We investigate two modifications to the traditional rules of Wythoff Nim, a com- binatorial game. In Wythoff Nim, players take turns removing stones from a pair of piles. In each turn, a player chooses to either (1) take any number of stones from one pile or (2) take an equal number from both. The player removing the last stone wins. A seminal result of Wythoff states that at any point in the game, the current player is in a P-position — that is, guaranteed to lose assuming the other player plays optimally — if and only if the pair of pile sizes is of the form ([øn], [ø²n]) for some n  ∈ ℕ, where ø = 1.618 … represents the golden ratio.

This thesis introduces a variant of Wythoff Nim, which we call W(a, b), characterized by positive integers a and b, where players' moves are constrained to either removing a multiple of a from the first pile, a multiple of b from the second, or an equal number from both. We prove that the P-positions of this variant also follow sloping "beams", but with slopes determined by a and b.

51 pages

Thesis Committee:
Danny Sleator (Chair)
Klaus Sutner

Srinivasan Seshan, Head, Computer Science Department
Martial Hebert, Dean, School of Computer Science

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