# 206 geometry/tiling/seven.cubes.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 206 geometry/tiling/seven.cubes.p

Consider 7 cubes of equal size arranged as follows. Place 5 cubes so

that they form a Swiss cross or a + (plus) (4 cubes on the sides and

1 in the middle). Now place one cube on top of the middle cube and the

seventh below the middle cube, to effectively form a 3-dimensional

Swiss cross.

Can a number of such blocks (of 7 cubes each) be arranged so that they

are able to completely fill up a big cube (say 10 times the size of

the small cubes)? It is all right if these blocks project out of the

big cube, but there should be no holes or gaps.

geometry/tiling/seven.cubes.s

Let n be a positive integer. Define the function f from Z^n to Z by

f(x) = x_1+2x_2+3x_3+...+nx_n. For x in Z^n, say y is a neighbor of x

if y and x differ by one in exactly one coordinate. Let S(x) be the

set consisting of x and its 2n neighbors. It is easy to check that

the values of f(y) for y in S(x) are congruent to 0,1,2,...,2n+1 (mod

2n+1) in some order. Using this, it is easy to check that every y in

Z^n is a neighbor of one and only one x in Z^n such that f(x) is

congruent to 0 (mod 2n+1). So Z^n can be tiled by clusters of the

form S(x), where f(x) is congruent to 0 mod 2n+1.

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