	0.47%	0.19%	0.20%
\wmoveKf1	0.13%	0.38%	0.12%	0.63%	1.72%	1.18%	0.41%	0.87%
\wmoveRd5	3.77%	0.27%	2.25%	1.07%	0.19%	0.07%	0.14%	0.65%
\wmoveQd1	0.12%	0.38%	0.27%	0.52%	1.35%	0.80%	0.28%	0.28%
\wmoveQc3	0.10%	0.51%	0.70%	3.10%	2.00%	1.10%	0.45%	0.82%
\wmovee4	0.54%	0.49%	0.91%	3.02%	0.92%	0.08%	0.10%	0.22%
\wmoveQa2	0.03%	1.39%	1.96%	2.76%	2.71%	1.38%	1.00%	1.24%
\wmoveNh5	2.21%	0.49%	0.05%	0.18%	0.51%	0.39%	0.16%	0.08%
\wmovef3	0.17%	0.12%	0.20%	0.60%	1.83%	0.13%	0.11%	0.08%
\wmoveNe2	1.09%	0.13%	0.90%	0.13%	0.05%	0.07%	0.27%	0.21%
\wmoveRd3	0.11%	0.20%	1.19%	0.63%	0.16%	0.08%	0.17%	0.08%
\wmoveKh1	0.34%	0.59%	0.71%	1.13%	0.27%	1.10%	0.59%	0.41%
\wmoveg4	1.09%	0.48%	0.13%	0.69%	0.69%	0.11%	0.25%	0.25%
\wmoveRd1	0.16%	0.17%	0.26%	0.32%	0.41%	1.06%	0.53%	0.54%
\wmoveQd5	1.04%	0.23%	1.04%	0.50%	0.40%	0.22%	0.22%	0.26%
\wmoveg3	0.17%	0.41%	0.09%	0.30%	0.80%	0.10%	0.12%	0.24%

Table 12. Table 12 : Move probabilities by layer for puzzle 0KBPy (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveNg6+	3.14%	10.83%	14.71%	16.18%	35.31%	53.13%	33.17%	70.65%
\wmoveRc4	33.25%	14.94%	13.26%	5.06%	1.01%	0.19%	0.28%	0.18%
\wmoveQg8+	1.51%	1.85%	5.92%	9.55%	1.02%	0.73%	0.52%	0.16%
\wmoveNe6	8.39%	12.37%	10.91%	10.09%	10.47%	9.26%	12.17%	1.46%
\wmoveh4	0.04%	0.04%	0.01%	0.04%	1.05%	1.58%	17.72%	4.76%
\wmoveQa4	2.40%	3.49%	1.38%	2.76%	2.78%	0.47%	0.31%	0.14%
\wmoveQxb5	1.99%	4.93%	2.81%	2.12%	0.64%	0.62%	1.30%	0.16%
\wmoveRd7	1.08%	0.99%	1.22%	2.37%	3.61%	1.34%	0.26%	0.21%
\wmoveQc2	1.09%	2.72%	5.97%	6.75%	1.36%	0.28%	0.21%	0.16%
\wmoveQe6	8.67%	13.05%	13.50%	13.51%	8.71%	2.56%	3.14%	0.48%
\wmoveRa4	6.37%	4.91%	0.40%	2.20%	1.72%	0.23%	0.12%	0.15%
\wmoveQb4	0.48%	0.85%	0.70%	0.98%	0.74%	0.77%	0.58%	0.71%
\wmoveRd6	0.46%	0.31%	0.45%	0.50%	0.67%	0.48%	0.14%	0.18%
\wmoveQd3	1.20%	1.38%	2.80%	4.34%	7.01%	9.79%	6.29%	4.04%
\wmoveNd3	2.31%	2.88%	4.05%	2.28%	3.70%	3.28%	9.63%	0.49%
\wmoveRb4	0.04%	0.06%	0.04%	0.09%	0.10%	0.09%	0.05%	0.15%
\wmoveNd5	0.95%	0.92%	0.48%	0.29%	0.24%	0.24%	0.50%	0.15%
\wmoveQc4	1.41%	1.06%	0.93%	0.54%	0.30%	0.24%	0.22%	0.17%
\wmoveRe4	6.63%	3.24%	4.44%	4.90%	3.39%	1.89%	0.50%	0.19%
\wmovea4	0.18%	0.08%	0.01%	0.03%	0.09%	0.03%	1.02%	5.80%
\wmoveKh2	5.48%	3.97%	2.81%	2.75%	0.81%	2.87%	3.23%	5.54%
\wmoveRd8	2.37%	5.16%	1.86%	1.57%	2.43%	0.85%	0.37%	0.17%
\wmoveRd2	1.06%	0.74%	1.34%	0.25%	0.14%	0.19%	0.42%	0.25%
\wmoveQf7	0.41%	0.76%	0.87%	1.69%	1.04%	0.30%	0.31%	0.20%
\wmoveKf1	1.68%	2.08%	2.11%	1.84%	4.25%	3.32%	0.32%	0.15%
\wmoveRd5	0.94%	0.62%	0.33%	0.35%	0.64%	0.62%	0.42%	0.22%
\wmoveQd1	1.01%	0.93%	1.71%	1.89%	3.13%	1.05%	0.62%	0.46%
\wmoveQc3	0.63%	0.55%	0.65%	0.87%	0.45%	0.29%	0.17%	0.15%
\wmovee4	0.31%	0.30%	0.50%	0.64%	0.46%	0.48%	0.61%	0.18%
\wmoveQa2	1.54%	1.18%	1.05%	0.78%	0.45%	0.15%	0.34%	0.27%
\wmoveNh5	0.17%	0.17%	0.05%	0.19%	0.11%	0.16%	0.37%	0.18%
\wmovef3	0.30%	0.17%	0.21%	0.21%	0.17%	0.12%	0.57%	0.18%
\wmoveNe2	0.47%	0.30%	0.50%	0.38%	0.50%	0.67%	1.46%	0.18%
\wmoveRd3	0.10%	0.09%	0.05%	0.08%	0.06%	0.08%	0.24%	0.19%
\wmoveKh1	0.54%	0.79%	0.83%	0.95%	0.48%	0.69%	0.45%	0.19%
\wmoveg4	0.29%	0.24%	0.11%	0.17%	0.23%	0.13%	0.25%	0.40%
\wmoveRd1	0.57%	0.36%	0.47%	0.15%	0.09%	0.30%	0.76%	0.25%
\wmoveQd5	0.45%	0.64%	0.55%	0.63%	0.63%	0.47%	0.68%	0.16%
\wmoveg3	0.11%	0.05%	0.03%	0.03%	0.01%	0.06%	0.26%	0.29%

Table 13. Table 13 : Move probabilities by layer for puzzle 09SS5 (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQe8+	1.73%	0.11%	0.16%	0.24%	1.28%	0.02%	0.05%	1.11%
\wmovegxf7+	13.74%	2.54%	29.80%	33.00%	15.57%	37.26%	36.43%	43.50%
\wmoveQxf7+	4.30%	31.00%	39.29%	28.88%	32.40%	35.43%	32.63%	30.50%
\wmoveQxa3	0.43%	33.82%	15.83%	23.85%	25.18%	13.23%	13.23%	10.04%
\wmoveQd7	30.62%	0.06%	0.18%	0.05%	0.03%	0.03%	0.02%	0.02%
\wmoveRxa3	0.06%	26.89%	10.22%	8.50%	21.20%	12.43%	15.65%	11.68%
\wmoveQc7	20.69%	0.06%	0.04%	0.11%	0.06%	0.04%	0.02%	0.02%
\wmoveQe4	0.01%	0.05%	0.06%	0.12%	0.06%	0.05%	0.06%	0.03%
\wmoveRh3	0.01%	0.13%	0.08%	0.16%	0.22%	0.03%	0.38%	1.42%
\wmoveQf6	6.82%	0.05%	0.14%	0.15%	0.10%	0.06%	0.15%	0.11%
\wmoveRf3	0.04%	0.08%	0.23%	0.15%	0.09%	0.02%	0.18%	0.29%
\wmoveQb7	1.29%	0.10%	0.05%	0.04%	0.02%	0.02%	0.02%	0.02%
\wmoveQd6	4.19%	0.05%	0.18%	0.05%	0.04%	0.03%	0.03%	0.02%
\wmoveQe6	3.74%	0.06%	0.08%	0.33%	0.08%	0.03%	0.05%	0.03%
\wmoveRe4	0.05%	0.07%	0.06%	0.07%	0.12%	0.03%	0.12%	0.15%
\wmoveQg5	0.43%	2.70%	0.81%	0.21%	0.21%	0.09%	0.11%	0.07%
\wmoveQb4	2.24%	0.20%	0.20%	1.01%	0.38%	0.08%	0.03%	0.03%
\wmoveQf8+	1.79%	0.10%	0.74%	0.36%	1.50%	0.08%	0.10%	0.23%
\wmoveQd8+	1.67%	0.10%	0.21%	0.06%	0.09%	0.02%	0.02%	0.03%
\wmoveQh4	1.51%	0.27%	0.21%	0.82%	0.10%	0.05%	0.03%	0.02%
\wmovef4	0.42%	0.20%	0.07%	0.09%	0.07%	0.08%	0.04%	0.06%
\wmoveRe6	1.18%	0.07%	0.06%	0.06%	0.06%	0.03%	0.03%	0.03%
\wmoveRg3	0.03%	0.11%	0.10%	0.11%	0.09%	0.02%	0.18%	0.18%
\wmoveQa7	0.59%	0.08%	0.06%	0.05%	0.05%	0.04%	0.02%	0.01%
\wmoveQc5	0.75%	0.07%	0.04%	0.43%	0.14%	0.06%	0.03%	0.03%
\wmoveRe5	0.60%	0.08%	0.16%	0.07%	0.05%	0.04%	0.05%	0.05%
\wmoveQe5	0.59%	0.06%	0.11%	0.22%	0.09%	0.06%	0.04%	0.03%
\wmoveKg2	0.03%	0.06%	0.24%	0.39%	0.28%	0.06%	0.09%	0.08%
\wmoveRc3	0.03%	0.13%	0.08%	0.05%	0.12%	0.15%	0.04%	0.04%
\wmoveRb3	0.03%	0.20%	0.11%	0.09%	0.12%	0.12%	0.03%	0.03%
\wmoveKh1	0.08%	0.09%	0.08%	0.06%	0.04%	0.03%	0.03%	0.02%
\wmoveKh2	0.05%	0.09%	0.09%	0.11%	0.09%	0.05%	0.05%	0.05%
\wmoveRd3	0.04%	0.09%	0.06%	0.04%	0.03%	0.05%	0.03%	0.03%
\wmoveRe2	0.11%	0.06%	0.06%	0.03%	0.03%	0.05%	0.02%	0.02%
\wmoveRe1	0.05%	0.11%	0.07%	0.03%	0.02%	0.12%	0.02%	0.02%
\wmovef3	0.04%	0.06%	0.01%	0.01%	0.00%	0.00%	0.00%	0.00%

Table 14. Table 14 : Move probabilities by layer for puzzle 09SS5 (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQe8+	0.46%	3.87%	4.40%	7.59%	16.43%	15.84%	22.31%	58.86%
\wmovegxf7+	41.53%	37.46%	42.34%	36.43%	27.24%	36.29%	20.04%	1.46%
\wmoveQxf7+	29.73%	27.14%	28.31%	19.52%	15.49%	5.55%	1.83%	0.27%
\wmoveQxa3	10.11%	11.00%	8.21%	7.69%	4.35%	2.92%	6.87%	0.45%
\wmoveQd7	0.02%	0.01%	0.01%	0.01%	0.02%	0.02%	0.17%	0.24%
\wmoveRxa3	15.22%	16.47%	13.12%	19.88%	19.22%	20.38%	12.69%	9.17%
\wmoveQc7	0.03%	0.02%	0.04%	0.02%	0.02%	0.01%	0.26%	0.21%
\wmoveQe4	0.03%	0.07%	0.19%	0.30%	0.54%	0.87%	8.36%	9.86%
\wmoveRh3	0.93%	1.98%	1.26%	3.77%	6.90%	6.86%	6.19%	6.35%
\wmoveQf6	0.15%	0.06%	0.06%	0.07%	0.08%	0.04%	0.14%	0.18%
\wmoveRf3	0.21%	0.27%	0.19%	1.53%	5.85%	6.66%	5.66%	1.98%
\wmoveQb7	0.02%	0.01%	0.02%	0.02%	0.04%	0.20%	6.48%	4.20%
\wmoveQd6	0.03%	0.02%	0.03%	0.02%	0.02%	0.01%	0.08%	0.18%
\wmoveQe6	0.03%	0.02%	0.05%	0.03%	0.03%	0.04%	0.08%	0.18%
\wmoveRe4	0.11%	0.15%	0.07%	0.29%	1.69%	1.86%	2.77%	0.75%
\wmoveQg5	0.11%	0.06%	0.05%	0.05%	0.03%	0.02%	0.08%	0.20%
\wmoveQb4	0.07%	0.05%	0.10%	0.07%	0.05%	0.01%	0.09%	0.17%
\wmoveQf8+	0.19%	0.24%	0.27%	1.02%	0.21%	0.03%	0.41%	0.22%
\wmoveQd8+	0.02%	0.04%	0.04%	0.07%	0.21%	0.02%	0.07%	0.25%
\wmoveQh4	0.03%	0.05%	0.06%	0.06%	0.07%	0.06%	0.46%	0.71%
\wmovef4	0.15%	0.30%	0.38%	0.75%	0.40%	0.56%	1.42%	0.16%
\wmoveRe6	0.04%	0.03%	0.04%	0.05%	0.06%	0.04%	0.11%	0.24%
\wmoveRg3	0.14%	0.19%	0.10%	0.27%	0.42%	1.08%	1.05%	0.54%
\wmoveQa7	0.02%	0.01%	0.02%	0.02%	0.01%	0.01%	0.94%	0.77%
\wmoveQc5	0.04%	0.03%	0.06%	0.05%	0.04%	0.02%	0.07%	0.22%
\wmoveRe5	0.04%	0.05%	0.04%	0.05%	0.14%	0.18%	0.16%	0.21%
\wmoveQe5	0.03%	0.02%	0.02%	0.02%	0.02%	0.01%	0.08%	0.18%
\wmoveKg2	0.10%	0.07%	0.10%	0.06%	0.02%	0.01%	0.10%	0.18%
\wmoveRc3	0.12%	0.05%	0.10%	0.04%	0.09%	0.08%	0.16%	0.35%
\wmoveRb3	0.07%	0.05%	0.08%	0.06%	0.08%	0.07%	0.26%	0.19%
\wmoveKh1	0.02%	0.02%	0.03%	0.03%	0.09%	0.11%	0.13%	0.21%
\wmoveKh2	0.04%	0.05%	0.04%	0.05%	0.02%	0.01%	0.11%	0.19%
\wmoveRd3	0.06%	0.07%	0.11%	0.10%	0.09%	0.10%	0.16%	0.18%
\wmoveRe2	0.04%	0.03%	0.05%	0.02%	0.03%	0.02%	0.10%	0.17%
\wmoveRe1	0.03%	0.02%	0.02%	0.01%	0.01%	0.01%	0.12%	0.17%
\wmovef3	0.02%	0.00%	0.01%	0.00%	0.00%	0.00%	0.01%	0.15%

Table 15. Table 15 : Move probabilities by layer for puzzle 7UUlN (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveBh2+	0.08%	0.02%	0.54%	0.16%	0.28%	0.29%	1.79%	14.62%
\wmoveRxe3	15.44%	30.90%	82.76%	82.54%	86.84%	86.62%	76.43%	43.60%
\wmovea4	57.52%	43.79%	0.40%	0.10%	3.44%	0.02%	5.46%	3.59%
\wmoveRe5	0.49%	0.01%	0.07%	0.13%	0.44%	2.98%	9.25%	31.57%
\wmoveh5	0.20%	13.90%	1.77%	0.11%	0.02%	0.00%	0.01%	0.01%
\wmoveb5	2.34%	8.17%	0.40%	0.36%	0.01%	0.02%	0.08%	0.05%
\wmoveRe4	5.54%	0.02%	1.78%	0.61%	1.55%	0.30%	0.40%	0.25%
\wmoveBc5	1.33%	0.02%	0.99%	4.38%	0.59%	1.23%	0.27%	0.12%
\wmovef5	3.35%	0.57%	0.13%	0.02%	0.02%	0.00%	0.01%	0.04%
\wmoveKh8	0.63%	0.11%	1.07%	2.96%	1.91%	2.12%	0.38%	0.18%
\wmoveRf6	0.37%	0.02%	2.26%	0.32%	0.16%	0.37%	0.33%	1.45%
\wmoveg5	2.25%	0.46%	0.22%	0.05%	0.01%	0.01%	0.05%	0.08%
\wmovef6	0.18%	0.13%	0.03%	0.06%	0.00%	0.01%	0.02%	0.13%
\wmoveh6	0.05%	0.73%	0.13%	0.08%	0.02%	0.03%	0.07%	0.11%
\wmoveBg3	0.75%	0.03%	0.34%	0.26%	0.16%	0.30%	0.18%	0.16%
\wmoveRa8	0.74%	0.12%	0.17%	1.38%	0.30%	0.38%	0.30%	0.27%
\wmoveRg6	0.31%	0.02%	1.38%	0.28%	0.33%	0.18%	0.13%	0.11%
\wmoveRh6	0.10%	0.02%	1.29%	0.36%	0.36%	0.57%	0.40%	0.67%
\wmoveBe7	1.22%	0.01%	0.08%	0.15%	0.16%	0.18%	0.14%	0.04%
\wmoveRee8	0.39%	0.18%	1.14%	0.88%	0.30%	0.09%	0.27%	0.09%
\wmoveBe5	1.10%	0.01%	0.21%	1.13%	0.47%	0.91%	0.37%	0.57%
\wmoveBa3	1.10%	0.02%	0.21%	0.12%	0.61%	0.24%	0.75%	0.26%
\wmoveBf4	0.99%	0.02%	0.22%	0.58%	0.17%	0.41%	0.39%	0.43%
\wmoveRce8	0.28%	0.26%	0.99%	0.48%	0.45%	0.10%	0.45%	0.13%
\wmoveg6	0.31%	0.22%	0.03%	0.04%	0.00%	0.01%	0.01%	0.01%
\wmoveRe7	0.79%	0.01%	0.44%	0.10%	0.14%	0.31%	0.25%	0.19%
\wmoveBb4	0.60%	0.02%	0.12%	0.31%	0.34%	0.32%	0.62%	0.19%
\wmoveRb8	0.53%	0.04%	0.09%	0.69%	0.19%	0.74%	0.44%	0.57%
\wmoveKf8	0.26%	0.06%	0.23%	0.39%	0.31%	0.57%	0.29%	0.16%
\wmoveRd8	0.34%	0.05%	0.21%	0.50%	0.17%	0.37%	0.25%	0.24%
\wmoveRf8	0.39%	0.05%	0.23%	0.27%	0.13%	0.17%	0.12%	0.07%
\wmoveBf8	0.04%	0.02%	0.09%	0.18%	0.11%	0.15%	0.11%	0.05%

