MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 113
Figure 4.16: Rep-4 Non-Polygons: The Snail, The Lamp, and The Carpenter’s
Plane [125]
Recreation 17 (MacMahon’s 24 Color Triangles [128]). In 1921, Major
Percy A. MacMahon, a noted combinatorialist, introduced a set of 24 color
triangles [213], the edges of which are colored with one of four colors, that
are pictured in Figure 4.19 [128]. (Rotations of triangles are not considered
different but mirror-image pairs are considered distinct.) The pieces are to
be fitted together with adjacent edges matching in color to form symmetrical
polygons, the border of which must all be of the same color.
It is known that all polygons so assembled from the 24 color triangles
must have perimeters of 12, 14, or 16 unit edges. Also, only one polygon, the
regular hexagon, has the minimum perimeter of 12. Its one-color border can
be formed in six different ways, each with an unknown number of solutions.
For each type of border, the hexagon can be solved with the three triangles of
solid color (necessarily differing in color from the border) placed symmetrically
around the center of the hexagon. Since each solid-color triangle must be
surrounded by triangular segments of the same color, the result is three smaller
regular hexagons of solid color situated symmetrically at the center of the larger
hexagon. Figure 4.20 displays a hexagon solution for each of the six possible
border patterns [128]. As previously noted, it is not known how many solutions
there are of these six types although the total number of solutions has been
estimated to be in the thousands.