A map of the P-positions in 'Nim With a Pass' played on heap sizes of at most four

Perhaps the most famous combinatorial game is Nim, which was completely analyzed by C.L. Bouton in 1902. Since then, countless variants of Nim have been the subject of many research papers. In Guy and Nowakowski's Unsolved Problems in Combinatorial Games, the following entry is found: "David Gale would like to see an analysis of Nim played with the option of a single pass by either of the players, which may be made at any time up to the penultimate move. It may not be made at the end of the game. Once a player has passed, the game is as in ordinary Nim. The game ends when all heaps have vanished." In this paper, we determine all of the P-positions (second-player wins) in this particular variant of Nim played on heap sizes of at most four. Copyright © 2018 Elsevier B.V.