3 ms·
I agree, don't think it works out. Consider a 3x3...x3 puzzle (with N dimensions). There are 2^N corners but 3^N pieces so the corner-to-piece ratio decreases e
by tgb 6y ago
I agree, don't think it works out. Consider a 3x3...x3 puzzle (with N dimensions). There are 2^N corners but 3^N pieces so the corner-to-piece ratio decreases exponentially in N.
However, what does "corner" mean in an N dimensional puzzle? I was taking it to mean that it has N sides without pegs or holes. But if you took it to mean at least 2 sides without pegs or wholes, then indeed the ratio would increase (I think there are 3^N - 2^N - 1 of those) with N. But I think this is the less-satisfactory way to generalize "corner" pieces and these really should be thought of as edge pieces.
- bo1024 6y agoAh, this is a nice point, it does work for the number of edge pieces. In this case (3x3x...x3) every piece but one is an edge piece! More generally, with k pieces on a side, there are k^N pieces total, and (k-2)^N internal pieces, so in the limit, the ratio of internal to total is (1-2/k)^N -> e^{-2N/k}. So if the number of dimensions gets to be bigger the grid size, almost all pieces are "edge" (i.e. outer) pieces. Even if N=k/2, something like 70% of the pieces are edge pieces.