A Variant of Bogus Nim -Experimental Mathematics Approach

Authors

  • Ryohei Miyadera, Nazuki Terakawa, Keito Tanemura, Akira Murakami, Mariko Kashiwagi, Ryuhei Utsuhara

Abstract

The authors present the result of research on a winning strategy of mathematical game
calculated by computer. They used “Bogus Nim with a push” as their topic. In Bogus Nim, there are coins
on a strip of squares, and you have at most one coin on a square. Two players take turns, and slide a coin
leftward as far as possible, up to, but not onto or over, the next coin, not off the end of the strip. The player
loses the game when he or she cannot move a coin anymore. “Bogus Nim with a push” is a game proposed
by the authors. In this game, players can move one of the coins at a time as in Bogus Nim or two coins
simultaneously by pushing a coin with another coin when they are adjacent.
When this game use two coins, the authors discovered formulas of Grundy numbers by computer
calculations, and confirmed these formulas for the game on a strip of certain size. With formulas of Grundy
number, we have a complete understanding of mathematical structure of the game, and hence they have a
winning strategy. As for the game of three coins, they could not find formulas of Grundy numbers, but they
discovered formulas for P-positions that are previous player’s winning positions.
Since the mathematical proof is not complete, the result is a discovery in “experimental mathematics”.
In this study, we present sample calculation code using computer algebra system Mathematica.

Published

2020-01-31

Issue

Section

Articles