Table 16. Table 16 : Move probabilities by layer for puzzle 7UUlN (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveBh2+	16.54%	23.41%	22.23%	23.31%	23.15%	51.16%	67.02%	92.87%
\wmoveRxe3	30.72%	50.54%	35.04%	42.78%	31.82%	5.67%	3.56%	0.17%
\wmovea4	1.12%	0.35%	0.10%	0.27%	0.90%	1.69%	5.71%	1.08%
\wmoveRe5	48.65%	21.68%	38.76%	29.97%	35.14%	35.87%	3.34%	0.17%
\wmoveh5	0.01%	0.01%	0.01%	0.01%	0.02%	0.02%	0.37%	0.26%
\wmoveb5	0.10%	0.12%	0.05%	0.07%	0.37%	0.03%	1.04%	0.15%
\wmoveRe4	0.06%	0.12%	0.06%	0.12%	0.29%	0.14%	0.75%	0.16%
\wmoveBc5	0.07%	0.20%	0.15%	0.13%	0.15%	0.18%	0.72%	0.16%
\wmovef5	0.15%	0.44%	0.06%	0.06%	0.30%	0.17%	0.73%	0.71%
\wmoveKh8	0.12%	0.19%	0.29%	0.35%	0.77%	0.78%	1.39%	0.19%
\wmoveRf6	0.61%	0.29%	0.53%	0.34%	1.86%	0.40%	0.37%	0.16%
\wmoveg5	0.07%	0.08%	0.07%	0.06%	0.24%	0.11%	0.63%	0.16%
\wmovef6	0.09%	0.06%	0.03%	0.05%	0.10%	0.06%	1.98%	0.18%
\wmoveh6	0.04%	0.01%	0.02%	0.02%	0.18%	0.06%	1.74%	0.14%
\wmoveBg3	0.07%	0.14%	0.21%	0.27%	0.64%	0.69%	1.46%	0.19%
\wmoveRa8	0.08%	0.07%	0.07%	0.06%	0.07%	0.06%	0.26%	0.30%
\wmoveRg6	0.03%	0.04%	0.03%	0.04%	0.31%	0.09%	0.50%	0.15%
\wmoveRh6	0.24%	0.12%	0.14%	0.13%	0.80%	0.18%	0.49%	0.15%
\wmoveBe7	0.02%	0.08%	0.07%	0.08%	0.09%	0.23%	0.53%	0.16%
\wmoveRee8	0.03%	0.03%	0.02%	0.01%	0.13%	0.09%	0.47%	0.21%
\wmoveBe5	0.19%	0.42%	0.27%	0.23%	0.80%	0.52%	0.71%	0.16%
\wmoveBa3	0.15%	0.25%	0.26%	0.16%	0.23%	0.13%	0.89%	0.16%
\wmoveBf4	0.25%	0.49%	0.59%	0.63%	0.40%	0.62%	1.07%	0.18%
\wmoveRce8	0.08%	0.08%	0.10%	0.11%	0.17%	0.16%	0.27%	0.39%
\wmoveg6	0.01%	0.00%	0.00%	0.01%	0.04%	0.01%	0.81%	0.14%
\wmoveRe7	0.04%	0.10%	0.09%	0.08%	0.21%	0.10%	0.49%	0.17%
\wmoveBb4	0.08%	0.18%	0.25%	0.13%	0.14%	0.28%	0.78%	0.15%
\wmoveRb8	0.13%	0.11%	0.15%	0.09%	0.15%	0.08%	0.40%	0.17%
\wmoveKf8	0.11%	0.15%	0.10%	0.14%	0.19%	0.14%	0.44%	0.17%
\wmoveRd8	0.06%	0.07%	0.11%	0.09%	0.12%	0.08%	0.28%	0.27%
\wmoveRf8	0.04%	0.06%	0.08%	0.12%	0.18%	0.15%	0.44%	0.17%
\wmoveBf8	0.03%	0.10%	0.07%	0.06%	0.03%	0.07%	0.36%	0.14%

Table 17. Table 17 : Move evaluation for puzzle ANklq : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+\infty$	—	$1.4\%$	$2.6\%$	$96.0\%$	—
\wmoveQxf1+ $\star$	$+\infty$	—	$99.8\%$	$0.1\%$	$0.0\%$	$+98.4\%$
\wmoveBa8	$-4.34$	$-\infty$	$0.1\%$	$0.2\%$	$99.7\%$	$-1.3\%$
\wmoveRd7	$-4.36$	$-\infty$	$0.3\%$	$0.6\%$	$99.1\%$	$-1.1\%$
\wmoveBc8	$-4.51$	$-\infty$	$0.3\%$	$0.6\%$	$99.1\%$	$-1.1\%$
\wmoveQc8	$-4.87$	$-\infty$	$0.3\%$	$0.7\%$	$98.9\%$	$-1.1\%$
\wmoveQd7	$-4.91$	$-\infty$	$0.2\%$	$0.5\%$	$99.3\%$	$-1.2\%$
\wmoveRb8	$-4.95$	$-\infty$	$0.2\%$	$0.3\%$	$99.5\%$	$-1.2\%$
\wmoveNc5	$-4.97$	$-\infty$	$0.1\%$	$0.1\%$	$99.8\%$	$-1.3\%$
\wmoveBh1	$-5.23$	$-\infty$	$0.1\%$	$0.2\%$	$99.7\%$	$-1.3\%$
\wmoveBe4	$-5.34$	$-\infty$	$0.1\%$	$0.2\%$	$99.7\%$	$-1.3\%$
\wmoveBd5	$-5.37$	$-\infty$	$0.1\%$	$0.2\%$	$99.6\%$	$-1.3\%$
\wmoveNxb4	$-5.37$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveNc7	$-5.83$	$-\infty$	$0.0\%$	$0.1\%$	$99.8\%$	$-1.4\%$
\wmoveNb8	$-6.19$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveBf3	$-6.21$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveBg2	$-6.44$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-1.4\%$
\wmoveQg4	$-6.53$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-1.4\%$
\wmoveRf8	$-6.78$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-1.4\%$
\wmoveRc8	$-6.78$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveQe6	$-6.82$	$-\infty$	$0.0\%$	$0.1\%$	$99.8\%$	$-1.4\%$
\wmoveRd6	$-6.87$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveBc6	$-6.90$	$-\infty$	$0.0\%$	$0.2\%$	$99.8\%$	$-1.4\%$
\wmovef6	$-6.92$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveKh8	$-6.93$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveKf8	$-6.95$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveg6	$-6.95$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-1.4\%$
\wmoveKh7	$-6.97$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-1.4\%$
\wmoveRd3	$-7.06$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveRd5	$-7.07$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveRa8	$-7.21$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveRd2	$-7.30$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmovef5	$-7.32$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveQxh4	$-7.72$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveRd4	$-7.94$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.4\%$
\wmoveRd1	$-8.33$	$-\infty$	$0.0\%$	$0.4\%$	$99.6\%$	$-1.4\%$
\wmoveQxg3+	$-8.48$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveQg2+	$-8.48$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveQxh2+	$-8.71$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveQf5	$-9.18$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveRe8	$-9.33$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$
\wmoveg5	$-9.38$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.4\%$

Table 18. Table 18 : Move probabilities by layer for puzzle ANklq (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQxf1+	0.46%	45.73%	49.14%	46.74%	44.85%	47.54%	66.46%	47.93%
\wmoveRd2	41.09%	0.07%	0.04%	0.15%	0.44%	0.81%	0.08%	0.11%
\wmoveQc8	0.10%	0.02%	0.04%	0.03%	0.03%	0.13%	0.01%	0.02%
\wmoveQxh2+	0.51%	23.84%	13.22%	25.78%	30.20%	31.29%	23.55%	31.01%
\wmoveQg2+	26.47%	0.03%	0.37%	0.34%	0.67%	0.04%	0.08%	0.57%
\wmoveQd7	0.01%	0.01%	0.03%	0.02%	0.04%	0.05%	0.02%	0.02%
\wmoveQxh4	0.18%	13.43%	24.41%	3.60%	1.43%	0.72%	0.68%	0.59%
\wmoveQxg3+	1.06%	15.02%	9.23%	17.08%	15.67%	9.74%	5.29%	12.33%
\wmoveRd7	0.04%	0.02%	0.03%	0.04%	0.04%	0.10%	0.02%	0.04%
\wmoveRd3	12.98%	0.03%	0.02%	0.11%	0.63%	1.50%	0.27%	0.41%
\wmoveNc7	0.18%	0.19%	0.01%	0.02%	0.14%	0.05%	0.45%	0.37%
\wmoveRd1	5.56%	0.06%	0.06%	0.08%	1.16%	0.07%	0.16%	0.88%
\wmoveBc8	0.10%	0.01%	0.01%	0.01%	0.01%	0.02%	0.01%	0.01%
\wmoveNxb4	4.36%	0.04%	1.58%	3.76%	1.96%	1.45%	0.97%	0.26%
\wmoveBg2	0.11%	0.03%	0.18%	0.22%	0.27%	0.04%	0.27%	2.32%
\wmoveRd4	2.29%	0.06%	0.04%	0.11%	0.26%	0.83%	0.11%	0.23%
\wmovef5	1.00%	0.06%	0.01%	0.02%	0.01%	0.01%	0.01%	0.01%
\wmoveBd5	0.13%	0.01%	0.01%	0.02%	0.05%	0.03%	0.06%	0.06%
\wmoveBh1	0.04%	0.05%	0.08%	0.16%	0.41%	0.04%	0.57%	1.02%
\wmoveBf3	0.29%	0.16%	0.34%	0.09%	0.27%	0.07%	0.05%	0.14%
\wmoveRb8	0.17%	0.03%	0.01%	0.03%	0.05%	1.18%	0.08%	0.29%
\wmoveQg4	0.48%	0.02%	0.08%	0.05%	0.08%	0.10%	0.03%	0.04%
\wmoveRd5	0.01%	0.02%	0.05%	0.05%	0.04%	0.10%	0.03%	0.02%
\wmoveNc5	0.09%	0.20%	0.37%	0.76%	0.44%	0.46%	0.16%	0.28%
\wmoveBa8	0.02%	0.02%	0.02%	0.01%	0.01%	0.02%	0.01%	0.01%
\wmoveQe6	0.00%	0.02%	0.08%	0.05%	0.05%	0.05%	0.01%	0.01%
\wmoveRa8	0.23%	0.05%	0.01%	0.04%	0.07%	0.49%	0.04%	0.05%
\wmoveRd6	0.01%	0.02%	0.03%	0.08%	0.08%	0.80%	0.08%	0.24%
\wmoveg6	0.09%	0.05%	0.01%	0.01%	0.01%	0.01%	0.00%	0.00%
\wmoveBe4	0.33%	0.01%	0.01%	0.02%	0.03%	0.04%	0.04%	0.03%
\wmoveBc6	0.05%	0.01%	0.01%	0.01%	0.02%	0.03%	0.03%	0.02%
\wmoveRc8	0.11%	0.02%	0.05%	0.03%	0.04%	0.73%	0.02%	0.03%
\wmoveQf5	0.01%	0.02%	0.04%	0.05%	0.04%	0.04%	0.02%	0.03%
\wmoveg5	0.68%	0.09%	0.11%	0.20%	0.15%	0.02%	0.04%	0.16%
\wmovef6	0.09%	0.03%	0.01%	0.00%	0.00%	0.00%	0.00%	0.00%
\wmoveRe8	0.10%	0.02%	0.03%	0.02%	0.04%	0.33%	0.06%	0.12%
\wmoveNb8	0.07%	0.38%	0.07%	0.05%	0.05%	0.45%	0.13%	0.23%
\wmoveRf8	0.12%	0.02%	0.04%	0.03%	0.04%	0.27%	0.03%	0.05%
\wmoveKh7	0.11%	0.02%	0.04%	0.04%	0.08%	0.13%	0.02%	0.01%
\wmoveKh8	0.19%	0.05%	0.05%	0.03%	0.05%	0.10%	0.02%	0.02%
\wmoveKf8	0.08%	0.02%	0.04%	0.03%	0.08%	0.15%	0.02%	0.03%

Table 19. Table 19 : Move probabilities by layer for puzzle ANklq (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQxf1+	53.31%	51.84%	56.08%	59.27%	60.25%	88.28%	30.59%	5.79%
\wmoveRd2	0.10%	0.06%	0.12%	0.04%	0.08%	0.05%	0.60%	0.22%
\wmoveQc8	0.02%	0.02%	0.03%	0.04%	0.06%	0.07%	2.40%	37.04%
\wmoveQxh2+	30.50%	27.80%	26.32%	25.86%	23.28%	3.29%	3.05%	0.18%
\wmoveQg2+	0.50%	2.37%	1.78%	0.47%	0.58%	0.16%	1.41%	0.21%
\wmoveQd7	0.03%	0.05%	0.14%	0.30%	0.59%	0.31%	7.38%	24.66%
\wmoveQxh4	0.28%	0.19%	0.11%	0.10%	0.22%	0.25%	2.00%	0.26%
\wmoveQxg3+	6.65%	9.07%	8.91%	8.23%	6.30%	2.32%	11.04%	0.17%
\wmoveRd7	0.01%	0.01%	0.01%	0.00%	0.00%	0.02%	0.94%	13.27%
\wmoveRd3	0.47%	0.37%	0.55%	0.34%	0.52%	0.12%	0.36%	0.21%
\wmoveNc7	0.23%	0.23%	0.32%	0.20%	0.38%	0.35%	12.47%	2.19%
\wmoveRd1	1.45%	2.10%	0.87%	1.02%	3.03%	0.38%	0.80%	0.32%
\wmoveBc8	0.01%	0.02%	0.02%	0.07%	0.14%	0.35%	2.62%	4.44%
\wmoveNxb4	0.39%	0.35%	0.50%	0.29%	0.17%	0.23%	1.36%	0.30%
\wmoveBg2	2.99%	2.86%	1.29%	0.80%	0.30%	0.22%	1.28%	0.23%
\wmoveRd4	0.17%	0.19%	0.46%	0.15%	0.31%	0.24%	0.40%	0.20%
\wmovef5	0.01%	0.01%	0.02%	0.07%	0.11%	0.14%	2.19%	0.21%
\wmoveBd5	0.12%	0.16%	0.19%	0.23%	0.45%	0.60%	2.13%	0.88%
\wmoveBh1	0.68%	0.28%	0.35%	0.94%	0.44%	0.18%	1.80%	1.07%
\wmoveBf3	0.17%	0.16%	0.25%	0.31%	0.33%	0.10%	1.48%	0.20%
\wmoveRb8	0.40%	0.14%	0.07%	0.01%	0.02%	0.01%	0.27%	1.23%
\wmoveQg4	0.04%	0.05%	0.06%	0.04%	0.09%	0.06%	1.13%	0.24%
\wmoveRd5	0.02%	0.01%	0.01%	0.00%	0.00%	0.02%	1.06%	0.42%
\wmoveNc5	0.27%	0.45%	0.17%	0.24%	0.32%	0.45%	0.70%	1.05%
\wmoveBa8	0.01%	0.01%	0.01%	0.01%	0.01%	0.02%	0.33%	1.02%
\wmoveQe6	0.02%	0.05%	0.05%	0.13%	0.28%	0.55%	1.01%	0.31%
\wmoveRa8	0.06%	0.05%	0.12%	0.09%	0.15%	0.16%	0.97%	0.20%
\wmoveRd6	0.18%	0.07%	0.16%	0.05%	0.10%	0.04%	0.87%	0.21%
\wmoveg6	0.01%	0.01%	0.01%	0.01%	0.02%	0.04%	0.86%	0.20%
\wmoveBe4	0.04%	0.04%	0.05%	0.06%	0.11%	0.06%	0.83%	0.59%
\wmoveBc6	0.03%	0.03%	0.04%	0.07%	0.11%	0.09%	0.76%	0.24%
\wmoveRc8	0.04%	0.06%	0.07%	0.02%	0.02%	0.03%	0.24%	0.23%
\wmoveQf5	0.02%	0.02%	0.03%	0.04%	0.09%	0.13%	0.71%	0.19%
\wmoveg5	0.05%	0.06%	0.02%	0.04%	0.05%	0.04%	0.50%	0.27%
\wmovef6	0.01%	0.01%	0.01%	0.00%	0.01%	0.01%	0.67%	0.21%
\wmoveRe8	0.37%	0.59%	0.56%	0.21%	0.64%	0.28%	0.39%	0.17%
\wmoveNb8	0.19%	0.12%	0.07%	0.03%	0.05%	0.04%	0.62%	0.36%
\wmoveRf8	0.12%	0.05%	0.10%	0.05%	0.10%	0.06%	0.53%	0.21%
\wmoveKh7	0.02%	0.02%	0.02%	0.03%	0.05%	0.05%	0.49%	0.21%
\wmoveKh8	0.02%	0.02%	0.03%	0.08%	0.10%	0.09%	0.45%	0.22%
\wmoveKf8	0.03%	0.03%	0.04%	0.04%	0.12%	0.09%	0.30%	0.19%

Table 20. Table 20 : Move evaluation for puzzle ID 58Ib0 : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+6.18$	—	$20.0\%$	$8.2\%$	$71.8\%$	—
\wmoveRxg7+ $\star$	$+6.15$	$-3$	$99.5\%$	$0.4\%$	$0.1\%$	$+79.5\%$
\wmoveKf1	$-2.47$	$-865$	$0.9\%$	$7.9\%$	$91.1\%$	$-19.1\%$
\wmoveQe3	$-3.01$	$-919$	$0.4\%$	$2.5\%$	$97.1\%$	$-19.6\%$
\wmoveQh2	$-3.29$	$-947$	$0.5\%$	$2.9\%$	$96.6\%$	$-19.5\%$
\wmoveRg2	$-3.64$	$-982$	$0.3\%$	$1.5\%$	$98.2\%$	$-19.7\%$
\wmoveQe1	$-4.53$	$-1071$	$0.2\%$	$0.7\%$	$99.1\%$	$-19.8\%$
\wmoveRf3	$-7.32$	$-1350$	$0.1\%$	$0.5\%$	$99.4\%$	$-19.9\%$
\wmoveQxa7	$-7.41$	$-1359$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRh3	$-8.01$	$-1419$	$0.0\%$	$0.1\%$	$99.9\%$	$-20.0\%$
\wmoveQg2	$-8.17$	$-1435$	$0.0\%$	$0.1\%$	$99.9\%$	$-20.0\%$
\wmoveQc5	$-8.60$	$-1478$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveQf3	$-8.60$	$-1478$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveQd4	$-8.64$	$-1482$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveQd2	$-8.87$	$-1505$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveQe2	$-9.30$	$-1548$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRg6	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRg5	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRg4	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRe3	$-\infty$	$-\infty$	$0.0\%$	$0.2\%$	$99.8\%$	$-20.0\%$
\wmoveRd3	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveRc3	$-\infty$	$-\infty$	$0.0\%$	$0.3\%$	$99.6\%$	$-19.9\%$
\wmoveQb6	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmoveQf1	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-20.0\%$
\wmovef5	$-\infty$	$-\infty$	$0.0\%$	$0.3\%$	$99.6\%$	$-19.9\%$
\wmovea5	$-\infty$	$-\infty$	$0.1\%$	$0.4\%$	$99.6\%$	$-19.9\%$
\wmoveb4	$-\infty$	$-\infty$	$0.0\%$	$0.3\%$	$99.6\%$	$-19.9\%$
\wmovec3	$-\infty$	$-\infty$	$0.1\%$	$0.8\%$	$99.1\%$	$-19.9\%$
\wmovec4	$-\infty$	$-\infty$	$0.0\%$	$0.2\%$	$99.8\%$	$-20.0\%$

Table 21. Table 21 : Move probabilities by layer for puzzle ID 58Ib0 (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveRxg7+	2.93%	52.97%	34.97%	56.12%	51.65%	54.48%	63.68%	64.57%
\wmovea5	59.08%	3.06%	0.04%	0.01%	0.02%	0.01%	0.02%	0.05%
\wmoveKf1	0.31%	0.06%	0.34%	0.29%	0.13%	0.17%	0.26%	0.45%
\wmoveQxa7	11.45%	27.21%	52.41%	31.89%	27.95%	31.24%	33.11%	29.93%
\wmoveQg2	0.39%	10.35%	2.08%	0.34%	0.19%	0.18%	0.04%	0.05%
\wmoveQb6	6.17%	0.13%	0.19%	2.31%	8.67%	2.10%	0.45%	0.37%
\wmovef5	0.69%	1.39%	2.83%	3.29%	5.62%	0.12%	0.44%	1.04%
\wmoveRg4	0.41%	0.06%	0.09%	0.63%	2.00%	5.14%	0.71%	1.27%
\wmoveQc5	4.78%	0.10%	0.85%	0.12%	0.02%	0.06%	0.04%	0.11%
\wmoveQe3	0.92%	0.07%	0.21%	0.22%	0.07%	0.31%	0.03%	0.04%
\wmoveRg2	0.15%	2.88%	3.08%	0.20%	0.15%	0.24%	0.05%	0.08%
\wmoveb4	2.64%	0.58%	0.12%	0.14%	0.01%	0.01%	0.01%	0.02%
\wmoveQh2	0.48%	0.08%	0.15%	0.07%	0.02%	0.05%	0.02%	0.02%
\wmovec4	2.27%	0.23%	0.11%	0.10%	0.00%	0.04%	0.07%	0.26%
\wmoveRg6	1.81%	0.08%	0.50%	0.91%	0.52%	0.21%	0.29%	0.24%
\wmovec3	0.40%	0.06%	0.05%	0.02%	0.00%	0.00%	0.01%	0.05%
\wmoveRg5	0.92%	0.05%	0.19%	1.21%	1.45%	1.40%	0.15%	0.51%
\wmoveRh3	0.07%	0.05%	0.09%	0.34%	0.31%	1.19%	0.15%	0.22%
\wmoveQf1	0.02%	0.07%	0.28%	0.09%	0.03%	0.08%	0.03%	0.02%
\wmoveRd3	0.26%	0.06%	0.09%	0.48%	0.34%	0.97%	0.06%	0.11%
\wmoveQd4	0.85%	0.09%	0.19%	0.22%	0.09%	0.15%	0.04%	0.10%
\wmoveQe2	0.81%	0.06%	0.20%	0.10%	0.06%	0.13%	0.02%	0.02%
\wmoveQf3	0.75%	0.06%	0.13%	0.22%	0.25%	0.23%	0.06%	0.13%
\wmoveQd2	0.66%	0.06%	0.16%	0.11%	0.08%	0.33%	0.03%	0.04%
\wmoveRc3	0.21%	0.06%	0.22%	0.13%	0.03%	0.05%	0.03%	0.03%
\wmoveRf3	0.21%	0.05%	0.09%	0.19%	0.20%	0.53%	0.12%	0.21%
\wmoveRe3	0.23%	0.05%	0.22%	0.13%	0.07%	0.40%	0.05%	0.06%
\wmoveQe1	0.15%	0.07%	0.12%	0.14%	0.07%	0.16%	0.02%	0.02%

