Emperor nim and emperor sum: a new sum of impartial games

Abstract

The emperor sum of combinatorial games is discussed in this study. In this sum, a player moves arbitrarily many times in one component. For every other component, the player moves once at most. The $\mathcal {P}$-positions of emperor sums are characterized using a parameter referred to as $\mathcal {P}$-position length. An emperor sum is a $\mathcal {P}$-position if and only if every component is a $\mathcal {P}$-position and the nim-sum of the $\mathcal {P}$-position lengths of all components is 0. This is similar to using the nim-sum of $\mathcal {G}$-values to characterize the $\mathcal {P}$-positions of the disjunctive sum of games.