Some Chip Transfer Games

Thomas S. Ferguson


Four related impartial combinatorial games are investigated. The first game is called Empty & Transfer . Given k boxes each containing at least one chip, a move consists of emptying a box and transferring to it at least one chip, but not all chips, from one of the other boxes. Players move alternately and play continues until no more moves are possible. Under the normal ending rule, the last player to move wins; under the misère rule, the last player to move loses. The normal form of this game is solved for k<5. For k=2, the Sprague-Grundy function is found. The second game is called Empty-All-But-One . A move consists of emptying all but one of the boxes and transferring chips from the remaining box so that there is at least one chip in each box. The third game is called Empty & Redistribute . A move empties only one box but the chips in the remaining boxes may be redistributed among all boxes in an arbitrary manner. These Two games are solved for arbitrary k in both normal and misère forms. The last game is called Entropy Reduction . Two boxes are chosen and made more nearly equal in size by transferring chips from one box to the other. This game is solved for k=3 in both the normal and the misère versions.