Consider 7 cubes of equal size arranged as follows.

geometry/tiling/seven.cubes

Problem

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.

Show solution

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.