Rrr 1 - 2012 on the Repetition - Free Subtraction Games and Vile –

RRR 1-2012 ON THE REPETITION-FREE SUBTRACTION GAMES AND VILE–DOPEY INTEGERS, Vladimir Gurvich Given two integers n ≥ 0 and k ≥ 3, two players alternate turns taking stones from a pile of n stones. By one move a player is allowed to take any number of stones k′ ∈ {1, . . . , k − 1}. In particular, it is forbidden to pass. Furthermore, it is not allowed to take the same number of stones as the opponent by the previous move. The player who takes the last stone wins in the normal version of the game and loses in the misère version; (s)he also wins, in both cases, when the opponent has no legal move, that is, after the move from 2 to 1. An integer k ∈ Z≥0 is called vile if the maximum power of 2 that is still a divisor of k is even, or in other words, if the binary representation of k ends with an even number of zeros; otherwise, if this number is odd, k is called dopey. In this short note, we will solve (both the normal and misère versions of) the game when k is vile and obtain partial results when k is dopey. In the normal version, the set of P-positions is an arithmetic progression ak, where a = 0, 1, . . . , if k is vile. When k is dopey, for the P-positions n0, n1, . . . , we have n0 = 0, n1 = k + 1, and ni+1 − ni is either k or k + 1 for all i ∈ ZZ≥0. Yet, in the latter case, it seems not easy to choose among k and k+ 1. We conjecture that the differences ni+1 − ni between the successive P-positions form a periodical sequence. In the misère version of the game, the P-positions are shifted by +1 with respect to the P-positions of the corresponding normal version.