MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 125
Recreation 24 (Eternity Puzzle [242]). The Eternity Puzzle [242] is a
jigsaw puzzle comprised of 209 pieces constructed from 12 hemi-equilateral
(30 ◦ − 60 ◦ − 90 ◦ ) triangles (Figure 4.28(a)). These pieces must be assembled
into an almost-regular dodecagon on a game board with a triangular grid
(Figure 4.28(b)).
In June 1999, the inventor of the puzzle, Christopher Monckton, offered a
£1M prize for its solution. In May 2000, two mathematicians, Alex Selby and
Oliver Riordan, claimed the prize with their solution shown in Figure 4.29(a).
In July 2000, Günter Stertenbrink presented the independent solution shown
in Figure 4.29(b). As these two solutions do not conform to the six clues
provided by Monckton, his solution, which remains unknown, is presumably
different. This is not surprising since it is estimated that the Eternity Puzzle
has on the order of 10 95 solutions (it is estimated that there are approximately
8×10 80 atoms in the observable universe), but these are the only two (three?)
that have been found!
Figure 4.30: Knight’s Tours on a Triangular Honeycomb [316]
Recreation 25 (Knight’s Tours on a Triangular Honeycomb [316]).
The traditional 8 × 8 square chessboard may be replaced by using hexagons
rather than squares and build chessboards, called triangular honeycombs by
their inventor Heiko Harborth of the Technical University of Braunschweig, in
the shape of equilateral triangles. Knight’s Tours for boards of orders 8 and 9
are on display in Figure 4.30 [316].
The subject of Knight’s Tours on the traditional chessboard have a rich
mathematical history [242]. The earliest recorded solution was provided by de
Moivre which was subsequently “improved” by Legendre. Euler was the first
to write a mathematical paper analyzing Knight’s Tours.