Table 22. Table 22 : Move probabilities by layer for puzzle ID 58Ib0 (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveRxg7+	62.81%	68.93%	82.50%	82.72%	80.06%	80.68%	50.84%	31.61%
\wmovea5	0.10%	0.09%	0.14%	0.08%	0.34%	0.07%	0.61%	0.26%
\wmoveKf1	0.31%	0.51%	0.55%	0.33%	1.22%	6.20%	30.42%	52.98%
\wmoveQxa7	32.68%	27.58%	14.43%	14.94%	13.42%	8.13%	2.23%	0.29%
\wmoveQg2	0.05%	0.05%	0.08%	0.08%	0.25%	0.35%	1.01%	0.33%
\wmoveQb6	0.60%	0.31%	0.17%	0.13%	0.13%	0.05%	0.27%	0.29%
\wmovef5	0.39%	0.28%	0.04%	0.05%	0.31%	0.07%	0.79%	0.23%
\wmoveRg4	0.76%	0.34%	0.29%	0.11%	0.11%	0.11%	0.29%	0.28%
\wmoveQc5	0.17%	0.15%	0.18%	0.11%	0.60%	0.23%	0.76%	0.30%
\wmoveQe3	0.07%	0.06%	0.06%	0.06%	0.16%	0.29%	1.40%	4.46%
\wmoveRg2	0.06%	0.07%	0.08%	0.07%	0.18%	0.14%	0.88%	1.53%
\wmoveb4	0.08%	0.07%	0.02%	0.02%	0.08%	0.03%	0.77%	0.24%
\wmoveQh2	0.04%	0.04%	0.04%	0.03%	0.10%	0.17%	1.58%	2.60%
\wmovec4	0.35%	0.22%	0.06%	0.05%	0.18%	0.13%	0.93%	0.24%
\wmoveRg6	0.28%	0.19%	0.19%	0.24%	0.44%	0.12%	0.31%	0.34%
\wmovec3	0.10%	0.05%	0.02%	0.03%	0.15%	0.23%	1.75%	0.19%
\wmoveRg5	0.35%	0.22%	0.30%	0.34%	0.36%	0.42%	0.45%	0.29%
\wmoveRh3	0.20%	0.09%	0.11%	0.07%	0.12%	0.21%	0.31%	0.28%
\wmoveQf1	0.04%	0.04%	0.05%	0.06%	0.23%	0.79%	0.97%	0.37%
\wmoveRd3	0.08%	0.05%	0.06%	0.04%	0.14%	0.19%	0.34%	0.26%
\wmoveQd4	0.14%	0.30%	0.18%	0.14%	0.47%	0.42%	0.71%	0.30%
\wmoveQe2	0.03%	0.03%	0.03%	0.03%	0.06%	0.12%	0.33%	0.29%
\wmoveQf3	0.07%	0.08%	0.10%	0.08%	0.30%	0.31%	0.30%	0.26%
\wmoveQd2	0.03%	0.03%	0.04%	0.02%	0.05%	0.07%	0.26%	0.29%
\wmoveRc3	0.03%	0.08%	0.05%	0.03%	0.18%	0.10%	0.56%	0.27%
\wmoveRf3	0.10%	0.06%	0.09%	0.07%	0.19%	0.20%	0.38%	0.45%
\wmoveRe3	0.06%	0.07%	0.10%	0.06%	0.10%	0.05%	0.16%	0.37%
\wmoveQe1	0.03%	0.03%	0.03%	0.02%	0.05%	0.12%	0.39%	0.37%

Table 23. Table 23 : Move evaluation for puzzle ID 6PIrs : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+7.17$	—	$1.8\%$	$22.3\%$	$75.9\%$	—
\wmoveg5+ $\star$	$+8.54$	$+137$	$98.3\%$	$1.5\%$	$0.2\%$	$+96.5\%$
\wmoveQd3	$-2.70$	$-987$	$0.8\%$	$9.7\%$	$89.5\%$	$-1.0\%$
\wmoveQc4	$-3.09$	$-1026$	$0.8\%$	$10.1\%$	$89.1\%$	$-1.0\%$
\wmoveQa2	$-3.09$	$-1026$	$0.5\%$	$3.6\%$	$95.9\%$	$-1.3\%$
\wmoveQf3	$-3.19$	$-1036$	$0.5\%$	$4.9\%$	$94.6\%$	$-1.3\%$
\wmoveQd2	$-3.24$	$-1041$	$0.8\%$	$8.3\%$	$90.9\%$	$-1.0\%$
\wmoveQa6	$-3.26$	$-1043$	$0.5\%$	$3.8\%$	$95.7\%$	$-1.3\%$
\wmoveQc2	$-3.45$	$-1062$	$0.6\%$	$5.1\%$	$94.3\%$	$-1.2\%$
\wmoveQd1	$-3.70$	$-1087$	$0.7\%$	$8.0\%$	$91.4\%$	$-1.1\%$
\wmoveQh2	$-3.72$	$-1089$	$0.2\%$	$0.9\%$	$98.8\%$	$-1.6\%$
\wmoveQf1	$-3.84$	$-1101$	$0.3\%$	$2.0\%$	$97.6\%$	$-1.5\%$
\wmoveQf2	$-4.05$	$-1122$	$0.3\%$	$1.5\%$	$98.3\%$	$-1.5\%$
\wmoveQg2	$-4.09$	$-1126$	$0.4\%$	$2.8\%$	$96.8\%$	$-1.4\%$
\wmoveQxe5	$-6.70$	$-1387$	$0.1\%$	$0.1\%$	$99.8\%$	$-1.7\%$
\wmoveKf8	$-12.22$	$-1939$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQb2	$-12.29$	$-1946$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveg6	$-12.97$	$-2014$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveKh8	$-\infty$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.8\%$
\wmoveKh7	$-\infty$	$-\infty$	$0.0\%$	$0.2\%$	$99.7\%$	$-1.8\%$
\wmoveKf7	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQh5+	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQb5	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQg4+	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQe4	$-\infty$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.8\%$
\wmoveQe3	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveQe1	$-\infty$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.8\%$
\wmoveb6	$-\infty$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.8\%$
\wmoveh5	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-1.8\%$
\wmoveb5	$-\infty$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-1.8\%$

Table 24. Table 24 : Move probabilities by layer for puzzle ID 6PIrs (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQxe5	0.03%	19.22%	16.74%	41.20%	46.85%	89.57%	57.18%	49.90%
\wmoveQg2	5.38%	51.44%	65.55%	2.96%	1.62%	0.13%	0.24%	0.19%
\wmoveg5+	0.22%	1.01%	0.11%	6.18%	6.27%	0.55%	18.46%	14.62%
\wmoveQd3	6.21%	0.13%	0.13%	0.39%	0.22%	0.15%	0.37%	0.50%
\wmoveQd2	25.48%	0.14%	0.36%	0.48%	0.17%	0.11%	0.12%	0.17%
\wmoveQc4	1.15%	0.17%	0.16%	0.55%	0.24%	0.14%	0.40%	0.59%
\wmoveQg4+	0.45%	0.40%	1.46%	4.21%	21.21%	0.84%	3.49%	10.09%
\wmoveQd1	5.23%	0.17%	0.33%	0.50%	0.35%	0.15%	0.61%	1.19%
\wmoveQc2	14.22%	0.16%	0.14%	0.36%	0.24%	0.14%	0.29%	0.29%
\wmoveQb5	2.00%	0.34%	0.46%	14.07%	2.75%	0.40%	0.67%	0.65%
\wmoveQf3	7.58%	0.24%	2.69%	3.25%	0.89%	0.18%	0.93%	1.16%
\wmoveh5	0.02%	10.08%	0.55%	6.22%	9.38%	2.67%	8.66%	4.21%
\wmoveQf2	8.87%	0.17%	0.43%	1.03%	0.21%	0.12%	0.22%	0.21%
\wmoveQh5+	0.59%	1.09%	1.88%	3.04%	2.39%	0.81%	3.19%	6.33%
\wmoveQh2	1.73%	6.20%	4.46%	1.02%	0.61%	0.17%	0.14%	0.13%
\wmoveQe3	5.99%	0.16%	0.14%	0.71%	0.23%	0.15%	0.11%	0.12%
\wmoveQe1	5.84%	0.26%	0.27%	0.34%	0.23%	0.12%	0.17%	0.15%
\wmoveQf1	5.49%	1.69%	2.47%	1.32%	0.30%	0.14%	0.18%	0.17%
\wmoveKh7	0.03%	0.12%	0.11%	1.70%	1.30%	0.41%	0.99%	2.21%
\wmoveg6	0.03%	0.57%	0.06%	0.69%	0.11%	0.20%	1.66%	4.30%
\wmoveb5	0.19%	3.90%	0.28%	4.28%	1.88%	0.19%	0.37%	0.74%
\wmoveKf7	0.01%	0.09%	0.25%	1.68%	0.81%	0.88%	0.43%	0.66%
\wmoveQb2	1.57%	0.23%	0.14%	0.36%	0.16%	0.13%	0.11%	0.11%
\wmoveQe4	1.21%	0.17%	0.11%	0.75%	0.24%	0.16%	0.16%	0.49%
\wmoveb6	0.03%	0.99%	0.03%	0.07%	0.01%	0.00%	0.01%	0.01%
\wmoveQa6	0.10%	0.49%	0.28%	0.94%	0.18%	0.18%	0.26%	0.19%
\wmoveKf8	0.02%	0.10%	0.15%	0.87%	0.64%	0.69%	0.26%	0.28%
\wmoveKh8	0.06%	0.15%	0.09%	0.51%	0.35%	0.51%	0.16%	0.14%
\wmoveQa2	0.26%	0.11%	0.16%	0.32%	0.17%	0.12%	0.16%	0.18%

Table 25. Table 25 : Move probabilities by layer for puzzle ID 6PIrs (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQxe5	53.49%	59.50%	77.55%	55.82%	56.34%	11.45%	2.52%	0.49%
\wmoveQg2	0.13%	0.07%	0.08%	0.12%	0.10%	0.12%	0.18%	0.36%
\wmoveg5+	22.91%	22.21%	0.38%	0.34%	6.39%	0.77%	41.93%	11.17%
\wmoveQd3	0.88%	0.94%	2.92%	10.24%	9.81%	29.23%	13.54%	29.14%
\wmoveQd2	0.16%	0.15%	0.34%	1.66%	1.47%	4.29%	5.51%	9.92%
\wmoveQc4	1.00%	1.53%	1.49%	4.79%	5.07%	19.61%	14.48%	25.06%
\wmoveQg4+	3.88%	1.70%	0.76%	1.28%	1.32%	0.56%	0.19%	0.17%
\wmoveQd1	1.12%	0.94%	1.70%	5.72%	5.08%	11.34%	7.67%	14.27%
\wmoveQc2	0.27%	0.16%	0.20%	0.46%	0.43%	1.20%	1.15%	1.99%
\wmoveQb5	0.43%	0.88%	1.05%	1.15%	0.53%	0.47%	0.29%	0.17%
\wmoveQf3	1.21%	0.92%	0.90%	2.81%	2.88%	10.71%	3.29%	2.98%
\wmoveh5	1.38%	0.99%	1.02%	0.59%	0.17%	0.16%	0.21%	0.18%
\wmoveQf2	0.13%	0.07%	0.08%	0.14%	0.14%	0.13%	0.23%	0.21%
\wmoveQh5+	2.19%	1.67%	1.97%	4.22%	2.31%	0.60%	0.17%	0.17%
\wmoveQh2	0.12%	0.09%	0.11%	0.15%	0.12%	0.11%	0.21%	0.20%
\wmoveQe3	0.09%	0.07%	0.09%	0.18%	0.25%	0.20%	0.30%	0.18%
\wmoveQe1	0.13%	0.08%	0.11%	0.21%	0.23%	0.17%	0.17%	0.18%
\wmoveQf1	0.14%	0.08%	0.10%	0.16%	0.11%	0.11%	0.21%	0.25%
\wmoveKh7	3.22%	2.93%	4.00%	4.05%	2.22%	4.85%	2.57%	0.18%
\wmoveg6	4.15%	2.10%	1.11%	1.13%	0.15%	0.36%	1.07%	0.20%
\wmoveb5	0.96%	1.30%	2.17%	1.52%	1.93%	0.61%	1.09%	0.19%
\wmoveKf7	0.70%	0.43%	0.43%	0.62%	0.73%	0.91%	0.30%	0.18%
\wmoveQb2	0.10%	0.07%	0.10%	0.16%	0.15%	0.13%	0.26%	0.18%
\wmoveQe4	0.46%	0.41%	0.49%	1.19%	0.89%	0.25%	0.38%	0.18%
\wmoveb6	0.02%	0.02%	0.01%	0.01%	0.02%	0.01%	0.71%	0.17%
\wmoveQa6	0.17%	0.20%	0.20%	0.30%	0.22%	0.40%	0.60%	0.81%
\wmoveKf8	0.27%	0.21%	0.33%	0.42%	0.64%	0.80%	0.20%	0.17%
\wmoveKh8	0.16%	0.19%	0.22%	0.41%	0.17%	0.28%	0.28%	0.18%
\wmoveQa2	0.13%	0.09%	0.10%	0.16%	0.12%	0.16%	0.31%	0.48%

Table 26. Table 26 : Move evaluation for puzzle ID 1Egyn : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+\infty$	—	$12.3\%$	$5.4\%$	$82.3\%$	—
\wmoveQxc8+ $\star$	$+\infty$	—	$100.0\%$	$0.0\%$	$0.0\%$	$+87.7\%$
\wmoveQxa7	$-4.62$	$-\infty$	$0.3\%$	$0.5\%$	$99.2\%$	$-12.0\%$
\wmoveQe4	$-7.11$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQb4	$-7.23$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-12.3\%$
\wmoveQc6	$-7.39$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQb6	$-7.64$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-12.3\%$
\wmoveQf3	$-7.67$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-12.3\%$
\wmoveQb5	$-7.70$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveh4	$-8.96$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveh3	$-9.11$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQe7	$-9.17$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoved4	$-9.18$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmovec4	$-9.22$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveb4	$-9.33$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveg3	$-9.33$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQc7	$-9.35$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRf1	$-9.44$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveg4	$-9.46$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveKh1	$-9.47$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmovec3	$-9.52$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmovef3	$-9.55$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRd1	$-9.66$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveKf1	$-9.68$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmovef4	$-10.00$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQa8	$-10.26$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQb8	$-10.29$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQxf7+	$-11.09$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQa6	$-11.11$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQd7	$-12.92$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveQd5	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRxe6	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRe5	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRe4	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRe3	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRe2	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRc1	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRb1	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$
\wmoveRa1	$-\infty$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-12.3\%$

Table 27. Table 27 : Move probabilities by layer for puzzle ID 1Egyn (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQxa7	0.13%	58.43%	72.93%	38.65%	12.37%	3.65%	3.45%	2.67%
\wmoveQxc8+	5.06%	35.05%	9.71%	36.67%	49.39%	54.67%	72.84%	48.69%
\wmoveRxe6	1.67%	0.17%	0.20%	0.40%	3.85%	14.24%	8.29%	13.42%
\wmoveQd7	27.16%	0.05%	0.33%	0.68%	0.40%	0.11%	0.10%	0.09%
\wmoveQe7	26.77%	0.05%	0.49%	0.36%	0.62%	0.20%	0.09%	0.20%
\wmovec4	0.33%	0.04%	0.38%	0.20%	0.31%	0.32%	0.84%	2.51%
\wmoveQc7	11.15%	0.16%	0.10%	0.08%	0.16%	0.15%	0.11%	0.15%
\wmoveQb8	7.99%	0.11%	0.37%	0.53%	1.13%	0.67%	0.40%	0.88%
\wmoveRc1	0.07%	0.04%	0.29%	1.46%	5.14%	4.71%	3.90%	7.98%
\wmoveRb1	0.10%	0.05%	0.34%	3.68%	6.77%	3.81%	2.34%	5.70%
\wmoveh3	0.01%	0.05%	0.20%	0.57%	1.23%	0.19%	0.38%	4.53%
\wmoveQxf7+	2.32%	1.37%	2.62%	2.94%	2.18%	3.33%	1.76%	0.84%
\wmoveQc6	4.90%	0.05%	0.15%	0.36%	0.54%	0.21%	0.15%	0.12%
\wmoveQa8	4.40%	2.71%	1.75%	1.08%	0.67%	0.42%	0.37%	1.33%
\wmoveRa1	0.10%	0.15%	0.60%	4.11%	3.70%	3.51%	1.24%	2.08%
\wmoveRe2	0.03%	0.04%	0.24%	0.44%	1.20%	1.75%	0.41%	3.42%
\wmoveRd1	0.07%	0.05%	0.32%	0.49%	2.01%	2.44%	1.32%	1.62%
\wmovef3	0.05%	0.05%	0.37%	0.15%	0.17%	0.06%	0.04%	0.09%
\wmoveQb6	1.74%	0.04%	0.17%	0.28%	0.37%	0.24%	0.27%	0.25%
\wmoveRe5	1.63%	0.04%	0.20%	0.74%	1.06%	1.35%	0.31%	1.64%
\wmoveQd5	1.61%	0.05%	1.44%	0.27%	0.42%	0.11%	0.06%	0.03%
\wmoveRe4	0.01%	0.04%	0.41%	0.20%	0.27%	1.05%	0.11%	0.29%
\wmoved4	0.15%	0.06%	0.73%	1.03%	0.57%	0.08%	0.08%	0.08%
\wmovec3	0.06%	0.09%	0.96%	0.39%	0.27%	0.06%	0.02%	0.02%
\wmoveh4	0.03%	0.12%	0.17%	0.27%	0.35%	0.08%	0.08%	0.18%
\wmoveKh1	0.12%	0.21%	0.87%	0.29%	0.49%	0.39%	0.08%	0.11%
\wmoveQb5	0.37%	0.04%	0.24%	0.37%	0.69%	0.22%	0.14%	0.10%
\wmoveRe3	0.00%	0.04%	0.20%	0.39%	0.36%	0.67%	0.05%	0.07%
\wmoveQb4	0.37%	0.07%	0.47%	0.51%	0.60%	0.14%	0.10%	0.05%
\wmoveQa6	0.03%	0.05%	0.15%	0.27%	0.59%	0.34%	0.25%	0.30%
\wmoveb4	0.38%	0.09%	0.36%	0.46%	0.34%	0.08%	0.07%	0.11%
\wmoveg4	0.38%	0.09%	0.57%	0.17%	0.11%	0.03%	0.01%	0.01%
\wmovef4	0.57%	0.05%	0.29%	0.17%	0.13%	0.03%	0.02%	0.04%
\wmoveQf3	0.03%	0.05%	0.37%	0.34%	0.44%	0.13%	0.07%	0.07%
\wmoveKf1	0.05%	0.09%	0.24%	0.31%	0.42%	0.27%	0.05%	0.07%
\wmoveg3	0.06%	0.04%	0.25%	0.15%	0.23%	0.07%	0.07%	0.10%
\wmoveQe4	0.03%	0.04%	0.33%	0.36%	0.27%	0.14%	0.08%	0.14%
\wmoveRf1	0.07%	0.05%	0.19%	0.19%	0.15%	0.05%	0.03%	0.03%

