The Game of End-Nim

Abstract

In the game of End-Nim two players take turns in removing one or more boxes from a string of non-empty stacks. At each move boxes may only be taken from the two stacks which form the ends of the string (unless only one stack remains!). We give a solution for both impartial and partizan versions of the game and explain the significance of the mystic hieroglyphs: $$ \begin{array}{|cc|} \hline \uparrow & \downarrow \\ \downarrow & \uparrow \\ \hline \end{array} $$