\magnification=1200 \hsize=4in \nopagenumbers \noindent % % {\bf Joseph Kahane and Alexander J. Ryba} % % \medskip \noindent % % {\bf The Hexad Game } % % \vskip 5mm \noindent % % % % We give a non-enumerative proof that the $P$-positions of the hexad game give the Steiner system $S(5,6,12)$. We show that the distributions of nim sums and Welter values of these $P$-positions have simple and surprising regularities. \bye .