Table 28. Table 28 : Move probabilities by layer for puzzle ID 1Egyn (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQxa7	1.84%	6.79%	5.08%	3.98%	5.37%	12.37%	23.09%	63.67%
\wmoveQxc8+	47.17%	30.34%	31.97%	42.62%	58.43%	61.50%	54.05%	28.66%
\wmoveRxe6	13.02%	25.77%	29.43%	29.42%	20.49%	10.78%	3.06%	0.16%
\wmoveQd7	0.06%	0.07%	0.20%	0.12%	0.09%	0.06%	0.12%	0.21%
\wmoveQe7	0.10%	0.11%	0.19%	0.08%	0.07%	0.06%	0.23%	0.22%
\wmovec4	3.99%	9.02%	11.81%	16.30%	10.39%	8.67%	1.58%	0.23%
\wmoveQc7	0.10%	0.09%	0.21%	0.13%	0.08%	0.04%	0.13%	0.20%
\wmoveQb8	1.38%	0.85%	1.00%	0.39%	0.43%	0.23%	0.16%	0.19%
\wmoveRc1	6.78%	0.34%	1.03%	0.51%	0.47%	0.54%	0.14%	0.22%
\wmoveRb1	7.56%	4.31%	3.81%	0.66%	0.49%	0.39%	0.25%	0.20%
\wmoveh3	7.37%	7.22%	2.36%	1.75%	1.06%	3.46%	6.99%	0.23%
\wmoveQxf7+	1.35%	5.57%	2.49%	1.07%	0.35%	0.28%	1.52%	0.19%
\wmoveQc6	0.07%	0.08%	0.16%	0.09%	0.07%	0.05%	0.45%	0.21%
\wmoveQa8	1.16%	0.45%	1.08%	0.41%	0.33%	0.16%	0.10%	0.21%
\wmoveRa1	2.89%	3.44%	1.64%	0.32%	0.23%	0.24%	0.18%	0.22%
\wmoveRe2	1.88%	0.45%	0.79%	0.13%	0.06%	0.09%	0.27%	0.22%
\wmoveRd1	0.76%	1.58%	2.81%	0.33%	0.19%	0.06%	0.13%	0.21%
\wmovef3	0.35%	0.31%	0.24%	0.02%	0.03%	0.04%	1.76%	0.20%
\wmoveQb6	0.21%	0.19%	0.28%	0.11%	0.07%	0.04%	0.19%	0.23%
\wmoveRe5	0.58%	0.98%	1.41%	0.63%	0.77%	0.26%	0.37%	0.21%
\wmoveQd5	0.02%	0.03%	0.05%	0.04%	0.02%	0.04%	0.17%	0.21%
\wmoveRe4	0.11%	0.07%	0.14%	0.03%	0.03%	0.05%	0.62%	0.21%
\wmoved4	0.06%	0.23%	0.07%	0.03%	0.02%	0.02%	0.32%	0.21%
\wmovec3	0.02%	0.04%	0.08%	0.06%	0.03%	0.02%	0.12%	0.24%
\wmoveh4	0.24%	0.33%	0.23%	0.06%	0.08%	0.15%	0.94%	0.22%
\wmoveKh1	0.09%	0.13%	0.33%	0.36%	0.09%	0.08%	0.25%	0.18%
\wmoveQb5	0.07%	0.09%	0.14%	0.05%	0.03%	0.03%	0.16%	0.23%
\wmoveRe3	0.03%	0.04%	0.04%	0.01%	0.00%	0.02%	0.12%	0.22%
\wmoveQb4	0.03%	0.07%	0.05%	0.01%	0.01%	0.01%	0.15%	0.22%
\wmoveQa6	0.19%	0.11%	0.25%	0.04%	0.03%	0.02%	0.21%	0.23%
\wmoveb4	0.13%	0.39%	0.10%	0.02%	0.01%	0.01%	0.58%	0.22%
\wmoveg4	0.01%	0.01%	0.02%	0.01%	0.01%	0.02%	0.13%	0.19%
\wmovef4	0.04%	0.08%	0.18%	0.08%	0.05%	0.08%	0.35%	0.18%
\wmoveQf3	0.09%	0.19%	0.12%	0.03%	0.03%	0.02%	0.24%	0.23%
\wmoveKf1	0.05%	0.05%	0.05%	0.03%	0.01%	0.02%	0.15%	0.23%
\wmoveg3	0.08%	0.07%	0.07%	0.01%	0.03%	0.05%	0.39%	0.19%
\wmoveQe4	0.07%	0.09%	0.08%	0.04%	0.04%	0.04%	0.26%	0.24%
\wmoveRf1	0.05%	0.02%	0.02%	0.01%	0.00%	0.02%	0.08%	0.25%

Table 29. Table 29 : Move evaluation for puzzle ID DsaGi : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+5.76$	—	$4.5\%$	$23.1\%$	$72.4\%$	—
\wmoveRxg4+ $\star$	$+6.10$	$+34$	$97.9\%$	$1.9\%$	$0.2\%$	$+93.4\%$
\wmoveRf1+	$-0.05$	$-581$	$0.8\%$	$25.6\%$	$73.6\%$	$-3.7\%$
\wmoveRh1	$-0.15$	$-591$	$0.9\%$	$35.8\%$	$63.3\%$	$-3.7\%$
\wmoveKg7	$-2.46$	$-822$	$0.6\%$	$8.9\%$	$90.5\%$	$-4.0\%$
\wmoveRc1	$-2.75$	$-851$	$0.3\%$	$3.7\%$	$96.0\%$	$-4.2\%$
\wmoveRd1	$-2.77$	$-853$	$0.1\%$	$0.9\%$	$99.0\%$	$-4.4\%$
\wmoveRa1	$-2.81$	$-857$	$0.2\%$	$1.9\%$	$97.9\%$	$-4.3\%$
\wmoveRb1	$-2.86$	$-862$	$0.2\%$	$1.3\%$	$98.5\%$	$-4.3\%$
\wmoveBc3	$-3.28$	$-904$	$0.2\%$	$2.4\%$	$97.3\%$	$-4.3\%$
\wmoveBb2	$-3.48$	$-924$	$0.3\%$	$4.1\%$	$95.6\%$	$-4.2\%$
\wmoveBg7	$-3.50$	$-926$	$0.2\%$	$1.1\%$	$98.7\%$	$-4.4\%$
\wmoveBh8	$-3.56$	$-932$	$0.1\%$	$0.9\%$	$98.9\%$	$-4.4\%$
\wmoveKf8	$-3.63$	$-939$	$0.1\%$	$0.7\%$	$99.1\%$	$-4.4\%$
\wmoveRg2	$-3.64$	$-940$	$0.3\%$	$5.0\%$	$94.7\%$	$-4.2\%$
\wmoveBa1	$-3.75$	$-951$	$0.2\%$	$1.1\%$	$98.8\%$	$-4.4\%$
\wmovea4	$-3.75$	$-951$	$0.5\%$	$13.1\%$	$86.4\%$	$-4.1\%$
\wmovec4	$-3.88$	$-964$	$0.2\%$	$1.4\%$	$98.4\%$	$-4.4\%$
\wmoveBe5+	$-7.19$	$-1295$	$0.0\%$	$0.0\%$	$99.9\%$	$-4.5\%$
\wmoveRg3	$-7.42$	$-1318$	$0.0\%$	$0.1\%$	$99.9\%$	$-4.5\%$
\wmoveRe1	$-7.42$	$-1318$	$0.0\%$	$0.1\%$	$99.9\%$	$-4.5\%$
\wmoveBf6	$-7.49$	$-1325$	$0.0\%$	$0.1\%$	$99.8\%$	$-4.5\%$
\wmoveBe3+	$-7.88$	$-1364$	$0.0\%$	$0.1\%$	$99.9\%$	$-4.5\%$
\wmoveBf2	$-9.17$	$-1493$	$0.0\%$	$0.1\%$	$99.9\%$	$-4.5\%$

Table 30. Table 30 : Move probabilities by layer for puzzle ID DsaGi (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmovea4	15.39%	88.66%	14.83%	15.77%	10.39%	0.24%	1.31%	0.93%
\wmoveRxg4+	0.38%	0.66%	11.54%	22.23%	29.10%	71.66%	80.50%	75.47%
\wmoveRf1+	3.45%	0.12%	1.81%	7.09%	4.04%	0.47%	0.29%	0.60%
\wmoveKg7	0.12%	0.31%	25.84%	10.54%	5.47%	1.05%	2.15%	1.43%
\wmoveRh1	5.32%	0.13%	2.62%	2.47%	3.66%	0.35%	0.57%	1.02%
\wmoveBh8	0.13%	0.06%	0.54%	0.51%	18.57%	1.73%	8.11%	17.35%
\wmoveRg3	7.11%	0.10%	4.71%	10.33%	14.05%	6.37%	0.68%	0.31%
\wmoveRg2	13.65%	0.19%	2.17%	3.87%	3.22%	4.06%	0.38%	0.28%
\wmoveBf2	12.36%	0.08%	1.08%	4.40%	0.77%	1.30%	0.27%	0.21%
\wmovec4	1.25%	8.55%	10.83%	2.58%	0.51%	0.05%	0.51%	0.43%
\wmoveBe3+	9.81%	0.05%	1.11%	2.74%	0.73%	0.76%	0.12%	0.05%
\wmoveBc3	9.74%	0.07%	1.04%	0.38%	0.49%	0.37%	1.61%	0.20%
\wmoveRa1	3.41%	0.07%	5.73%	2.58%	0.84%	0.75%	0.18%	0.11%
\wmoveRb1	4.89%	0.14%	2.50%	1.50%	1.51%	2.38%	0.56%	0.18%
\wmoveBe5+	0.42%	0.05%	2.37%	4.46%	0.68%	1.05%	0.15%	0.06%
\wmoveBg7	0.45%	0.29%	3.97%	0.22%	0.58%	0.22%	0.98%	0.37%
\wmoveRe1	3.29%	0.08%	0.60%	1.26%	1.12%	1.86%	0.27%	0.22%
\wmoveRd1	3.17%	0.11%	0.97%	1.12%	0.92%	1.85%	0.31%	0.16%
\wmoveRc1	1.29%	0.08%	2.23%	1.43%	0.90%	1.35%	0.28%	0.19%
\wmoveBa1	2.02%	0.04%	1.18%	0.96%	1.16%	0.74%	0.29%	0.15%
\wmoveBb2	1.62%	0.06%	0.74%	1.25%	0.42%	0.65%	0.09%	0.03%
\wmoveKf8	0.00%	0.07%	1.25%	1.61%	0.70%	0.48%	0.28%	0.19%
\wmoveBf6	0.73%	0.04%	0.34%	0.69%	0.16%	0.24%	0.11%	0.03%

Table 31. Table 31 : Move probabilities by layer for puzzle ID DsaGi (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmovea4	0.96%	0.48%	0.04%	0.20%	0.86%	0.64%	1.17%	0.72%
\wmoveRxg4+	72.17%	73.29%	83.14%	88.00%	78.25%	86.89%	61.03%	12.14%
\wmoveRf1+	1.23%	2.08%	0.56%	2.04%	4.41%	3.45%	16.89%	40.43%
\wmoveKg7	3.84%	3.85%	2.77%	2.18%	3.21%	3.16%	5.34%	8.01%
\wmoveRh1	0.84%	1.16%	0.30%	0.87%	1.89%	1.41%	4.99%	23.00%
\wmoveBh8	17.62%	14.62%	9.44%	3.12%	1.55%	0.36%	0.52%	0.50%
\wmoveRg3	0.40%	0.47%	0.85%	0.85%	1.88%	0.49%	0.26%	0.19%
\wmoveRg2	0.32%	0.37%	0.27%	0.35%	0.51%	0.20%	0.52%	5.38%
\wmoveBf2	0.24%	0.53%	0.78%	1.01%	4.98%	1.25%	0.60%	0.23%
\wmovec4	0.35%	0.06%	0.02%	0.04%	0.16%	0.20%	0.59%	0.31%
\wmoveBe3+	0.05%	0.06%	0.04%	0.03%	0.02%	0.00%	0.15%	0.27%
\wmoveBc3	0.05%	0.13%	0.05%	0.02%	0.02%	0.03%	0.12%	1.32%
\wmoveRa1	0.08%	0.09%	0.05%	0.04%	0.08%	0.03%	0.12%	1.05%
\wmoveRb1	0.24%	0.29%	0.07%	0.24%	0.43%	0.25%	2.95%	0.36%
\wmoveBe5+	0.06%	0.10%	0.05%	0.03%	0.04%	0.01%	0.09%	0.24%
\wmoveBg7	0.52%	1.22%	0.67%	0.26%	0.24%	0.28%	1.36%	0.50%
\wmoveRe1	0.21%	0.27%	0.39%	0.22%	0.56%	0.61%	0.17%	0.20%
\wmoveRd1	0.16%	0.19%	0.08%	0.11%	0.16%	0.12%	2.18%	0.32%
\wmoveRc1	0.21%	0.20%	0.13%	0.10%	0.23%	0.15%	0.29%	2.40%
\wmoveBa1	0.08%	0.20%	0.13%	0.05%	0.14%	0.13%	0.13%	0.28%
\wmoveBb2	0.02%	0.03%	0.02%	0.02%	0.03%	0.04%	0.13%	1.62%
\wmoveKf8	0.31%	0.28%	0.15%	0.22%	0.35%	0.30%	0.27%	0.27%
\wmoveBf6	0.04%	0.04%	0.02%	0.01%	0.01%	0.02%	0.14%	0.27%

Table 32. Table 32 : Move evaluation for puzzle ID 00aDl : Stockfish evaluation at depth 20 and model WDL prediction for resulting positions

Move	Stockfish	$\Delta$ (cp)	Win	Draw	Loss	$\Delta$ Win
Current position	$+\infty$	—	$8.3\%$	$10.6\%$	$81.1\%$	—
\wmoveQxb1+ $\star$	$+\infty$	—	$99.3\%$	$0.5\%$	$0.2\%$	$+91.0\%$
\wmoveb5	$-4.12$	$-\infty$	$0.2\%$	$0.4\%$	$99.4\%$	$-8.1\%$
\wmoveR2a4	$-4.57$	$-\infty$	$0.2\%$	$0.3\%$	$99.5\%$	$-8.1\%$
\wmoveR2a7	$-4.72$	$-\infty$	$0.1\%$	$0.1\%$	$99.8\%$	$-8.2\%$
\wmoveNa5	$-4.74$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveR2a6	$-4.77$	$-\infty$	$0.1\%$	$0.1\%$	$99.8\%$	$-8.2\%$
\wmovec5	$-4.78$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveNd6	$-5.07$	$-\infty$	$0.1\%$	$0.2\%$	$99.7\%$	$-8.2\%$
\wmoveKg7	$-5.40$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveb6	$-5.42$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmovee5	$-5.45$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveKh8	$-5.49$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmovef5	$-5.49$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveg5	$-5.53$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveR2a5	$-5.55$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveR8a4	$-5.56$	$-\infty$	$0.1\%$	$0.2\%$	$99.8\%$	$-8.2\%$
\wmoveR8a6	$-5.60$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveR8a5	$-5.60$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveRc8	$-5.63$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveRd8	$-5.75$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveR8a7	$-5.75$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveRb8	$-5.76$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveKf8	$-5.82$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveKh7	$-5.87$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveNe5	$-5.96$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-8.3\%$
\wmoveRf8	$-5.98$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-8.3\%$
\wmoveRe8	$-6.17$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveNe3	$-6.29$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-8.3\%$
\wmoveNb6	$-6.78$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-8.3\%$
\wmoveNxb2	$-7.09$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$
\wmoveQxb2+	$-7.11$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-8.3\%$
\wmoveRxb2	$-7.23$	$-\infty$	$0.0\%$	$0.1\%$	$99.8\%$	$-8.3\%$
\wmoveNa3	$-7.65$	$-\infty$	$0.0\%$	$0.0\%$	$99.9\%$	$-8.3\%$
\wmoveNd2	$-7.77$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-8.3\%$
\wmoveR2a3	$-8.16$	$-\infty$	$0.0\%$	$0.0\%$	$100.0\%$	$-8.3\%$
\wmoveR8a3	$-8.34$	$-\infty$	$0.0\%$	$0.1\%$	$99.9\%$	$-8.3\%$

Table 33. Table 33 : Move probabilities by layer for puzzle ID 00aDl (Part 1: Input to Layer 6)

Move	Input	Layer 0	Layer 1	Layer 2	Layer 3	Layer 4	Layer 5	Layer 6
\wmoveQxb1+	1.80%	62.90%	47.54%	29.77%	23.48%	18.87%	27.94%	27.42%
\wmoveNd2	56.64%	0.41%	0.72%	0.18%	2.89%	0.03%	1.47%	0.89%
\wmoveb5	0.43%	0.31%	0.18%	0.03%	0.37%	0.13%	2.56%	12.77%
\wmoveNxb2	1.28%	0.66%	0.69%	24.30%	29.59%	35.56%	31.40%	28.96%
\wmoveNe3	26.36%	0.10%	0.37%	0.02%	0.04%	0.06%	0.02%	0.02%
\wmoveQxb2+	8.75%	11.39%	23.80%	24.03%	20.94%	22.84%	19.88%	13.99%
\wmoveRxb2	0.35%	17.28%	18.29%	20.38%	20.92%	20.66%	13.81%	13.55%
\wmoveNd6	0.09%	0.11%	0.09%	0.05%	0.17%	0.08%	0.57%	0.33%
\wmoveNa5	0.58%	0.59%	0.87%	0.03%	0.27%	0.10%	1.49%	0.23%
\wmoveb6	0.06%	0.06%	0.12%	0.02%	0.01%	0.05%	0.14%	0.62%
\wmoveNa3	0.14%	0.35%	1.00%	0.03%	0.16%	0.05%	0.05%	0.18%
\wmoveR2a4	0.03%	0.08%	0.13%	0.15%	0.09%	0.03%	0.02%	0.06%
\wmoveR8a4	0.07%	0.13%	0.73%	0.04%	0.01%	0.02%	0.01%	0.03%
\wmoveKf8	0.05%	2.16%	0.25%	0.03%	0.09%	0.08%	0.02%	0.04%
\wmoveR8a3	0.05%	0.14%	1.61%	0.15%	0.04%	0.11%	0.03%	0.06%
\wmoveR8a5	0.05%	0.20%	0.90%	0.05%	0.03%	0.07%	0.04%	0.09%
\wmoveR2a6	0.12%	0.12%	0.15%	0.03%	0.05%	0.12%	0.02%	0.02%
\wmovef5	0.80%	0.05%	0.07%	0.01%	0.02%	0.03%	0.03%	0.06%
\wmoveKg7	0.04%	0.24%	0.05%	0.25%	0.15%	0.07%	0.03%	0.04%
\wmoveR2a3	0.01%	0.07%	0.12%	0.03%	0.06%	0.08%	0.03%	0.08%
\wmoveR2a7	0.09%	0.09%	0.12%	0.04%	0.04%	0.05%	0.01%	0.02%
\wmovec5	0.39%	0.26%	0.14%	0.07%	0.03%	0.02%	0.02%	0.05%
\wmoveR2a5	0.09%	0.08%	0.07%	0.03%	0.07%	0.07%	0.05%	0.05%
\wmoveKh8	0.14%	0.23%	0.19%	0.04%	0.08%	0.09%	0.03%	0.04%
\wmoveR8a6	0.03%	0.13%	0.35%	0.04%	0.04%	0.17%	0.03%	0.06%
\wmovee5	0.21%	0.04%	0.15%	0.00%	0.04%	0.03%	0.04%	0.06%
\wmoveRf8	0.08%	0.52%	0.13%	0.02%	0.01%	0.02%	0.01%	0.02%
\wmoveKh7	0.08%	0.46%	0.05%	0.05%	0.13%	0.10%	0.03%	0.03%
\wmoveg5	0.49%	0.41%	0.29%	0.01%	0.04%	0.01%	0.03%	0.05%
\wmoveRb8	0.05%	0.05%	0.10%	0.02%	0.02%	0.06%	0.01%	0.02%
\wmoveR8a7	0.04%	0.09%	0.17%	0.02%	0.02%	0.05%	0.01%	0.02%
\wmoveNb6	0.31%	0.09%	0.14%	0.02%	0.02%	0.10%	0.09%	0.06%
\wmoveRd8	0.06%	0.05%	0.10%	0.02%	0.02%	0.04%	0.01%	0.02%
\wmoveRe8	0.06%	0.05%	0.09%	0.02%	0.01%	0.03%	0.01%	0.01%
\wmoveRc8	0.05%	0.06%	0.13%	0.02%	0.02%	0.06%	0.01%	0.02%
\wmoveNe5	0.12%	0.04%	0.10%	0.01%	0.01%	0.05%	0.03%	0.04%

Table 34. Table 34 : Move probabilities by layer for puzzle ID 00aDl (Part 2: Layer 7 to Final)

