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$m$-Modular Wythoff

Tanya Khovanova, Shuheng Niu

Abstract

We introduce a variant of Wythoff's Game that we call $m$-Modular Wythoff's Game. In the original Wythoff's Game, players can take a positive number of tokens from one pile, or they can take a positive number of tokens from both piles if the number of tokens they take from the first pile is equal to the number of tokens they take from the second. In our variant, we weaken this equality condition to one of equivalence modulo $m$. We characterize the P-positions of our $m$-Modular variant as a finite subset of the P-positions of the known P-positions of the original Wythoff's Game.

$m$-Modular Wythoff

Abstract

We introduce a variant of Wythoff's Game that we call -Modular Wythoff's Game. In the original Wythoff's Game, players can take a positive number of tokens from one pile, or they can take a positive number of tokens from both piles if the number of tokens they take from the first pile is equal to the number of tokens they take from the second. In our variant, we weaken this equality condition to one of equivalence modulo . We characterize the P-positions of our -Modular variant as a finite subset of the P-positions of the known P-positions of the original Wythoff's Game.

Paper Structure

This paper contains 12 sections, 6 theorems, 13 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Wythoff's Game
  3. Rules of Wythoff's Game
  4. P-positions of Wythoff's Game
  5. Introducing m-Modular Wythoff
  6. Example: 2-Modular Wythoff
  7. More examples: m-Modular Wythoff for m ≥ 5
  8. P-positions of m-Modular Wythoff
  9. Further directions
  10. Grundy numbers for m-Modular Wythoff
  11. Extensions with finite P-positions
  12. Modular variants of other games

Key Result

Lemma 1

Given positions $(q_1,q_2)\succ(s_1,s_2)$ with $q_1-q_2 \equiv s_1-s_2 \pmod{m}$, there is always a move from $(q_1,q_2)$ to $(s_1,s_2)$.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 2 more