A Generalized Diagonal Wythoff Nim
Urban Larsson

TL;DR
This paper investigates a family of two-pile take-away games called Generalized Diagonal Wythoff Nim, exploring how the set of P-positions behaves when extending Wythoff Nim with generalized diagonal moves, and provides proofs for specific cases.
Contribution
It introduces the concept of generalized diagonal moves in Wythoff Nim and proves conjectures about the structure of P-positions for certain parameter pairs.
Findings
Proved the conjecture for (p,q) = (1,2) (splitting pair).
Established conditions under which the ratio of P-position coordinates converges to the golden ratio.
Included experimental data to guide future research on GDWN games.
Abstract
In this paper we study a family of 2-pile Take Away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose sets of options and -positions are and respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} as options, the new game is Wythoff Nim. It is well-known that the -positions of this game lie on two 'beams' originating at the origin with slopes and . Hence one may think of this as if, in the process of going from Nim to Wythoff Nim, the set of -positions has \emph{split} and landed some distance off the main diagonal. This geometrical observation has motivated us to ask the following intuitive question. Does this splitting of the set of -positions continue in some meaningful way if we, to…
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Taxonomy
TopicsArtificial Intelligence in Games · Gambling Behavior and Treatments · Auction Theory and Applications
