Question 492259: How many cuts are necessary to cut an n x n x n cube into 1 x 1 x 1 cubes? Existing cuboids may be stacked together for cutting. So one cut may go through multiple existing cuboids.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! If you cannot stack, the number of cuts would be 3(n-1), n-1 cuts in any direction.
However, we can stack cubes, so the problem becomes much trickier. Suppose we have an 8x8x8 cube. We can cut parallel to the xy-plane, in which we cut into two 4x8x8 prisms, then stack them, cut into four 2x8x8 prisms, then cut again into eight 1x8x8 prisms. Repeating this twice would yield nine cuts.
Instead we can try small cases. With n=2, we need three cuts. With n=3, we need six cuts. With n=4, we can cut into two halves, stack them, cut again, repeat twice for a total of six cuts. If n=5, we can cut the cube into a 2x5x5 and a 3x5x5 prism. Stack the 2x5x5 prism on top, cut through, then cut the 3x5x5 once more. We can repeat this twice to get nine cuts. n=6 also yields nine cuts because we can cut into half, stack those halves on top of each other, and cut twice.
Here I will have to assume that the number of cuts is always a multiple of 3, because we cut parallel to three distinct planes (I would be shocked if there was a better method otherwise). To find the number of ways to split an nxnxn cube into n 1xnxn sheets, you have to factor n. If n is a perfect square you can use sqrt(n) - 1 cuts, then stack and use another sqrt(n) - 1 cuts, but it gets much more difficult when n is prime or composite, but not perfect square. In any case, look for a pattern! Maybe you can even induct on n.
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