Semester Award Granted

Summer 2025

Submission Date

July 2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Thesis/Dissertation Advisor [Chair]

Stephen Locke

Abstract

Split-S-Nim is a variant of Nim in which each player on their turn can choose to make any legal move in the traditional version of Nim or add q coins to a heap, where q is some element of a predetermined subset of integers, S. We classify all sets that have the property in which the winning strategy is equivalent to the winning strategy for a version of the game with a set that has cardinality 1.

If S has a smallest non-negative even value, q, then we conclude that it must either have a winning strategy identical to a version of the game with the set S = {q} or must have a general winning strategy that differs from any general winning strategy appropriate for a version of the game such that S has one element. This winning strategy is found by calculating the Sprague-Grundy numbers for the game with one heap.

If S has no non-negative even elements, we show that the winning strategy is the same as the traditional game of Nim. Similarly, we show that if there is an odd integer in S greater than or equal to −1, then the winning strategy must differ, with the exception of the odd value of 1 when 0 is in S.

Finally, we show subsets with smallest non-negative even elements of the form 2n + 4s or 2n + 4s + 2 will only have a winning strategy identical to a singleton set version of the game if all other even values have certain properties.

We complete our study of Split-S-Nim by considering the winning strategy for a game where S = Z. This version of the game is a logical extension of the game Nim in which the player can replace a heap of any size with up to two heaps of a size smaller than the original heap.

Chomp is a combinatorial game attributed to Frederik Shue and David Gale in which players take turns removing rectangular pieces from an n × m grid. While a winning strategy for the first player has been shown to exist, the strategy is not known. We analyze the case where the starting position has 2 rows, finding the Sprague-Grundy numbers for all subgames of that starting position.

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