document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #335001 by richard1234(7193) $\"\"$ $\"About$

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If you cannot stack, the number of cuts would be 3(n-1), n-1 cuts in any direction.\r
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\n" ); document.write( "\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

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