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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