Move	Layer 7	Layer 8	Layer 9	Layer 10	Layer 11	Layer 12	Layer 13	Final
\wmoveQxb1+	25.28%	27.69%	24.79%	33.23%	40.50%	41.78%	56.96%	35.99%
\wmoveNd2	0.68%	0.26%	7.69%	1.87%	0.16%	0.17%	0.21%	0.32%
\wmoveb5	16.94%	26.56%	17.03%	25.45%	35.78%	33.58%	8.85%	36.12%
\wmoveNxb2	31.09%	17.79%	12.84%	12.49%	4.64%	0.73%	0.70%	0.35%
\wmoveNe3	0.02%	0.01%	0.04%	0.03%	0.04%	0.04%	0.20%	0.32%
\wmoveQxb2+	10.75%	11.81%	11.53%	8.22%	3.26%	0.36%	0.50%	0.29%
\wmoveRxb2	9.97%	10.30%	8.75%	6.71%	2.18%	0.32%	0.49%	0.44%
\wmoveNd6	1.16%	1.37%	5.72%	2.96%	5.18%	10.97%	9.70%	5.41%
\wmoveNa5	0.95%	0.83%	4.04%	1.76%	0.93%	2.08%	5.94%	1.85%
\wmoveb6	1.28%	1.19%	1.84%	1.99%	2.01%	0.87%	2.58%	0.62%
\wmoveNa3	0.46%	0.31%	2.31%	0.77%	0.49%	0.84%	1.79%	0.94%
\wmoveR2a4	0.16%	0.24%	0.46%	0.35%	0.63%	1.68%	2.24%	2.28%
\wmoveR8a4	0.07%	0.19%	0.27%	0.29%	0.29%	0.92%	1.06%	2.18%
\wmoveKf8	0.03%	0.05%	0.11%	0.22%	0.25%	0.21%	0.16%	0.40%
\wmoveR8a3	0.11%	0.20%	0.47%	0.32%	0.19%	0.60%	0.74%	0.64%
\wmoveR8a5	0.08%	0.12%	0.11%	0.13%	0.26%	0.22%	0.45%	1.35%
\wmoveR2a6	0.02%	0.02%	0.05%	0.04%	0.08%	0.22%	0.65%	0.93%
\wmovef5	0.14%	0.17%	0.06%	0.25%	0.50%	0.21%	0.49%	0.54%
\wmoveKg7	0.04%	0.03%	0.05%	0.08%	0.02%	0.04%	0.20%	0.77%
\wmoveR2a3	0.10%	0.16%	0.42%	0.19%	0.21%	0.72%	0.64%	0.65%
\wmoveR2a7	0.02%	0.02%	0.05%	0.06%	0.12%	0.27%	0.47%	0.71%
\wmovec5	0.04%	0.07%	0.10%	0.25%	0.15%	0.34%	0.68%	0.70%
\wmoveR2a5	0.06%	0.08%	0.16%	0.11%	0.32%	0.51%	0.67%	0.52%
\wmoveKh8	0.06%	0.06%	0.24%	0.66%	0.35%	0.17%	0.29%	0.41%
\wmoveR8a6	0.03%	0.03%	0.05%	0.07%	0.06%	0.18%	0.27%	0.61%
\wmovee5	0.11%	0.10%	0.14%	0.09%	0.07%	0.08%	0.41%	0.54%
\wmoveRf8	0.02%	0.01%	0.03%	0.07%	0.04%	0.08%	0.18%	0.40%
\wmoveKh7	0.04%	0.04%	0.16%	0.35%	0.35%	0.12%	0.22%	0.51%
\wmoveg5	0.10%	0.10%	0.10%	0.32%	0.40%	0.45%	0.41%	0.41%
\wmoveRb8	0.02%	0.01%	0.03%	0.04%	0.03%	0.08%	0.16%	0.47%
\wmoveR8a7	0.02%	0.02%	0.04%	0.07%	0.07%	0.28%	0.26%	0.47%
\wmoveNb6	0.07%	0.07%	0.14%	0.13%	0.18%	0.27%	0.43%	0.32%
\wmoveRd8	0.02%	0.01%	0.03%	0.07%	0.03%	0.09%	0.26%	0.42%
\wmoveRe8	0.02%	0.01%	0.03%	0.11%	0.04%	0.12%	0.23%	0.39%
\wmoveRc8	0.02%	0.01%	0.04%	0.13%	0.04%	0.15%	0.27%	0.39%
\wmoveNe5	0.04%	0.06%	0.07%	0.11%	0.15%	0.27%	0.23%	0.34%

Table 35. Table 35 : Summary of chess concepts from Stockfish 8’s evaluation function taken from (McGrath et al. , 2022 ) . Concepts are enumerated as <concept_name> <side> <game_phase> , where side is [mine|opponent|t] for current player, opponent, or total (difference), and game phase is [mg|eg|ph] for midgame, endgame, or phase-interpolated value.

Concept	Description
material t [mg|eg|ph]	Material score, where each piece on the board has a predefined value that changes depending on the phase of the game.
imbalance t [mg|eg|ph]	Imbalance score that compares the piece count of each piece type for both colours. E.g., it awards having a pair of bishops vs a bishop and a knight.
pawns t [mg|eg|ph]	Evaluation of the pawn structure. E.g., the evaluation considers isolated double, connected, backward, blocked, weak, etc. pawns.
knights [mine|opponent|t] [mg|eg|ph]	Evaluation of knights. E.g., extra points are given to knights that occupy outposts protected by pawns.
bishops [mine|opponent|t] [mg|eg|ph]	Evaluation of bishops. E.g., bishops that occupy the same color squares as pawns are penalised.
rooks [mine|opponent|t] [mg|eg|ph]	Evaluation of rooks. E.g., rooks that occupy open or semi-open files have higher valuation.
queens [mine|opponent|t] [mg|eg|ph]	Evaluation of queens. E.g., queens that have relative pin or discovered attack against them are penalized.
mobility [mine|opponent|t] [mg|eg|ph]	Evaluation of piece mobility score. It depends on the number of squares attacked by the pieces.
king safety [mine|opponent|t] [mg|eg|ph]	A complex concept related to king safety. It depends on the number and type of pieces that attack squares around the king, shelter strength, number of pawns around the king, penalties for being on pawnless flank, etc.
threats [mine|opponent|t] [mg|eg|ph]	Evaluation of threats to pieces, such as whether a pawn can safely advance and attack an opponent’s higher value piece, hanging pieces, possible xray attacks by rooks, etc.
passed pawns [mine|opponent|t] [mg|eg|ph]	Evaluates bonuses for passed pawns. The closer a pawn is to the promotion rank, the higher is the bonus.
space [mine|opponent|t] [mg|eg|ph]	Evaluation of the space. It depends on the number of safe squares available for minor pieces on the central four files on ranks 2 to 4.
total t [mg|eg|ph]	The total evaluation of a given position. It encapsulates all the above concepts.

Equations

$$\Delta c_{\ell }=\sum _{m\in \text{legal}(s)}{\pi }_{\ell }(m)\cdot \Delta c_m={\mathbb{E}}_{{\pi }_{\ell }}[\Delta c_m]$$

$${\mathbf{h}}_{\ell }^{\prime }={\mathbf{h}}_{\ell }\pm \alpha \cdot \mathbb{E}[{\nabla }_{{\mathbf{h}}_{\ell }}\Delta c_L]\cdot \frac{\mathbb{E}[\Vert {\mathbf{h}}_{\ell }\Vert ]}{\Vert \mathbb{E}[{\nabla }_{{\mathbf{h}}_{\ell }}\Delta c_L]\Vert }$$

$${\mathbf{h}}_{\ell }^{\prime }$$

$${\mathbf{h}}_{\ell }$$

$${\mathbf{h}}_{\ell }^{\prime }$$

$${\mathbf{h}}_{\ell }$$

$${\mathbf{h}}_{\ell }^{\prime }$$

$${\mathbf{h}}_{\ell }$$

$${\mathbf{h}}_{L,t}$$

$${\mathbf{i}}_{L,t}$$

$${\mathbf{z}}_{L,t}^{\mathrm{MHA}}$$

$${\mathbf{z}}_{L,t}^{\mathrm{FFN}}$$

$${\mathbf{b}}_{L,t}$$

$$+\sum _{\ell =1}^L{\alpha }^{2L-2\ell +1}\cdot \frac{{⨀}_{{\ell }^{\prime }=\ell }^L{\gamma }_{{\ell }^{\prime }}^{\mathrm{MHA}}\odot {\gamma }_{{\ell }^{\prime }}^{\mathrm{FFN}}}{{\prod }_{{\ell }^{\prime }=\ell }^L{\sigma }_{{\ell }^{\prime },t}^{\mathrm{MHA}}{\sigma }_{{\ell }^{\prime },t}^{\mathrm{FFN}}}\odot b_{\ell }^{\mathrm{MHA},O}$$

$$+\sum _{\ell =1}^L{\alpha }^{2L-2\ell }\cdot \frac{{\gamma }_{\ell }^{\mathrm{FFN}}{⨀}_{{\ell }^{\prime }=\ell +1}^L{\gamma }_{{\ell }^{\prime }}^{\mathrm{MHA}}\odot {\gamma }_{{\ell }^{\prime }}^{\mathrm{FFN}}}{{\sigma }_{\ell ,t}^{\mathrm{FFN}}{\prod }_{{\ell }^{\prime }=\ell +1}^L{\sigma }_{{\ell }^{\prime },t}^{\mathrm{MHA}}{\sigma }_{{\ell }^{\prime },t}^{\mathrm{FFN}}}\odot b_{\ell }^{\mathrm{FFN},O}$$

$$+\sum _{\ell =1}^L{\alpha }^{2L-2\ell +1}\cdot \frac{{\gamma }_{\ell }^{\mathrm{FFN}}{⨀}_{{\ell }^{\prime }=\ell +1}^L{\gamma }_{{\ell }^{\prime }}^{\mathrm{MHA}}\odot {\gamma }_{{\ell }^{\prime }}^{\mathrm{FFN}}}{{\sigma }_{\ell ,t}^{\mathrm{FFN}}{\prod }_{{\ell }^{\prime }=\ell +1}^L{\sigma }_{{\ell }^{\prime },t}^{\mathrm{MHA}}{\sigma }_{{\ell }^{\prime },t}^{\mathrm{FFN}}}\odot {\beta }_{\ell }^{\mathrm{MHA}}$$

$$+\sum _{\ell =1}^L{\alpha }^{2L-2\ell }\cdot \frac{{⨀}_{{\ell }^{\prime }=\ell +1}^L{\gamma }_{{\ell }^{\prime }}^{\mathrm{MHA}}\odot {\gamma }_{{\ell }^{\prime }}^{\mathrm{FFN}}}{{\prod }_{{\ell }^{\prime }=\ell +1}^L{\sigma }_{{\ell }^{\prime },t}^{\mathrm{MHA}}{\sigma }_{{\ell }^{\prime },t}^{\mathrm{FFN}}}{\beta }_{\ell }^{\mathrm{FFN}}$$

$${\mathbf{m}}_{L,t}$$

$$+\sum _{\ell =1}^L{\alpha }^{2L-2\ell }\cdot \frac{{\gamma }_{\ell }^{\mathrm{FFN}}{⨀}_{{\ell }^{\prime }=\ell +1}^L{\gamma }_{{\ell }^{\prime }}^{\mathrm{MHA}}\odot {\gamma }_{{\ell }^{\prime }}^{\mathrm{FFN}}}{{\sigma }_{\ell ,t}^{\mathrm{FFN}}{\prod }_{{\ell }^{\prime }=\ell +1}^L{\sigma }_{{\ell }^{\prime },t}^{\mathrm{MHA}}{\sigma }_{{\ell }^{\prime },t}^{\mathrm{FFN}}}\odot {\mu }_{\ell ,t}^{\mathrm{FFN}}1$$

$$\Delta c_m=c(s^{\prime })-c(s)$$

$$\Delta c_{\ell }=\sum _{m\in \text{legal moves}}{\pi }_{\ell }(m)\cdot \Delta c_m={\mathbb{E}}_{{\pi }_{\ell }}[\Delta c_m]$$

$${\hat{\mathbf{v}}}_{c,\ell }=\frac{\mathbb{E}[{\nabla }_{{\mathbf{h}}_{\ell }}\Delta c_L]}{\Vert \mathbb{E}[{\nabla }_{{\mathbf{h}}_{\ell }}\Delta c_L]\Vert }$$

$${\mathbf{h}}_{\ell }^{\prime }={\mathbf{h}}_{\ell }+\alpha \cdot \mathbb{E}[\Vert {\mathbf{h}}_{\ell }{\Vert }_2]\cdot {\hat{\mathbf{v}}}_{c,\ell }$$

Full text

1 Introduction

When mechanistic analysis identifies a learned algorithm inside a neural network, does that algorithm determine the network’s output? Several works have provided evidence for algorithmic computation within neural networks, from modular arithmetic (Nanda et al., 2023) and search in synthetic tasks (Brinkmann et al., 2024) to planning in Sokoban (Bush et al., 2025; Taufeeque et al., 2026) and look-ahead in a chess-playing neural network (Jenner et al., 2024). These works establish that learned algorithms exist and are causally active, focusing on how they function when they succeed. Less is known about how they can fail: can a model internally compute the correct solution through a learned algorithm only to output the wrong answer—and if so, what overrides it?

We investigate this by extending the logit lens (nostalgebraist, 2020)—which projects intermediate representations through a model’s output head (jastrzębski2018residualconnectionsencourageiterative) to reveal what the network would predict at each layer—to the move-selecting policy network of Leela Chess Zero (Leela Chess Zero team, ), the model in which Jenner et al. (2024) identified learned look-ahead and the strongest neural chess engine available today (TCEC, 2026). Leela’s encoder-only transformer architecture provides distinct interpretability advantages: it treats each chessboard square as a token and, unlike decoder-only language models where the logit lens is limited to decoding from a single token position, uses all tokens simultaneously for prediction, giving the logit lens direct access to every intermediate activation contributing to the model’s output.

Applying the logit lens reveals interpretable policies at every layer, with move preferences following a structured progression through recognizable chess heuristics of increasing complexity rather than converging smoothly on the final answer (Figure 2). But the layer-wise view also exposes a surprising phenomenon: in chess puzzles with a single objectively winning move, the correct solution often appears in intermediate layers only to be overridden in the final output, including immediate checkmates where even a single step of look-ahead should suffice (Figure 1).

Do these forgotten puzzles arise despite look-ahead functioning normally, or does the algorithm simply fail in these positions? By replicating the analyses of Jenner et al. (2024) on forgotten puzzles, we find that look-ahead operates as expected: future moves of the correct continuation are represented, causally important, and linearly decodable. If the algorithm is not at fault, what overrides it? Characterization of these positions reveals that the model overwhelmingly replaces tactical solutions, such as sacrifices, with quiet positional alternatives, suggesting that learned priors for safe play may be responsible. Such priors would reflect sensible generalizations from the training distribution that conflict with the correct move in tactical positions. To quantify this, we measure which chess concepts, such as king safety and threats, each layer prioritizes, and find that late layers increasingly favor safer concepts over more aggressive ones. Gradient-based steering against these preferences recovers $61.7\%$ of forgotten puzzles with minimal regressions, providing causal evidence that safety priors override algorithmically computed solutions.

Our contributions are: (1) We extend the logit lens to the Post-LN architecture of Leela’s policy network, yielding highly interpretable intermediate policies whose playing strength improves consistently with depth. (2) Using this, we identify forgotten puzzles—positions where intermediate layers find the correct solution but the full model does not—and verify that look-ahead functions normally in these positions. (3) We provide causal evidence that learned safety priors override these algorithmically computed solutions, recovering a majority of forgotten puzzles through gradient-based steering against safety concepts. Since both the algorithm and the priors emerge from the same likelihood-based training objective, these results raise the question of whether such training can produce models that reliably prefer algorithmically computed solutions over distributional heuristics, even in domains with verifiable ground truth. More broadly, our findings imply that models may possess capabilities their outputs do not reflect, and that mechanistic understanding can identify such failure modes and suggest principled interventions to address them. Our code will be released.

2 Background

2.1 Model architecture

We analyze the T82-768x15x24h transformer model from Leela Chess Zero, the strongest neural chess engine available today (Jenner et al., 2024). This model uses a Post-LN architecture similar to the original transformer (Vaswani et al., 2017) with DeepNorm scaling (Wang et al., 2024), featuring a $15$-layer transformer encoder with $768$-dimensional embeddings and $24$ attention heads per layer. The shared encoder feeds two output heads: a policy head that outputs a distribution over legal moves and a value head that predicts game outcomes. Chess positions are encoded as $8\times 8$ grids where each square corresponds to a token position.

Leela is trained using the AlphaZero (Silver et al., 2018) paradigm and normally functions in tandem with Monte Carlo tree search (MCTS) as a chess engine. However, we focus solely on the policy network, which already demonstrates strong play even without external search (lepned, 2024). Architectural details are provided in Appendix D.

2.2 Encoder-only Post-LN logit lens

The logit lens (nostalgebraist, 2020) projects intermediate activations after layer $\ell$ through the final layer normalization and unembedding matrix to obtain layer-wise predictions. This approach works seamlessly for Pre-LN transformers, where layer normalization precedes each sublayer and leaves the residual stream free of normalization operations. Post-LN architectures create a challenge by applying normalization after residual connections instead, transforming the residual stream at each layer and creating non-linear dependencies that prevent straightforward application of the logit lens.

Functionally, the Pre-LN logit lens is equivalent to applying zero ablation to all sublayer outputs beyond a given layer $\ell$ (Belrose et al., 2023). We extend this principle to Post-LN models by applying the same zero ablation while preserving the subsequent layer normalizations. For consistency we also ablate layer normalization biases for layers beyond $\ell$, with further justification provided in Appendix C.

Since Leela is an encoder that uses representations from all tokens simultaneously, we project the intermediate representations of all $64$ squares through the policy head to obtain policies at each layer.

2.3 Evaluation setup

To verify that our Post-LN logit lens produces meaningful intermediate predictions, we evaluate the playing strength and puzzle-solving ability of layer-wise policies following Ruoss et al. (2024). Additionally, we track distributional properties of intermediate policies across layers.

Playing strength assessment

We conduct a round-robin tournament between policies derived from representations at the input and all $15$ transformer layers. Each pairing plays $200$ Encyclopedia of Chess Openings (Matanović, 1978) positions, with one game per side using argmax move selection and five games per side using a temperature of $\tau=1.0$. Elo ratings are computed using BayesElo (Coulom, 2008). We include the Leela policy network from Ruoss et al. (2024) as a fixed anchor using argmax selection to match their evaluation and obtain absolute Elo scores. Additionally, we deploy the layer-wise policies as bots on Lichess (Lichess.org, 2025) across multiple time controls, using a temperature of $1.0$ for the first five full-moves to introduce opening diversity.

Puzzle-solving performance

We evaluate tactical understanding using the general dataset of $10{,}000$ Lichess puzzles from Ruoss et al. (2024), each constructed with a single clear winning line—the principal variation—while all other moves are significantly inferior. For each layer’s policy, we use argmax selection to predict moves and consider a puzzle solved if it reproduces all the player’s moves of the principal variation correctly.

Intermediate policy dynamics

To characterize how layer-wise policies evolve across depth, we track policy entropy and Kendall’s $\tau$ ranking correlation between intermediate and final move rankings at each layer. Additional metrics are provided in Appendix E.

2.4 Learned look-ahead

Jenner et al. (2024) demonstrated that Leela internally represents future moves of correct puzzle continuations and that these representations are causally important for its output. They analyzed positions where the model correctly solves tactical puzzles with a clear principal variation and presented three lines of evidence, which we briefly summarize below. We replicate their analyses on forgotten puzzles to determine whether this forgetting reflects a failure of the look-ahead mechanism itself or whether look-ahead operates normally but is overridden.

Dataset

To study look-ahead, they filter for positions where the model is especially likely to employ look-ahead rather than simple heuristics—tactical puzzles with a unique best line that are difficult to evaluate without considering future moves. They focus on puzzles the model solves correctly, yielding $\sim 22{,}500$ positions. We apply identical filtering criteria but select for forgotten puzzles, yielding $7{,}901$ positions (the forgotten dataset), constituting a substantial fraction relative to the solved cases and indicating that forgotten puzzles are far from a rare edge case. To ensure these represent genuine forgetting rather than absence of look-ahead, we also require the full model to assign $>50\%$ probability to all the player’s subsequent moves in the principal variation, given the preceding correct moves, confirming that the model demonstrably understands the winning continuation. Neither dataset filters for the presence of look-ahead mechanisms; any effects observed are emergent rather than selected for.

Activation patching

To measure the causal importance of individual square representations, Jenner et al. (2024) employ activation patching. For each position, a corrupted board state is generated by applying a small modification—such as adding or removing a single piece—selected to substantially affect the model’s output while minimally affecting a weaker model, thereby aiming to target mechanisms beyond simple heuristics. Representations from the corrupted forward pass are then patched into the clean forward pass at each layer one square at a time. The resulting change in log-odds of the correct move measures each square’s causal importance at that depth. They find that target squares of future moves in the principal variation exhibit outsized causal effects, indicating that look-ahead information is actively used. Although the model does not select the correct move in forgotten puzzles, it still assigns some probability to it. If this residual probability originates from look-ahead, causal interventions targeting look-ahead should reduce it, and log-odds reduction remains a meaningful metric in these positions.

Attention head ablation

Jenner et al. (2024) identified specific attention heads that transfer information between squares involved in future moves—in particular, heads that attend from the first move’s target square to the third move’s target square, effectively relaying look-ahead information backward along the principal variation. To test the causal role of this information flow, they zero-ablate the single attention entry corresponding to this query-key pair and find that it has an outsized effect on the log-odds of the correct move.

Probing

To test whether information about future moves is explicitly encoded in the model’s representations, Jenner et al. (2024) train bilinear probes to predict the third move in the principal variation in two steps: first predicting the target square of the third move from activations at the first move’s target square, then predicting the source square conditioned on this predicted target. Full methodological details for all look-ahead analyses are provided in Appendix L.

3 Results

3.1 Capability progression and policy dynamics

Tournament strength

Table 1 reports Elo ratings across layers. Playing strength increases with depth but suggests a three-phase progression rather than uniform improvement. Early layers show rapid gains through layer $5$, middle layers form a performance plateau through approximately layer $10$, and late layers demonstrate sharp strengthening beginning around layer $11$. This pattern holds consistently under both deterministic $(\tau=0)$ and stochastic $(\tau=1)$ move selection. Real-world Lichess deployment shows similar trends with clear late-layer strengthening (Appendix F).

