Ha. That sounded close to me but it's not right, I think the reason is that the corners are very small. Suppose a perfect grid in a hypercube. In d dimensions

Ha. That sounded close to me but it's not right, I think the reason is that the corners are very small. Suppose a perfect grid in a hypercube. In d dimensions we have 2^d corner pieces. But if there are k pieces per side, then k^d total pieces (for example 10^d). If we consider pieces adjacent to a corner, including a diagonal, then these are little cubes of size 2^d sitting at the corners, so a total of (2^d)(2^d) = 4^d pieces close to a corner. In general taking little cubes out of the corners isn't enough to get most of the pieces. If we think about the diagonals of length k sqrt(d), we've only cut those a little shorter.