A

David

Darling

MacMahon squares

The concept of color-matching tiles based on all the permutations of colors on their edges dates from 1926, when Percy MacMahon invented and introduced Three-Color Squares and Four-Color Triangles as mathematical pastimes.

 

MacMahon 3-color squares

 

MacMahon divided squares and triangles into triangles to give each edge of a piece its own color, in all possible combinations. Each set contains 24 different tiles, and MacMahon discovered that they could form a single figure with all adjacent edges matching and just one color all around the outside border.

 

MacMahon 4-color triangles

 

The most extensive research into these sets, over three decades, was done by the American engineer Wade Philpott (1918–85), of Lima, Ohio, who identified all the possible symmetrical shapes that MacMahon squares and triangles could solve with both matching edge colors and uniform border color, and who calculated all the numbers of solutions for the MacMahon squares' 4 × 6 rectangle.