Puzzle solving capabilities

Figure 3 shows clear improvement in puzzle-solving ability across network layers within each Elo range, and consistent decrease in performance across increasing difficulty levels for all layers. Dashed lines and shading mark phase boundaries with annotated slope ratios indicating relative improvement rates between phases. While the early-to-middle phase distinction is less pronounced than in tournaments, the final-phase acceleration is clearly visible, particularly for harder puzzles where improvement rates exceed $24.8$ times the middle phase.

Distributional analysis

Kendall’s $\tau$ correlation between intermediate and final move rankings stays near zero through middle layers and rises sharply only in the final layers (Appendix E). Policy entropy remains approximately constant across layers, indicating that intermediate predictions reflect meaningful preference distributions. Constant entropy with changing rankings implies that the model reorganizes move preferences across layers rather than gradually sharpening them.

Solution discovery and forgetting

Figure 4 tracks three metrics across layers: the current solve rate at each layer, the cumulative solve rate across all layers up to the current one, and the rate of puzzles solved for the first time at each layer. The gap between current and cumulative rates—highlighted by the hatched region—shows that solutions are frequently discovered and subsequently discarded. The discovery rate follows the three-phase pattern from playing strength, with reduced discoveries during the middle-layer plateau and renewed acceleration in late layers where many puzzles are solved for the first time. The gap between the final cumulative and current solve rates represents forgotten puzzles: solutions discovered at intermediate layers but lost in the output.

3.2 Forgotten puzzles and the limits of learned look-ahead

Having established that our logit lens produces meaningful intermediate policies, we turn to forgotten puzzles—and find a systematic pattern rather than random failures. Of $7{,}901$ positions in the forgotten dataset, $88.6\%$ involve sacrifices compared to $26.4\%$ of solved puzzles in the general dataset. The correct moves are overwhelmingly forcing ($77.4\%$ checks, $79.6\%$ captures), yet the model replaces them with quiet moves in $67.9\%$ of cases. The model’s own value head assigns higher win probability to the position resulting from the correct move in $95.8\%$ of forgotten puzzles. Notably, $48.5\%$ are mate puzzles, half of which are mate-in-two positions where even a single step of reliable look-ahead should guarantee playing the correct move.

Look-ahead analysis

Could these puzzles simply be solved by shallow heuristics or chance in intermediate layers, rather than through genuine look-ahead? The localization of the override suggests otherwise: $62.8\%$ of forgotten puzzles remain solved after the final attention sublayer, indicating that the override is concentrated in the final feed-forward layer. To test this directly, we apply the methodology described in Section 2.4 to the forgotten dataset. Figure 5 shows that all three lines of evidence yield qualitatively similar patterns to those reported by Jenner et al. (2024) on solved puzzles, albeit with somewhat smaller effect sizes for the causal interventions, as forgotten puzzles have lower baseline probability on the correct move and thus less log-odds to lose. Activation patching reveals that the 3rd move target square exhibits outsized causal effects in middle to late layers, diminishing after layer $11$ as look-ahead information is transferred to the 1st move target square. Zero-ablating the single attention entry in L12H12 that relays information from the 3rd to the 1st move target square reduces the log-odds of the correct move by more than $1.5$ in more than $10\%$ of forgotten puzzles, and in $63.4\%$ of cases this single entry has a larger effect than ablating all remaining $4095$ entries simultaneously. For the example in Figure 1, ablating this single scalar drops the correct move’s probability from $5.8\%$ to $1.0\%$, corresponding to a log-odds reduction of $1.79$. Bilinear probes decode the third move from intermediate representations with $80\%$ accuracy at layer $12$, compared to $92\%$ on solved puzzles, though probes on randomly initialized models also show lower accuracy on forgotten puzzles. Results for additional look-ahead heads identified by Cruz (2025) and the corresponding findings on solved puzzles from Jenner et al. (2024) are in Appendix L.

Taken together, the evidence demonstrates that the look-ahead mechanism operates normally in forgotten puzzles—future moves are represented, causally important, and linearly decodable. The model computes the correct tactical solution but does not use it. The presence of algorithmic structure does not guarantee algorithmic behavior. If the failure is not in look-ahead, what overrides it?

3.3 Safety priors override tactical solutions

The forgotten puzzle characterization revealed that the model systematically replaces forcing tactical solutions with safe, positional alternatives. To quantify whether this pattern reflects learned safety priors, we analyze how chess concepts prioritized by layer-wise policies evolve across network depth. Since the model’s internal representation of safety is unknown, we rely on handcrafted evaluation terms as human-interpretable proxies, focusing on concepts with a direct relationship to safe play.

Layer-wise concept preferences

To examine which chess concepts each layer prioritizes, we analyze how layer-wise policies weight moves by their conceptual effects. Following McGrath et al. (2022), who trained linear concept probes on AlphaZero’s intermediate representations, we use Stockfish 8’s (Stockfish Developers, 2016) handcrafted continuous evaluation terms as human-interpretable concepts. Rather than probing for concept representation, we measure concept preference directly from layer-wise move probabilities.

For each move $m$ from position $s$ to resulting position $s^{\prime}$, and each concept $c$, we compute $\Delta c_{m}=c(s^{\prime})-c(s)$, representing the change in $c$ caused by $m$. All evaluations are from the perspective of the current player, so positive values indicate an improvement. At each layer $\ell$, we compute:

[TABLE]

where $\pi_{\ell}(m)$ is the move probability assigned by layer $\ell$’s policy. This represents the expected concept change when sampling moves according to $\pi_{\ell}$. Full details and plots are provided in Appendix M.

The left panel of Figure 6 shows the evolution of four safety-related concepts across layers. Early and middle layers favor aggressive over defensive concepts, with higher $\Delta c_{\ell}$ for opponent king vulnerability and own threats, while later layers increasingly prioritize safe play, increasing own king safety and reducing opponent threats, with all four concepts converging to similar values.

Causal validation via concept steering

As a proof of concept for the hypothesis that safety priors contribute to the forgotten puzzle phenomenon, we perform activation steering (Dathathri et al., 2020; Subramani et al., 2022) using a first-order approximation in activation space with respect to the concept preference metric. For a given concept $c$ at layer $\ell$ with steering strength $\alpha$, we modify the residual stream activations $\mathbf{h}_{\ell}$ as:

[TABLE]

where $\Delta c_{L}$ is the concept preference of the full model and the gradient is normalized and scaled to the layer’s mean activation norm. Gradients are computed on positions sampled from general play, independent of any puzzle dataset. We sweep over steering strengths $\alpha$ and intervention layers $\ell$ for the four safety-related concepts individually, selecting the best-performing strength at each layer. We evaluate on the general puzzle dataset using solve rate as the primary metric, as it captures both recovery of forgotten puzzles and potential regressions.

The right panel of Figure 6 shows steering effects across layers. The best-performing single concept—steering the model to place less emphasis on mitigating opponent threats at layer $13$ with $\alpha=-0.5$—achieves a solve rate of $92.4\%$ compared to the $88.6\%$ baseline, solving $510$ additional puzzles with $127$ regressions. Notably, steering recovers $61.7\%$ of forgotten puzzles while also solving $33.5\%$ of puzzles the model never solved at any layer. The same configuration recovers $65\%$ of previously-unsolved puzzles on the held-out forgotten dataset. The optimal steering direction aligns with the concept preference analysis: for all four concepts, performance improves when placing less emphasis on safety, specifically reducing the weight on own king safety and opponent threats against us while increasing emphasis on threats we pose and opponent king vulnerability. For three of four concepts, steering yields improvements across nearly all layers, substantially exceeding the random baseline, with the strongest effect at layer $13$, just before the override.

The observed regressions indicate that not all puzzles benefit from reduced safety preferences. Nevertheless, puzzles are adversarial by construction—the winning move is frequently a sacrifice or forcing sequence that conflicts with conservative heuristics learned from the training distribution, explaining why reducing safety preferences yields a substantial net improvement. That a simple gradient-based intervention targeting safety-related concepts suffices to recover a majority of forgotten puzzles supports the hypothesis that learned safety priors are a primary mechanism underlying the forgotten puzzle phenomenon. While other concepts also improve performance above the random baseline, the most directly safety-related concepts yield the largest and most consistent effects; complete preference and steering analyses for all concepts are provided in Appendix N.

4 Related work

Chess models and their interpretability

The success of AlphaZero (Silver et al., 2018) and Leela Chess Zero (Leela Chess Zero team, ) has motivated a growing body of interpretability work on neural chess engines, including concept probing (McGrath et al., 2022), concept discovery (Schut et al., 2025), and evidence of learned look-ahead (Jenner et al., 2024; Cruz, 2025). Separately, Karvonen (2024) studied world models in language models trained on chess move sequences, and Ruoss et al. (2024) trained a transformer to play chess via next-token prediction without external search. These models differ in training paradigm but are architecturally similar, suggesting the tension between learned algorithms and distributional priors we identify may extend to them as well.

Intermediate decoding

nostalgebraist (2020) introduced the logit lens for decoder-only language models, with subsequent work extending it to improve prediction accuracy (Belrose et al., 2023; Yom Din et al., 2024), to encoder-decoder architectures (Langedijk et al., 2024), and to anticipating future tokens (Pal et al., 2023). The logit lens has been applied to Leela Zero, a superhuman Go model, though limited to divergence metrics (Du, 2023), and used to identify distinct computational phases in language models (Lad et al., 2025), a pattern we also observe in Leela.

Learned algorithms in neural networks

Beyond language models, where circuit discovery has become a major research direction, recent work has mechanistically identified learned algorithms in other domains including search in synthetic tasks (Brinkmann et al., 2024) and planning in Sokoban (Bush et al., 2025; Taufeeque et al., 2026). These works demonstrate that identified circuits are causally relevant for the model’s output. In contrast, Nikankin et al. (2025) find that neural networks can solve seemingly algorithmic tasks like arithmetic through ensembles of heuristics rather than generalizable algorithms. Our work sits between these findings: algorithmically computed solutions can be present and causally active yet still be overridden in the final output by competing heuristics.

5 Conclusion

We have shown that the presence of learned algorithmic structure in a neural network does not guarantee that the network’s output will reflect it. In Leela Chess Zero, look-ahead computes correct tactical solutions that are systematically overridden by safety priors in the final layers. Concept preference analysis reveals the shift toward conservative evaluation, and gradient-based steering against these priors recovers a majority of forgotten puzzles, providing causal evidence for this mechanism. More broadly, our results suggest that circuit discovery alone is insufficient for predicting model behavior: identified algorithms may not determine the output, and incorrect outputs may not indicate absent capabilities. Our analysis demonstrates how mechanistic methods can identify such failure modes and suggest principled interventions to address them.

Limitations

The main limitations of our work are: (1) While we identify safety priors as a mechanism behind forgotten puzzles, we do not provide a precise mechanistic description of how these priors are implemented. (2) We verify the forgotten puzzle phenomenon across multiple Leela networks (Appendix G), but our causal analyses are conducted on a single network. (3) Our concept analysis uses Stockfish’s evaluation terms as proxies and is restricted to one-step position evaluations; the model likely encodes features that do not map cleanly onto these concepts.

Impact

There has been growing confidence that identifying algorithmic circuits in neural networks yields understanding of model behavior. Our results introduce a caveat: identified algorithms may be overridden by competing priors, meaning that behavioral predictions based on circuit analysis alone may be incomplete. While our analysis focuses on a chess-playing model, the tension between learned algorithms and priors likely exists in any model trained on broad distributions. Indeed, related phenomena have been observed in language models, where internal representations encode correct answers that the model does not output (Orgad et al., 2025) and sycophancy priors override factual knowledge in late layers (Wang et al., 2026). Together, these findings connect to broader concerns about hidden capabilities in deployed models—capabilities present in internal computations but not reflected in behavior. Such capabilities may surface under distribution shift or adversarial inputs and raise questions about how reliably outputs reflect underlying competence. Whether similar overrides affect algorithmic solutions in domains such as mathematical reasoning remains an open question.

Author contributions

Elias conceived, implemented, and conducted the study, performed all experiments, and wrote the manuscript. Sebastian and Wojciech provided supervision and feedback throughout the project.

Acknowledgments and Disclosure of Funding

We gratefully acknowledge Jenner et al. (2024) and Ruoss et al. (2024) for the codebase and inference framework on which this work is built. Without their open-source release, this research would not have been possible.

This work was supported by the Federal Ministry of Research, Technology and Space (BMFTR) as grants [BIFOLD (01IS18025A, 01IS180371I), xJuRAG (16IS25015B)]; the European Union’s Horizon Europe research and innovation programme (EU Horizon Europe) as grant ACHILLES (101189689); and the German Research Foundation (DFG) as research unit DeSBi [KI-FOR 5363] (459422098).

Appendix A Limitations and future directions

Scope of analysis

Our causal analyses focus on a single chess model with a Post-LN architecture, though we verify that the forgotten puzzle phenomenon itself replicates across other Leela networks (Appendix G). The findings function as an existence proof: algorithmically computed solutions can be overridden by competing priors in a deployed neural network. However, this single-domain analysis cannot disentangle whether the override reflects domain-specific characteristics of chess, architecture-specific properties of Post-LN transformers, or more general principles of neural network computation. Comparison across architectures and domains, particularly Pre-LN language models, would help isolate these effects. No Pre-LN Leela models exist for direct architectural comparison; the development team has chosen Post-LN over Pre-LN in their training paradigm (djedev, 2025).

Generalization and applications

While our analysis focuses on chess transformers, our methodology could inform interpretability research in other domains. First, concept preference analysis offers an alternative to representation probing: rather than identifying whether concepts are encoded, it tracks which concepts each layer prioritizes in driving outputs. Applied to language models, this could trace behavioral tendencies such as truthfulness or specificity across layers. Second, our observation that safety priors may override algorithmic solutions suggests that in tasks with verifiable ground truth, suppressing learned distributional biases—such as final-layer tendencies toward common tokens—might improve performance on tasks requiring precise reasoning.

Appendix B Compute resources and licensing

Compute resources

All experiments were conducted on a single workstation with 2$\times$ NVIDIA RTX 5090 (24 GB) and an AMD Threadripper PRO 7965WX (24 cores). Most individual experiments complete within a few hours on a single GPU. The most compute-intensive components are the internal tournaments ($\sim$1–3 days due to sequential game play), concept delta computation via Stockfish ($\sim$16 hours parallelized across CPU cores), and the full steering sweep across all concepts ($\sim$2 GPU-days). All main paper experiments can be reproduced in approximately 2–3 days on comparable hardware.

Licenses and data availability

The Lichess puzzle database is available under a Creative Commons CC0 license (Lichess.org, 2025). The CCRL dataset is publicly available (Leela Chess Zero team, 2018). The puzzle dataset, Encyclopedia of Chess Openings positions, and tournament code from Ruoss et al. (2024) are released under the Apache 2.0 license, with the Encyclopedia of Chess Openings originally compiled by Matanović (1978). The solved puzzle dataset and inference framework from Jenner et al. (2024), Leela Chess Zero (Leela Chess Zero team, ), and Stockfish (Stockfish Developers, 2016) are all released under the GPLv3 license.

Appendix C Post-LN Logit Lens Extension

The logit lens technique projects intermediate layer representations through the model’s output head to examine what the network would predict at different depths. This approach relies on the assumption that representations across layers exist in a shared basis that allows meaningful projection to output space—an assumption enabled by residual connections that create additive pathways for information flow (jastrzębski2018residualconnectionsencourageiterative). In transformer architectures, architectural inductive biases further encourage information about specific tokens to remain localized to their corresponding positions throughout the network, creating a privileged basis that facilitates cross-layer interpretation (Jenner et al., 2024).

In Pre-LN transformers, this involves taking intermediate representations, applying the final layer normalization, and projecting through the unembedding matrix. This approach works because layer normalization precedes each sublayer in Pre-LN models, leaving the residual stream unchanged and maintaining representational consistency across layers.

Post-LN architectures complicate this process by applying layer normalization after each sublayer’s output is added to the residual stream. Unlike Pre-LN models where only a final normalization is needed, Post-LN models have sequential normalization operations that directly transform the residual stream at each layer. These intermediate normalizations create dependencies between layers that prevent simply taking an intermediate representation and applying only the final layer normalization and output projection, as the intermediate representation has not undergone the normalization transformations it would experience in a complete forward pass.

Our goal is to develop an extension that maps intermediate layer representations to the representational basis expected by the policy head, accounting for the normalization transformations unique to Post-LN architectures.

C.1 Pre- vs Post-LN architectures and DeepNorm

The key difference between Pre-LN and Post-LN architectures lies in when layer normalization is applied relative to the residual connections. This placement affects how representations evolve through the network and impacts the applicability of interpretability techniques.

Pre-LN

In Pre-LN transformers, layer normalization is applied before each sublayer (attention and feed-forward), with the computation following the pattern:

[TABLE]

where $\mathbf{h}_{\ell}$ denotes the hidden representation at layer $\ell$, $\mathrm{MHA}_{\ell}$ is the multi-head attention operation, and $\mathrm{FFN}_{\ell}$ is the feed-forward network at layer $\ell$. This design ensures that the residual stream itself is never directly transformed by normalization operations, allowing intermediate representations to be projected through the final layer normalization and output head in a straightforward manner.

Post-LN

Post-LN models apply layer normalization after adding sublayer outputs to the residual stream:

[TABLE]

Each layer normalization operation directly transforms the accumulated representation, creating a sequence of transformations that intermediate representations must undergo to reach the final output space.

DeepNorm

Leela employs DeepNorm (Wang et al., 2024) scaling to stabilize Post-LN training, modifying the computation to include residual scaling factors:

[TABLE]

where $\alpha=(2N)^{1/4}\approx 2.34$ for $N=15$ layers. This scaling, combined with specialized weight initialization, enables stable training while preserving Post-LN’s representational advantages.

C.2 Zero ablation methodology for Post-LN architectures

The standard logit lens can be understood as performing zero ablation—setting all sublayer outputs to zero for layers beyond $\ell$, then applying the final layer normalization and projection to output space. We extend this principle to Post-LN models by applying the same zero ablation of sublayer outputs while preserving the normalization operations and $\alpha$ that would transform these representations.

Specifically, for examining layer $k$, we set $\mathrm{MHA}_{\ell}(\cdot)=0$ and $\mathrm{FFN}_{\ell}(\cdot)=0$ for all $\ell>k$ during a forward pass, while preserving the $\alpha$ and layer normalization scaling parameters ($\gamma$). We set layer normalization biases ($\beta$) to zero for layers beyond $k$ to maintain better consistency with the standard logit lens paradigm. The rationale for these choices will be explained in the subsection following the decomposition analysis C.4.

This approach ensures that representations from layers 1 through $k$ undergo the same sequence of transformations they would experience in the complete network, while removing contributions from later layers. Normalization statistics ($\mu$ and $\sigma$) are recomputed during this modified forward pass to reflect the actual distribution of the truncated representations, rather than using statistics computed on the full model output. This follows the same principle as Pre-LN logit lens implementations, which directly apply the final layer normalization to the intermediate activations being analyzed.

C.3 Transformer encoder decomposition

To provide intuition for why our proposed logit lens extension makes principled choices for Post-LN architectures, we present a decomposition of the transformer encoder output and reinterpret the logit lens in terms of this decomposition. Following Mickus et al. (2022), the final representation of the encoder at layer $L$ and token position $t$ can be decomposed into:

[TABLE]

where $\mathbf{h}_{0,t}$ is the initial input embedding, $\tilde{\mathbf{z}}_{\ell,t}^{\mathrm{MHA}}$ and $\tilde{\mathbf{z}}_{\ell,t}^{\mathrm{FFN}}$ are the raw unbiased outputs from multi-head attention and feed-forward networks at layer $\ell$, and $\alpha=(2N)^{1/4}$ is the DeepNorm scaling factor. The bias terms include $b_{\ell}^{\mathrm{MHA},V}$ (value projection bias), $W_{\ell}^{\mathrm{MHA},O}$ (output projection matrix), $b_{\ell}^{\mathrm{MHA},O}$ (attention output bias), and $b_{\ell}^{\mathrm{FFN},O}$ (feed-forward output bias). Layer normalization parameters are $\gamma_{\ell}$ (learned scales), $\sigma_{\ell,t}$ (computed standard deviations), $\mu_{\ell,t}$ (computed means), and $\beta_{\ell}$ (learned biases), with superscripts indicating MHA or FFN sublayers. The notation $\bigodot$ denotes element-wise multiplication.

The decomposition separates the final representation into the transformed input embedding ($\mathbf{i}_{L,t}$), accumulated sublayer contributions (${\mathbf{z}}^{\mathrm{MHA}}_{L,t}$, ${\mathbf{z}}^{\mathrm{FFN}}_{L,t}$), bias terms ($\mathbf{b}_{L,t}$), and mean centering effects ($\mathbf{m}_{L,t}$). Each component is scaled by normalization parameters from subsequent layers.

