De Bruijn's theorem

A coloring of the unit cubes in a box that may be used to prove the impossibility of packing it with bricks, as each brick covers 4 white and 4 black cubes but the box contains 8 more white than black cubes

In a 1969 paper, Dutch mathematician Nicolaas Govert de Bruijn proved several results about packing congruent rectangular bricks (of any dimension) into larger rectangular boxes, in such a way that no space is left over. One of these results is now known as de Bruijn's theorem. According to this theorem, a "harmonic brick" (one in which each side length is a multiple of the next smaller side length) can only be packed into a box whose dimensions are multiples of the brick's dimensions.[1]

  1. ^ de Bruijn, N. G. (1969), "Filling boxes with bricks", The American Mathematical Monthly, 76 (1): 37–40, doi:10.2307/2316785, JSTOR 2316785, MR 0234841.