The Game of End-wythoff

Given a vector of finitely many piles of finitely many tokens. In EndWythoff, two players alternate in taking a positive number of tokens from either end-pile, or taking the same positive number of tokens from both ends. The player first unable to move loses and the opponent wins. We characterize the P -positions (ai, K, bi) of the game for any vector K of middle piles, where ai, bi denote the sizes of the end-piles. A more succinct characterization can be made in the special case where K is a vector such that, for some n ∈ Z≥0, (K, n) and (n, K) are both P -positions. For this case the (noisy) initial behavior of the P -positions is described precisely. Beyond the initial behavior, we have bi − ai = i, as in the normal 2-pile Wythoff game.