The Games and Puzzles Journal, #5+6 - Mayhematics

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THE GAMES AND PVZZLES JOURNAL
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Polyhexes
{5-
$' Looking at a cube corner-on, its outline is, that of a hexagon. By this visualisation
we can see that the sum of lhe first n hexagonal numbers is n cubed. Henee the nth hexagonal
numUer is n3-l (n-1)3, which is equal to 3n(n-1)+1, jf;" formula previously given.
The first number both square and hexagonal is^169 132 = 83 - 7J. The first number both
triangular
=
and hexagonal is g1 = (13 x t+)/z = 63 - s3. Ttre answers to the other two questions
posed are provided by the diagrams:
\
F
F
r'
s?
d
39
\}
.s
&
sym metric
5-pieces in
hexagon minus
centre
These solutions are by Michael KELLER. and Leonard GoRDoN. Working independently
both came up with an identical solution to the hexagon problem_ - t!t" two pieces at the
bottom left -corner of this case can be rotated to give a fourth solution. M.K. provided
the artwork. He also coryects the total of 6-piece polyhexes to 82. Some remarkable
dissections using the whole set appear in No ? of his World Game Review.
Heptiamonds
Walter STEAD . s notebooks contain some fascinating dissections with 'polyiamonds"
i.e. shapes formed from regular triangles. There is only one of each of the sizes 1, 2 ond 3'
three oi a pieces, four of 5 pieces, twelve of 6 pieces and 24 of ? pieces. 'The Batr below'
comp.osed 30th lvtarch 196?, shows all the 43 one-sided heptiamonds (5 symmetric in the
centie and 19 pairs of asymmetrics - thus 19 + 5 = 24 the number noted above). The front
cover result dates from Aug-Sept 1966.
Batrl
Geometr[c Jigsauts
W.STEAD
ttFully interlocking" Jigsaw puzzles depend on pieces that have rrbobbles?tand trnibbles"
that fit together. If the body of each piece, apart from the bobbles and nibbles, is taken
to be square, how many different shapes of Jigsaw piece are possible? Can you pick out
sets of these pieces that will fit together to form a square or rectangle? No shape is
to bsused more than once.
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