If n-dimensional space is tiled by a lattice of parallel unit cubes, must some pair of them share a complete (n−1)-dimensional face?

Is it possible to tile a square with an odd number of triangles, all of which have the same area?

Is it possible to tile a square with 30°-60°-90° triangles?

For positive integers k and n a (k, n)-semicross consists of kn + 1 parallel n-dimensional unit cubes arranged as a corner cube with n arms of length k glued on to n non-opposite faces of the corner cube. (If n is 2, it resembles the letter L, and, if n is 3, a tripod.) For which values of k and n does the (k, n)-semicross tile space by translates?

The resolution of each of these questions quickly takes us away from geometry and places us in the world of algebra.

The first one, which grew out of Minkowski's work on diophantine approximation, ends up as a question about finite abelian groups, which is settled with the aid of the group ring, characters of abelian groups, factor groups, and cyclotomic fields.

Tiling by triangles of equal areas leads us to call on valuation theory and Sperner's lemma, while tiling by similar triangles turns out to involve isomorphisms of subfields of the complex numbers.

The semicross forces us to look at homomorphisms, cosets, factor groups, number theory, and combinatorics.