Winning strategies for aperiodic subtraction games

Details Winning strategies for aperiodic subtraction games Winning strategies for aperiodic subtraction games

2011-08-05

arxiv.org/abs/1108.1239v2

Mathematics

Combinatorics

6 pages, no figuress, added references, fixed typos, made proofs more efficient

Scientific paper

We provide a winning strategy for sums of games of MARK-t, an impartial game played on the nonnegative integers where each move consists of subtraction by an integer between 1 and t-1 inclusive, or division by t, rounding down when necessary. Our algorithm computes the Sprague-Grundy values for arbitrary n in quadratic time. This solves a problem posed by Aviezri Fraenkel. In addition, we characterize the P-positions and N-positions for the game in mis\`ere play.

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