[math-fun] Tiling puzzle

I have a nice, fun tiling puzzle: 1. Prove that there is no way to fill a 2*2*2 cube with four 2*1*1 dominoes, such that there is no 2*2*1 subset containing two dominoes. 2. Fill a 2*2*2*2 hypercube with eight 2*1*1*1 dominoes, such that there is no 2*2*1*1 subset containing two dominoes. 3. Prove that your solution is unique, up to automorphism of the hypercube. (This is inspired by the fact that in two dimensions, every possible domino tiling of a genus-0 (hole-less) polyomino can be reached from any other by rotating 2*2 blocks of two dominoes. This shows that it cannot be generalised to four dimensions.) Sincerely, Adam P. Goucher