Some Remarks on End-Nim

We reexamine Albert and Nowakowski's variation onthe game of Nim, called End-Nim, in which the players may onlyremove coins from the leftmost or rightmost piles. We reformulateAlbert and Nowakowski's solution to this game. We examine itsmisère version and a further variant where the winner is the playerwho reduces the game to a single pile; we call this Loop-End-Nim.We show that the three games, End-Nim, misère-End-Nim, andLoop-End-Nim, all have the same losing positions, except for thepositions where all the piles are of equal size. We also give somepartial results concerning the higher Sprague-Grundy values of thethree games