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The game of Nim is at least centuries old, possibly
originating in China, but noted in the 16th century
in European countries. It consists of several stacks
of tokens, and two players alternate taking one or
more tokens from one of the stacks, and the player
who cannot make a move loses.
The formal and intense study of Nim culminated in
the celebrated Sprague-Grundy Theorem, which is now
one of the centerpieces in the theory of impartial
combinatorial games.
We study a variation on Nim, played on a graph. Graph Nim, for which the theory of
Sprague-Grundy does not provide a clear strategy.
Graph Nim was originally developed at the University
of Colorado Denver.
Graph Nim was first played on graphs of three
vertices. The winning strategy and losing position
of three-vertex Graph Nim have been discovered,
but we will expand the game to four vertices and
develop the winning strategies for four-vertex
Graph Nim.
This work was published as a chapter in the
Master’s Thesis of Trevor Williams [8].