Cutting cheese (start with a 3x3x3 cube of cheese and a knife, how many straight cuts at minimum are needed to turn it into 27 1x1x1 cubes?): The obvious answe

Cutting cheese (start with a 3x3x3 cube of cheese and a knife, how many straight cuts at minimum are needed to turn it into 27 1x1x1 cubes?): The obvious answer is 6, by cutting it the obvious way (2 parallel slices along each dimension at the appropriate locations), so that seems almost certainly wrong. If the cheese remains convex, any straight cut can at most double the number of pieces of cheese, so it would be impossible to do in less than 5 cuts (2^4 = 16, which just can't give us enough pieces, whereas 2^5 = 32 does). That holds only if we can assume convexity, but for the purposes of the problem, I assume the cheese isn't bendable or anything like that, in which case any set of straight slices you could do to make cheese non-convex would be "wasting" cuts (to maximize cheese-block count, you should always cut all the way through along a plane, and you'll only end up with convex pieces that way). We might be tempted to go out on a limb and say that it must take 5 cuts, otherwise the problem isn't very interesting. Unfortunately, we'd be wrong. Think about the center piece of cheese in the 3x3x3 block. None of its faces are exposed in the original block, and there are six faces, none of which are coplanar. Unless we cheat pretty seriously (bending cheese), we cannot expose more than one of those faces with each cut, so there's no way to do it in less than six slices.