C.4 Implications for the logit lens

The decomposition framework provides direct justification for our zero ablation approach. The Post-LN logit lens can be understood as truncating the summations in ${\mathbf{z}}^{\mathrm{MHA}}_{L,t}$, ${\mathbf{z}}^{\mathrm{FFN}}_{L,t}$, and $\mathbf{b}_{L,t}$ to include only layers $\ell\leq k$, while recomputing the normalization statistics ($\mu$ and $\sigma$) based on the modified representations.

Preservation of scaling parameters

The decomposition reveals why preserving the $\gamma$ and $\alpha$ scaling factors is essential: all terms in the truncated representation—including the input embedding $\mathbf{i}_{L,t}$ and retained sublayer contributions—are transformed by these parameters from subsequent layers. Removing these transformations would fundamentally alter how the truncated representation maps to the output space of the encoder, although under our zero-ablation configuration the $\alpha$ factor is absorbed by layer normalization and has no numerical effect.

Treatment of bias terms

The bias terms $\beta$ from layer normalization appear in $\mathbf{b}_{L,t}$ alongside attention and feed-forward biases, indicating they are structurally equivalent. For consistency with the zero ablation principle of removing sublayer contributions beyond layer $k$, we ablate all bias terms including layer normalization biases. Empirically, preserving versus ablating these biases produces qualitatively similar results; we provide code to examine all configurations. Our approach uses the policy head unchanged, which contains its own bias terms that may contribute to biased outputs. Moreover, the decomposition only captures bias terms that can be linearly separated from the representation—biases that interact with activation functions cannot be decomposed into constant additive components.

Representational alignment versus learned priors

The treatment of bias terms reflects broader questions about their computational role. Bias terms may serve as representational alignment mechanisms that bridge coordinate system differences between layers, or they may encode learned priors about the task domain. The Tuned Lens (Belrose et al., 2023) learns affine transformations with bias terms to achieve representational alignment between intermediate and final layers in Pre-LN architectures, motivated by biased outputs observed with the standard logit lens. In practice, bias terms likely serve both functions simultaneously, and disentangling them remains an open problem.

Appendix D Complete model architecture

We analyze the T82-768x15x24h transformer model from Jenner et al. (2024), using their inference framework for our experiments. This model has $15$ transformer layers, $768$-dimensional representations, $24$ attention heads, and approximately $109$ million parameters. It processes chess positions through three stages: input encoding transforms board states into token representations (with each of the $64$ squares treated as a discrete token), a $15$-layer transformer encoder processes these representations, and task-specific output heads generate move probabilities, position evaluations, and game length predictions. We use the original model that incorporates historical board information for all experiments, as this represents the model deployed in practice rather than the variant fine-tuned for interpretability research by Jenner et al. (2024). For the look-ahead analyses (activation patching, head ablation, and probing), we use the fine-tuned variant without position history, as corrupted board states do not correspond to any valid game history. While minor differences exist on individual positions, the fine-tuned version is generally trained to behave equivalently, and both exhibit the forgotten puzzle phenomenon.

D.1 Input encoding

Leela’s input encoding transforms chess positions through binary feature planes, chess-specific positional encodings, and learned projections into the model’s embedding space.

Board representation

The board state is encoded using $112$ binary feature planes of size $8\times 8$, with the first $12$ planes representing current piece positions for each piece type and color. Historical context is incorporated through $8$ previous board positions ($96$ planes), plus auxiliary planes encoding castling rights ($4$ planes), side to move ($1$ plane), fifty-move clock ($1$ plane), and two constant planes ([math]s and $1$s) that are architectural remnants from CNNs without functional meaning.

Positional encoding

Leela employs chess-specific positional encodings that capture movement relationships between squares in a $64\times 64$ matrix. Each square receives a $64$-dimensional vector where position $(i,j)$ is set to 1 if any piece could legally move from square $i$ to square $j$ in one move regardless of current board state, [math] otherwise, and $-1$ for diagonal entries $(i=j)$ to distinguish self-reference.

Input preparation

The $112$ feature planes are reshaped from $112\times 8\times 8$ to $64\times 112$, where each of the $64$ entries corresponds to a board square with the $112$-dimensional feature vector for that square concatenated with its corresponding $64$-dimensional positional encoding vector, forming a $64\times 176$ tensor. This combined representation undergoes a linear transformation with Mish activation, followed by elementwise scaling and shifting operations, producing the final $64\times 768$ input for the transformer layers. This enriched preprocessing was motivated by observations that early attention layers contributed minimally to performance (Monroe, 2024). While no information mixing between tokens occurs here, this involves substantially more processing than typical language models (which simply use embedding matrices plus positional encodings), potentially explaining why our logit lens mappings yield meaningful results even before the first transformer layer.

D.2 Transformer encoder

The transformer encoder consists of $15$ identical layers with a model dimensionality of $768$, processing $64$ tokens (one per board square) through multi-head attention with Smolgen enhancement, feed-forward networks, and Post-LN normalization with DeepNorm scaling. Unlike autoregressive models, all tokens can attend to each other bidirectionally.

Attention with Smolgen

Each layer uses $24$ attention heads with $32$-dimensional head size, applying scaled dot-product attention to the $64$ board square tokens. The Smolgen module enhances standard self-attention by enabling attention scores to depend not only on individual square contents but also on the global board position. It compresses all square representations into a global vector, processes this through MLPs, then generates supplementary attention logits of shape $24\times 64\times 64$ that are added to the standard query-key dot products before softmax normalization.

Feed-Forward

Each FFN layer expands from $768$ to $1024$ dimensions with squared ReLU activation. Unlike other domains that benefit from $4\times$ expansion ratios, chess models show little improvement from larger feed-forward networks (Monroe, 2024).

Post-LN with DeepNorm

The model employs a Post-LN architecture, the original design used in early transformers (Vaswani et al., 2017), applying layer normalization after each sublayer (attention and feed-forward) within the residual stream rather than before sublayers as in modern Pre-LN variants, following empirical comparisons that demonstrated superior performance (djedev, 2025). To mitigate vanishing gradients, the model uses DeepNorm scaling (Wang et al., 2024), which applies a constant upscaling factor to residual connections and uses specialized weight initialization. See Appendix C.1 for complete equations.

D.3 Output heads

The transformer encoder’s final $64\times 768$ representations are processed by three specialized heads: the policy head generates move probability distributions, the value head predicts win/draw/loss outcomes, and the moves-left head estimates remaining game length.

Policy head

The policy head first processes each square’s $768$-dimensional representation through a shared MLP with Mish activation. These processed representations are then transformed via two separate linear projections: one creating “source” representations for squares where moves originate, and another creating “target” representations for destination squares. The source and target representations are matrix-multiplied along the $768$-dimensional axis, producing a $64\times 64$ matrix where each entry contains a scalar logit representing the likelihood of a move from the corresponding source square to target square. Promotion moves require additional processing through two specialized branches that handle move selection and promotion-type preferences separately. The final output combines standard move logits with promotion logits, which are then filtered to extract only legal moves for the current position before applying softmax normalization.

Value head

The value head processes each square’s representation through an MLP with Mish activation, reducing them to $32$ dimensions per square. These $64$ representations are flattened into a single $2048$-dimensional vector to enable global position assessment. A second MLP layer compresses this to $128$ dimensions, followed by a linear projection to $3$ logits, which are then converted to win, draw, and loss outcome probabilities via softmax normalization. Unlike AlphaZero’s single scalar output, this three-way classification provides more nuanced position evaluation.

Moves-left head

The moves-left head predicts the number of moves remaining until game termination. Each square’s representation is processed through an MLP, then flattened into a global vector for position-level assessment. Two additional MLP layers progressively reduce the dimensionality to produce a final scalar output estimating remaining game length. The output layer uses Mish activation because ReLU would provide zero gradients when pre-activation values are negative during training, even though the target output is always positive.

Appendix E Intermediate policy dynamics

To provide a deeper, quantitative view of the model’s inference process, we computed several policy metrics across the network’s depth. The following analyses were conducted on a sample of $10{,}000$ positions from the CCRL dataset (Leela Chess Zero team, 2018). Figure 7 presents five metrics that characterize the policy’s evolution. We plot the median, the $25$th-$75$th percentile range, and the $5$th-$95$th percentile range and indicate the proposed phases as shaded background.

Jensen-Shannon divergence

The Jensen-Shannon Divergence shows a median that remains high until the final layers, with the $5$th percentile being much lower. This shows that a small subset of positions converge early to the final policy, while the vast majority have different move preferences than the final output throughout most layers. The phase structure is again apparent, with higher variability and erratic changes in early layers, a more stable plateau in the middle phase, and a sharp decline in the final phase. In this metric, the transition to the final phase appears to occur slightly later—around layer $12$ rather than $11$—though consistent with the overall three-phase progression described in the main text.

Entropy

The policy entropy shows a relatively stable median with modest variation across layers, indicating that overall certainty changes only slightly with depth. The $95$th and $5$th percentiles are very high and very low respectively, indicating that for any layer there are positions where the layer is either very certain or very uncertain about what moves are good. The phase structure is again apparent, with more erratic fluctuations in the early layers, a stable middle phase, and increased variability toward the end.

Probability of final top move

The median probability assigned to the final top move is near zero for the first layers, then grows approximately linearly with a steeper increase in the final layers. The outliers are very high (between $0.6$ and $0.8$), indicating that all layers correctly predict the top move for some positions with high probability, having already converged. The phase structure is again visible, with slightly more stable behavior in the middle layers and a pronounced acceleration in the final phase, which here again appears to begin around layer $12$.

FFN output norms

Following Lad et al. (2025), we examine the L$2$ norm of each transformer FFN sublayer’s output during a standard forward pass to probe for their proposed residual sharpening mechanism. While the relatively constant policy entropy suggests no sharpening at the distributional level, the FFN output norms exhibit an extreme last-layer increase, suggesting a feature consolidation process analogous to that observed in language models. This is consistent with the final feed-forward network playing a dominant role in reshaping the policy, as observed in the forgotten puzzle override.

Ranking correlation (Kendall’s $\tau$)

To analyze the stability of move preferences, we compute Kendall’s $\tau$ rank correlation between intermediate and final policies. We calculate this metric in two ways: first using all legal moves, and second, to mitigate noise from low-probability moves that are never seriously considered, only over moves that appear in the top five at any layer. Both approaches yield nearly identical results, with the correlation being negative in early layers and remaining low until around the $12$th layer, where it increases sharply. The $5$th and $95$th percentiles in the top-$5$ moves analysis show that there are positions where intermediate layer move rankings are either very similar or very dissimilar to the final ranking until late in the network. The phase structure is again apparent, with high variability in early layers, relatively stable correlations across the middle phase up to layer $11$, and a sharp rise beginning around layer $12$, consistent with the transition to the late phase.

Appendix F Complete tables for the tournament and Lichess play

F.1 Internal tournament

We evaluate playing strength through internal round-robin tournaments processed with BayesElo (Coulom, 2008) using the default confidence parameter of $0.5$. Each model pairing played 200 distinct openings from the Encyclopedia of Chess Openings (Matanović, 1978), with one game per side at deterministic temperature $\tau=0$ and five games per side at stochastic temperature $\tau=1.0$. Tables 2 and 3 report the resulting Elo ratings, computed from all match outcomes. Each table lists the model’s Elo estimate with asymmetric confidence bounds, total score percentage (wins plus half draws), average opponent rating, and draw rate. The Lc0 anchor is evaluated under argmax in both regimes following Ruoss et al. (2024); in the $\tau=1$ tournament it therefore wins every game against the stochastic field, so the absolute Elo scale is meaningless and only relative ordering and gaps within the tournament are interpretable.The results show consistent improvement in playing strength with network depth, with clear early- and late-layer acceleration in both deterministic and stochastic settings, while middle layers exhibit a prolonged performance plateau.

F.2 Lichess bot performance

To validate our internal tournament findings in a real-world setting, we deployed layer-wise policies as bots on Lichess (Lichess.org, 2025) using the bot API framework (Lichess Bot Devs, 2025). Due to rate-limiting constraints, we distributed bots across multiple cloud instances and used policy sampling for the first five moves to introduce opening diversity. The bots participated in Bullet (1+0, 2+1), Blitz (3+0, 3+2, 5+0, 5+3), and Rapid (10+0, 10+5, 15+10) time controls until ratings stabilized. Several experimental constraints shaped the evaluation: (1) bots played exclusively against other bots per Lichess policy, (2) the limited pool of weak bots meant repeated matchups for early layers, and (3) deterministic play after the opening often led to repetitive game. Despite these limitations, the results largely corroborate our internal tournament findings, showing progressive improvement across layers with the most significant transitions occurring at similar points in the network depth. Table 4 presents the playing strength and performance statistics for our layer-wise Lichess bots, with bot names linked to their respective Lichess profiles.

Appendix G Forgotten puzzles across Lc0 networks

To verify that the forgotten puzzle phenomenon is not specific to the T82-768x15x24h network analyzed in the main paper, we apply the same logit lens to four additional Lc0 checkpoints from the publicly available Lc0 network repository, spanning three width classes ($256$, $512$, $768$), two depths ($10$ and $15$ encoder blocks), and multiple training generations.

Figure 8 reports puzzle solve rates across layers for each network, evaluated on the general dataset consisting of $10{,}000$ puzzles. The forgotten puzzle band – the gap between solved by any layer $\leq L$ and solved at $L$ – is visible in every network, indicating that solution forgetting is a general property rather than a single-checkpoint artifact.

Appendix H Complete probabilities for the example puzzle from Figure 2

This puzzle (Figure 9) features a knight sacrifice leading to mate (PV: 1. \bmoveNg3+ 2. \wmovehxg3 \bmoveRh6#). The input layer exhibits piece-specific bias toward queen moves, with multiple captures dominating early probabilities. The winning move \wmoveNg3+ first becomes the top choice at layer $6$ ($26.7\%$) but is then briefly overtaken by \wmoveQg1+, which dominates layers $9$–$10$. This temporary preference for \wmoveQg1+ likely reflects a learned heuristic that queen checks are tactically important, even though this particular check ultimately loses the queen without compensation. After layer $10$, \wmoveNg3+ regains its position as the preferred move, with the exception of layer $12$ where a rook lift \wmoveRh6 ($51.7\%$) briefly becomes the top candidate, potentially pinning the h-pawn and threatening the king along the h-file. The winning move’s probability follows a non-monotonic trajectory, fluctuating throughout the layers before ultimately surging to $49.2\%$ at layer $13$ and $87.6\%$ in the final output. This contrasts to some of the other cases in Appendix I, where the correct move is either identified immediately after early layers or remains unconsidered until the very late layers. Full probabilities are in Tables 5 and 6.

Appendix I Correctly solved puzzle case studies

I.1 Correct case study 1: knight fork and discovered attack

This puzzle (Figure 10) features a tactical sequence beginning with a knight check that sets up a discovered attack (PV: 1. \wmoveNd6+ \wmoveexd6 2. \wmoveBxc6+). After the knight is captured, the bishop delivers a check, forcing the black king to move and allowing White to capture the undefended queen. The input layer exhibits piece-specific bias toward knight moves, which are abandoned in early layers as the model shifts focus to queen captures in layer $1$. The queen trade \wmoveQxa5, with the queen being protected by a knight, dominates through layers $4$–$13$ with probabilities around $50\%$. The winning move \wmoveNd6+ maintains low probability until layer $12$ ($1.65\%$%), then suddenly jumps to $24.37\%$ at layer $13$ before becoming the decisive top choice in the final output. This sudden increase after layer $13$—the same layer Cruz (2025) identified as containing one of the “look-ahead” heads—may indicate the model requires multi-move analysis rather than simple tactical heuristics for this position. Full probabilities are in Tables 7 and 8.

I.2 Correct case study 2: early and consistent solution lock-in

This puzzle (Figure 11) features a back-rank mate initiated by a queen sacrifice (PV: 1. \wmoveQxe8+ \wmoveRxe8 2. \wmoveRxe8#). The winning move \wmoveQxe8+ becomes the top candidate immediately at layer [math] ($50.5\%$) and maintains dominance through nearly all subsequent layers. The only significant competitor is the pawn capture \wmovebxc3, which briefly overtakes the solution at layer $3$ ($56.3\%$ vs $40.7\%$) and layer $7$ ($52.9\%$ vs $42.9\%$), before declining steadily through late layers. All other moves remain below $1\%$ from layer $1$ onwards. The solution reaches $88.9\%$ in the final output. This early convergence contrasts with the delayed emergence seen in other case studies and may reflect the winning move combining both queen-move and capture biases present in initial layers. Full probabilities are in Tables 9 and 10.

I.3 Correct case study 3: knight sacrifice with rook mate

This puzzle (Figure 12), featured in Jenner et al. (2024), requires a knight sacrifice (PV: 1. \wmoveNg6+ \wmovehxg6 2. \wmoveRh4#). The input encoding favors specific pieces, while layer [math] is dominated by the queen check \wmoveQg8+ ($40.0\%$), which reappears briefly at layer $2$. Through middle layers the model favors the positional rook centralization \wmoveRc4, peaking at $69.4\%$ (layer $4$) and remaining the top candidate through layer $8$. Meanwhile, the queen maneuver \wmoveQe6 threatening the opposing queen while protected by the knight is also considered in layers $7$–$11$. The correct sacrifice \wmoveNg6+ remains below $7\%$ through layer $7$, then emerges from layer $8$ ($10.8\%$) and becomes the top candidate at layer $11$ ($35.3\%$), peaking at $53.1\%$ (layer $12$). After a brief dip at layer $13$ ($33.2\%$), where the quiet pawn push \wmoveh4 is considered, the solution recovers to dominate the final output at $70.7\%$. Full probabilities are in Tables 11 and 12.

I.4 Correct case study 4: queen sacrifice for rook mate

This puzzle (Figure 13) involves a queen sacrifice enabling a back-rank rook mate (PV: 1. \wmoveQe8+ \wmoveRxe8 2. \wmoveRxe8#). The input encoding favors quiet queen moves \wmoveQd7 ($30.6\%$) and \wmoveQc7 ($20.7\%$). From layer [math] onwards, the model shifts toward captures and checks: the queen capture \wmoveQxf7+ ($31.0\%$–$35.4\%$, layers [math]–$4$), the material capture \wmoveQxa3 ($33.8\%$ at layer [math], declining to $2.9\%$ by layer $12$), and the rook capture \wmoveRxa3 which persists throughout ($10\%$–$20\%$ across most layers). The pawn check \wmovegxf7+ dominates the middle layers, peaking at $43.5\%$ (layer $6$) and remaining the top candidate through layer $12$—a forcing move that delivers check but lacks the mating continuation. The winning move \wmoveQe8+ receives minimal probability ($<2\%$) through layer $7$, then emerges gradually from layer $8$ ($3.9\%$). The solution only narrowly leads at layer $13$ ($22.3\%$ vs $20.0\%$ for \wmovegxf7+), with the decisive separation occurring entirely in the final layer where it reaches $58.9\%$. Full probabilities are in Tables 13 and 14.

I.5 Correct case study 5: bishop sacrifice for discovered attack

This puzzle (Figure 14) features a bishop sacrifice winning material via a discovered attack (PV: 1. \wmoveBh2+ \wmoveKxh2 2. \wmoveRxc6). The input encoding strongly favors the pawn push \wmovea4 ($57.5\%$). From layer $1$, the immediate material capture \wmoveRxe3 dominates, peaking at $86.8\%$ (layer $3$) and remaining the top candidate through layer $6$. The rook centralization \wmoveRe5 then emerges as a competitor, and the two alternate as the top choice through layers $7$–$11$, with \wmoveRxe3 leading at layers $8$ and $10$ and \wmoveRe5 at layers $7$, $9$, and $11$. The winning move \wmoveBh2+ receives negligible probability ($<2\%$) through layer $5$, then builds steadily from layer $6$ ($14.6\%$) to \sim$$23\% across layers $8$–$11$, before overtaking all competitors at layer $12$ ($51.2\%$) and reaching $92.9\%$ in the final output, demonstrating a late-layer override of a simple material-gain heuristic in favor of a deeper sacrificial sequence. Full probabilities are in Tables 15 and 16.

