# The Winnability of Klondike Solitaire and Many Other Patience Games

## TL;DR

This paper introduces 'Solvitaire', an AI program that accurately determines the winnability percentages of 73 variants of patience games, significantly advancing knowledge and confidence in game outcomes.

## Contribution

The paper presents a general AI approach capable of precisely estimating winnability for numerous patience game variants, including the first correctness proofs of key dominances.

## Key findings

- Winnability of Klondike is approximately 81.945%.
- The AI achieves a 95% confidence interval of ±0.1%.
- Most results are new or significantly improved.

## Abstract

Our ignorance of the winnability percentage of the solitaire card game `Klondike' has been described as ``one of the embarrassments of applied mathematics''. Klondike, the game in the Windows Solitaire program, is just one of many single-player card games, generically called `patience' or `solitaire' games, for which players have long wanted to know how likely a particular game is to be winnable. A number of different games have been studied empirically in the academic literature and by non-academic enthusiasts. Here we show that a single general purpose Artificial Intelligence program named `Solvitaire' can be used to determine the winnability percentage of 73 variants of 35 different single-player card games with a 95% confidence interval of $\pm$ 0.1% or better. For example, we report the winnability of Klondike as 81.945% $\pm$ 0.084% (in the `thoughtful' variant where the player knows the rank and suit of all cards), a 30-fold reduction in confidence interval over the best previous result. The vast majority of our results are either entirely new or represent significant improvements on previous knowledge. Solvitaire uses depth-first search and exploits a number of AI techniques including transposition tables, symmetry breaking, dominances, and streamliners. We give the first correctness proofs of two key dominances for patience games.