AMC 10 · 2021 · #20

Three checks agree. First the checksum already run: 6 + 8 + 12 + 12 + 24 + 6 + 2 = 70, so the seven piles exactly exhaust the 70 nailed-down colorings. Second, a structural check on the pile sizes. Each pile is swept out by the 24 rotations, so its size must divide 24: the sizes 6, 8, 12, 12, 24, 6, 2 all do, and the size tells you how symmetric the shape is — the bent-triple-plus-loner pile has size 24, meaning that shape has no symmetry left at all, while the four separated cubes have a pile of size 2, meaning 12 rotations carry that shape onto itself, which matches the fact that those four cubes are the corners of a regular tetrahedron. Third, swapping the colors: recoloring every white cube blue and every blue cube white turns a build into a build, so the list of seven shapes must survive that swap, and it does — the complement of a slab is the opposite slab, the complement of a tripod is the tripod centered at the opposite corner, and the complement of each of the other shapes is a shape of the same kind. The wrong choices are all instances of splitting a pile that should not be split. The most tempting is 8: the two sets of four separated cubes look like mirror images of each other and feel like they should be different, but a single quarter turn swaps them, so they are one build. Choices 9, 10 and 11 come from separating still more shapes by an orientation that a rotation actually undoes. Failing the other way — forgetting that the staircase is handed — gives 6, which is not even offered, and that absence is itself a hint that handedness is the point of the problem.