Appendix J Complete probabilities for the forgotten example puzzle from Figure 1

This puzzle features a mate-in-two requiring a queen sacrifice (PV: 1. \bmoveQxf1+ 2. \wmoveKxf1 \wmoveRd1#). The winning move \wmoveQxf1+ becomes the top candidate immediately at layer [math] ($45.7\%$) and remains decisively the top choice through layer $12$ ($88.3\%$), with its primary competitor being the queen check \wmoveQxh2+ which maintains a persistent secondary role ($23\%$–$31\%$ across layers [math]–$11$). The solution peaks sharply at layer $12$ ($88.3\%$) as \wmoveQxh2+ drops to $3.3\%$, but is then abandoned: it falls to $30.6\%$ (layer $13$) and $5.8\%$ in the final output. The losing move \wmoveQc8—a quiet queen retreat to the back rank—receives negligible probability ($<0.2\%$) through layer $12$, then surges to $2.4\%$ (layer $13$) and $37.0\%$ in the final output. A second quiet queen move \wmoveQd7 follows the same pattern, rising from $0.3\%$ (layer $12$) to $24.7\%$ in the final output. The model replaces a winning queen sacrifice with quiet queen retreats, consistent with the safety-prior mechanism identified in the main text. The value head evaluates the current position as losing ($96.0\%$ loss) despite the mate-in-two available, yet through one-step look-ahead correctly assigns near-certain victory ($99.8\%$ win) to the position after \wmoveQxf1+ while evaluating all alternatives as losing ($99\%$–$100\%$ loss), including the model’s preferred \wmoveQc8 ($98.9\%$ loss) (Table 17). Full probabilities are in Tables 18 and 19.

Appendix K Forgotten puzzle case studies

K.1 Forgotten case study 1: rook sacrifice for queen trade

This puzzle (Figure 16) presents a rook sacrifice forcing a queen trade (PV: 1. \wmoveRxg7+ \wmoveKxg7 2. \wmoveQxh4). The winning move \wmoveRxg7+ dominates throughout nearly all layers, serving as the top candidate from layer [math] onwards (excluding layer $1$) with probabilities exceeding $50\%$ from layer $2$ and peaking at $80.68\%$ at layer $12$. Its primary competitor, the materially conservative queen capture \wmoveQxa7, maintains $28\%$-$53\%$ probability through layers [math]-$7$ before declining steadily, since this move leaves the rook hanging and gains insufficient compensation. The solution is abandoned in the final two layers in favor of \wmoveKf1—a king move that receives negligible probability ($<1\%$) throughout layers [math]-$11$, then surges to $6.20\%$ (layer $12$), $30.42\%$ (layer $13$), and $52.98\%$ (final output). Notably, the model’s value head evaluates the current position as unfavorable ($71.8\%$ loss probability), yet when performing a one-step look-ahead by evaluating resulting positions after each legal move, correctly distinguishes the forcing sequence: it assigns near-certain victory ($99.5\%$ win) to the position after \wmoveRxg7+ while evaluating all alternatives as losing positions ($91\%$-$100\%$ loss) (Table 20). Full probabilities are in Tables 21 and 22.

K.2 Forgotten case study 2: pawn sacrifice for queen capture

This puzzle (Figure 17) presents a pawn sacrifice forcing a queen capture (PV: 1. \wmoveg5+ \wmovefxg5 2. \wmoveQxe5). The winning move \wmoveg5+ exhibits highly non-monotonic behavior with substantial probability jumps between consecutive layers: $0.55\%$ to $18.46\%$ (layers $4$-$5$), $22.21\%$ to $0.38\%$ (layers $8$-$9$), and $0.77\%$ to $41.93\%$ (layers $12$-$13$). The solution briefly emerges as the top candidate at layer $13$ ($41.93\%$), but drops to fourth place in the final output ($11.17\%$). Through most middle layers ($2$-$11$), the model strongly favors the materially conservative queen trade \wmoveQxe5, peaking at $89.57\%$ (layer $4$). The final layer instead prioritizes queen retreats to safety: \wmoveQd3 ($29.14\%$), \wmoveQc4 ($25.06\%$), and \wmoveQd1 ($14.27\%$)—all receiving minimal probability through layers [math]-$11$. The sharp inter-layer transitions suggest algorithmic computation of forcing sequences at layer $13$ presumably through look-ahead rather than gradual accumulation of tactical heuristics. The model’s value head evaluates the current position as unfavorable ($75.9\%$ loss probability), yet through one-step look-ahead correctly assigns near-certain victory ($98.3\%$ win) to the position after \wmoveg5+ while evaluating all alternatives as losing positions ($89.1\%$-$100\%$ loss) (Table 23). Full probabilities are in Tables 24 and 25.

K.3 Forgotten case study 3: queen sacrifice to back rank mate

This puzzle (Figure 18) presents a forced mate-in-two sequence requiring a queen sacrifice (PV: 1. \wmoveQxc8+ \wmoveRxc8 2. \wmoveRe8#). The winning move \wmoveQxc8+ dominates from layers $3$-$12$, peaking at $72.9\%$ (layer $5$). However, this solution is abandoned in the final two layers, where \wmoveQxa7 surges to $63.7\%$ in the final output—a materially safe queen capture that doesn’t lead to sacrifice, receiving only $1.8\%$-$12.4\%$ probability in layers $4$-$12$. The reversal may reflect a safety prior against queen sacrifices overriding mid-layer tactical calculations. The model’s value head evaluates the current position as unfavorable ($82.3\%$ loss) despite the mate-in-two available, yet through one-step look-ahead correctly assigns near-certain victory ($100\%$ win) to the position after \wmoveQxc8+ while evaluating all alternatives as near-certain losses ($99\%$-$100\%$ loss) (Table 26). This illustrates a failure mode where sound mid-layer analysis is overwritten by conservative final-layer adjustments despite accurate position evaluation capabilities. Full probabilities are in Tables 27 and 28.

K.4 Forgotten case study 4: rook sacrifice for material gain

This puzzle (Figure 19) presents a rook sacrifice forcing material advantage (PV: 1. \bmoveRxg4+ 2. \wmoveKxg4 \wmoveKxe6). The winning move \wmoveRxg4+ dominates from layers $4$-$13$, maintaining probabilities above $61\%$ throughout and peaking at $88\%$ at layer $10$. However, the solution drops dramatically in the final layer to third place ($12.14\%$), overtaken by \wmoveRf1+ ($40.5\%$) and \wmoveRh1 ($23\%$)—two materially conservative rook moves that receive minimal probability ($<6\%$) through layers [math]-$12$. Unlike previous cases, these alternative moves are strategically sound rather than clearly losing: Stockfish evaluates \wmoveRf1+ at $-0.05$ pawns and \wmoveRh1 at $-0.15$ pawns, both near-neutral positions. The model’s value head reflects this nuance through one-step look-ahead evaluation: while correctly assigning near-certain victory ($97.9\%$ win) to the position after \wmoveRxg4+, it evaluates the alternatives as roughly even positions (\wmoveRf1+: $73.6\%$ loss, \wmoveRh1: $63.3\%$ loss), rather than the near-certain losses ($99\%$+) seen in previous puzzles (Table 29). This represents the most substantial final-layer reversal observed ($88\%$ to $12.14\%$), demonstrating that the safety prior against piece sacrifices can override even stronger mid-layer confidence than in previous cases. Full probabilities are in Tables 30 and 31.

K.5 Forgotten case study 5: narrowly forgotten queen sacrifice

This puzzle (Figure 20) presents a forced mate-in-two sequence requiring a queen sacrifice (PV: 1. \wmoveQxb1+ \wmoveRxb1 2. \wmoveRxb1#). The winning move \wmoveQxb1+ dominates through most layers, maintaining probabilities between $18.87\%$ and $62.90\%$ from layers [math]-$12$ and peaking at $56.96\%$ in layer $13$. However, in the final layer, the materially conservative pawn advance \wmoveb5 marginally overtakes the solution ($36.12\%$ vs $35.99\%$), despite receiving only $8.85\%$ probability in the preceding layer. This final-layer reversal demonstrates another instance of a safety prior against queen sacrifices overriding established tactical analysis, though the margin is narrower than in previous cases. Notably, the model’s value head evaluates the current position as unfavorable ($81.1\%$ loss probability) and appears to base this evaluation on the inferior move \wmoveb5 rather than the forcing sequence, yet when performing a one-step look-ahead by evaluating resulting positions after each legal move, correctly assigns near-certain victory ($99.3\%$ win) to the position after \wmoveQxb1+ while evaluating all alternatives as losing positions ($99.4\%$-$100\%$ loss) (Table 32). Full probabilities are in Tables 33 and 34.

Appendix L Look-ahead analysis details

This appendix provides full methodological details for the three look-ahead analyses replicated from Jenner et al. (2024) on forgotten puzzles. Error bars and confidence intervals follow their methodology throughout. Model variant details for look-ahead experiments are provided in Appendix D.

Dataset construction

We start from the full Lichess puzzle database of approximately $900{,}000$ puzzles and apply the same filtering criteria as Jenner et al. (2024): puzzles must have a principal variation of at least three moves, a weaker sparring model must assign less than $5\%$ probability to the first move (ensuring difficulty), and more than $70\%$ to the opponent’s response (ensuring the line is forcing). We use the same sparring model as Jenner et al. (2024), available in their public inference framework. Mate-in-one puzzles are excluded. We then select forgotten puzzles: positions where the correct first move is the top candidate at some intermediate layer but not at the final layer. To ensure these represent genuine forgetting rather than absence of look-ahead, we additionally require the full model to assign $>50\%$ probability to all subsequent player moves in the principal variation, confirming that the model demonstrably understands the winning continuation and thus plausibly employs look-ahead. This yields $7{,}901$ forgotten puzzles. For comparison, Jenner et al. (2024) apply the same filtering but select positions the model solves correctly, yielding approximately $22{,}000$ positions. Neither dataset directly filters for the presence of look-ahead mechanisms; any effects observed in the analyses below are emergent rather than selected for.

Corruption generation

Following Jenner et al. (2024), we generate corrupted board states by applying small modifications to the original position, such as adding or removing pawns or moving non-pawn pieces to empty squares. Illegal board states are discarded. Corruptions are selected to substantially affect the full model while minimally affecting the sparring model, thereby targeting mechanisms beyond simple heuristics. Although the model does not select the correct move in forgotten puzzles, it still assigns nonzero probability to it—consistent with the solution being computed but overridden—and log-odds reduction remains a meaningful metric for measuring causal importance. Specifically, a corruption is retained if (1) the log-odds of the correct move under the full model decrease by at least $1.0$, (2) the corruption worsens the position (value change $<-0.1$, where position value is $p_{\text{win}}-p_{\text{loss}}$), and (3) the Jensen-Shannon divergence of the sparring model’s policy remains below $0.2$. The log-odds criterion differs from Jenner et al. (2024), who require the correct move’s probability to drop below $10\%$; since the correct move already has low probability in forgotten puzzles, we use a relative decrease instead. When multiple corruptions pass these filters, we select the one with the lowest JSD on the sparring model. Of $7{,}901$ forgotten puzzles, $7{,}490$ ($94.8\%$) yield valid corruptions.

Activation patching

For each position-corruption pair, we run a clean forward pass and a corrupted forward pass. Token representations from the corrupted pass are patched into the clean pass one square at a time at each layer, and the effect on the log-odds of the correct move is measured. We report the mean effect across positions for four categories: the first move target square, the third move target square, the corrupted square(s), and the maximum effect over all remaining squares averaged across positions.

Attention head ablation

We replicate the attention ablation from Jenner et al. (2024). For each forgotten puzzle, we zero the attention score where the first move’s target square (query) attends to the third move’s target square (key), preventing information flow between these squares. We measure the log-odds reduction of the correct move. As a baseline, we simultaneously zero all remaining $4{,}095$ attention entries in the same head. Results for L12H12 are presented in the main text (Figure 5(b)). Figure 21 shows results for four additional heads identified by Cruz (2025). L12H17 and L13H3 show a similar pattern to L12H12, with the single look-ahead entry producing outsized effects compared to the baseline, while L11H13 and L11H10 show no visible effects on forgotten puzzles.

Probing

We train bilinear probes following the two-step architecture of Jenner et al. (2024): first predicting the target square of the third move from activations at the first move’s target square, then predicting the source square conditioned on the predicted target. On forgotten puzzles, we condition on the ground-truth first move target square rather than the model’s predicted move, since by definition the model does not select the correct first move. Probes are trained on a $70/30$ train/test split with five independent training runs. As a baseline, following Jenner et al. (2024), we train probes with the same architecture on a randomly initialized model.

Comparison with solved puzzles

For reference, Figure 22 reproduces the corresponding results from Jenner et al. (2024) on solved puzzles, enabling direct visual comparison with the forgotten puzzle results in Figure 5.

Appendix M Stockfish concept preference analysis

We analyze how different network layers prioritize chess concepts by measuring the expected change in concept values induced by each layer’s move distribution. Unlike McGrath et al. (2022), who probed for concept representations using linear classifiers on intermediate activations, we directly measure concept preferences from policy outputs. This approach reveals what concepts each layer prioritizes when selecting moves, complementing representation-based analyses.

Chess concepts

We use Stockfish 8’s (Stockfish Developers, 2016) handcrafted evaluation function as our source of chess concepts, following McGrath et al. (2022) to enable direct comparison with prior work from Sadler and Regan (2019). Stockfish decomposes position evaluation into interpretable components including material balance, piece-specific features, king safety, threats, mobility, passed pawns, and spatial control. Each concept is evaluated separately for midgame (mg), endgame (eg), and produces a phase-interpolated value (ph) computed as a weighted sum of midgame and endgame values based on the game phase. All concepts are represented as continuous values. Table 35 summarizes the main concept categories we analyze.

Perspective normalization

Stockfish evaluates all positions from White’s perspective, with positive values favoring White. To ensure consistency with Leela’s current-player perspective, we convert Stockfish’s evaluations to player-relative coordinates following McGrath et al. (2022). For concepts with side-specific values (knights, bishops, rooks, queens, mobility, king safety, threats, passed pawns, space), we relabel White/Black as Mine/Opponent based on the side to move. When White is to move, White concepts become Mine and Black concepts become Opponent, with values unchanged. When Black is to move, Black concepts become Mine and White concepts become Opponent, with all values negated to reflect the perspective change. For aggregate concepts (material, imbalance, pawns, total), we negate values when Black is to move. This ensures that positive concept values always represent advantages for the player to move.

Concept delta calculation

For each position $s$ with player $p$ to move, we compute concept preferences as follows:

Evaluate initial position: Use Stockfish 8 to obtain concept values $c(s)$ for all concepts $c$, normalized to player $p$’s perspective. 2. 2.

Generate and evaluate legal moves: For each legal move $m$ leading to position $s^{\prime}$, evaluate $c(s^{\prime})$ using Stockfish and normalize to player $p$’s perspective (accounting for the perspective flip after the move). 3. 3.

Calculate concept deltas: For each move-concept pair, compute:

[TABLE]

where both $c(s^{\prime})$ and $c(s)$ are evaluated from player $p$’s perspective, ensuring $\Delta c_{m}$ consistently represents concept change from the moving player’s viewpoint. 4. 4.

Compute layer-wise preferences: For each layer $\ell$, obtain the move probability distribution $\pi_{\ell}$ using the logit lens and calculate the expected concept delta:

[TABLE]

This weighted average represents the expected change in concept $c$ when sampling moves according to layer $\ell$’s policy.

Dataset and sampling

We sample $10{,}000$ positions from the CCRL dataset (Leela Chess Zero team, 2018), a standard benchmark consisting of $2.5$ million computer games from CCRL $40/40$ and $40/4$ tournaments. The dataset contains games between strong chess engines and provides diverse positions across different game phases and strategic themes.

Positions are sampled using a two-stage hierarchical process:

Sample 10% of games uniformly from the dataset 2. 2.

From each selected game, sample 10% of positions uniformly 3. 3.

Filter duplicate positions to ensure uniqueness

Implementation details

For each sampled position, we:

• •

Extract move policies $\pi_{\ell}$ for the input embedding layer plus all $15$ transformer layers, where the last layer corresponds to the final model output

• •

Evaluate the initial position and all legal move outcomes (typically $30$-$40$ moves per position) using Stockfish $8$

• •

Calculate concept deltas and probability-weighted averages for all $93$ Stockfish concepts across midgame, endgame, and phase-interpolated variants

• •

Store results for analysis and visualization

Figures in the main paper present phase-interpolated (ph) concept values, as these represent Stockfish’s actual evaluation for a given position. Comprehensive results showing all concept variants (mg, eg, ph) across all layers are provided in this appendix (Figures 23 to 29).

Statistical analysis

We report mean $\Delta c_{\ell}$ values across all $10{,}000$ positions with $95\%$ confidence intervals computed via the $t$-distribution. Many concepts have values of zero—and therefore zero deltas—in positions lacking relevant pieces (e.g., bishop concepts when no bishops are present, or passed-pawn bonuses without passed pawns) or where the concept is undefined. This produces zero-inflated distributions that bias means toward zero. We do not control for this effect, which makes absolute $\Delta c_{\ell}$ values across concepts skewed and not directly comparable. Nevertheless, relative trends across layers remain informative for understanding how concept preferences evolve with depth.

Interpretation of concept evolution

Analysis of concept preferences across layers reveals three distinct patterns corresponding to the computational phases identified in playing strength progression. In the early phase (layers Input to $5$), most concepts exhibit erratic, volatile shifts with substantial fluctuations in preference values. During the middle phase (layers $5$ to $10$), concept preferences stabilize significantly, with most concepts maintaining relatively constant values across this range, mirroring the performance plateau observed in tournament play and puzzle solving. In the late phase (layers $11$ to Final), concept preferences show consistent, smooth trends—typically monotonic increases or decreases for each concept, with minimal erratic behavior. This systematic evolution coincides with the sharp capability improvements and emergence of look-ahead mechanisms in late layers. The shift from aggressive to safety-oriented concepts occurs primarily in this phase: preferences for own king safety and opponent threats increase (prioritizing defensive considerations), while preferences for opponent king vulnerability and own threats decrease (de-emphasizing aggressive play). These patterns hold across most concepts, though some show different dynamics requiring deeper chess expertise to interpret. The consistency of trends in middle and late phases is particularly striking given the volatility of early layers, suggesting fundamentally different computational regimes across network depth.

Appendix N Concept steering details

This appendix provides complete details for the gradient-based concept steering experiments described in the main text. We group threats and king safety as safety-related concepts based on their intuitive connection to conservative play; this categorization is ours rather than intrinsic to Stockfish’s evaluation terms. The consistency of the effect across four independently defined safety-related concepts suggests the result reflects a genuine property of the model rather than an artifact of any single proxy. All steering experiments use the standard model with position history.

Gradient computation

For each concept $c$ and layer $\ell$, we compute the gradient of the concept preference of the complete model $\Delta c_{L}$ (Equation 1) with respect to the residual stream activations after layer $\ell$:

[TABLE]

Gradients are computed over $10{,}000$ positions sampled from the CCRL (Leela Chess Zero team, 2018) dataset using the same hierarchical sampling procedure described in Appendix M. Gradients are averaged across all $64$ squares and positions to obtain a single $768$-dimensional steering vector $\hat{\mathbf{v}}_{c,\ell}$ per concept per layer based on $640{,}000$ activations.

Steering strength

The steering strengths $\alpha\in\{\pm 0.25,\pm 0.5,\ldots,\pm 1.5\}$ reported in the main text are scaled by the per-layer mean activation norm $\mathbb{E}[\|\mathbf{h}_{\ell}\|_{2}]$, computed over the CCRL gradient positions, to ensure comparable intervention magnitude across layers, which have different activation scales. The applied perturbation is:

[TABLE]

where $\hat{\mathbf{v}}_{c,\ell}$ is the normalized gradient direction. Positive $\alpha$ steers toward increasing the concept preference; negative $\alpha$ steers toward decreasing it. The steering vector is applied uniformly to all $64$ square positions. We sweep over all strengths and all layers, evaluating each configuration on the full $10{,}000$ general puzzle dataset using full principal variation correctness.

For each layer in Figure 6 (right panel) from the main text, we report the solve rate at the best-performing strength for that layer. This selection protocol is applied identically to concept steering and the random baseline, ensuring a fair comparison.

Random baseline

To establish that the steering effects are concept-specific rather than artifacts of generic activation perturbation, we compare against $50$ random steering vectors drawn from an isotropic Gaussian distribution, each normalized to unit length and evaluated under the same sweep protocol. The reported random baseline shows the mean solve rate across these $50$ vectors with $95\%$ confidence intervals. Because the same selection protocol is applied to concept steering and the random baseline, the gap between them isolates the concept-specific effect from the selection bias introduced by sweeping over $\alpha$ and $\ell$.

Per-concept steering results

Figures 30–36 show the complete layer-by-strength sweep for all Stockfish concepts. Each plot displays the solve rate across layers at the best-performing strength per layer, with point color indicating the selected best strength. The unsteered baseline and random baseline are shown for reference. The four safety-related concepts—threats (opponent), threats (mine), king safety (mine), and king safety (opponent)—consistently produce the largest improvements, peaking at layer $13$ with solve rates exceeding $91\%$. While other concepts also improve performance above the random baseline, likely reflecting correlations between piece-specific concepts and tactical move types common in puzzles, the effects are smaller and less consistent across layers. Across most concepts, steering in middle layers has the least impact, again mirroring the computational phases identified in the main text. The most directly safety-related concepts yield the largest effects, supporting the interpretation that learned safety priors are a primary mechanism underlying the forgotten puzzle phenomenon.