Tiling with notched cubes

Golomb (J. Combin. Theory 1 (1966) 280–296) showed that any polyomino which tiles a rectangle also tiles a larger copy of itself. Although there is no compelling reason to expect the converse to be true, no counterexamples are known. In three dimensions, the analogous result is that any polycube that tiles a box also tiles a larger copy of itself. In this note, we exhibit a polycube (a ‘notched cube’) that tiles a larger copy of itself, but does not tile any box, and obtain several related results about tiling with this figure. We also obtain analogous results in all dimensions d⩾3.