the construction of this graph involves the unique (up to symmetry) 56-point cap set (a subset of points with no three in line) in formula_1, the five-dimensional projective geometry over a three-element field. the six-dimensional projective geometry, formula_2, can be partitioned into a six-dimensional affine space formula_3 and a copy of formula_1 (the points at infinity with respect to the affine space). the games graph has as its vertices the 729 points of the affine space formula_3. each line in the affine space goes through three of these points, and through a fourth point at infinity. the graph contains a triangle for every line of three affine points that passes through a point of the cap set.