Cutting a 4 by 4 by 4 cube when you may restack

Question

A wooden cube measures 4 inches on each side. A carpenter wants to cut it into 64 one-inch cubes using a saw that makes straight, planar cuts all the way through whatever is in front of it. Between cuts the carpenter may rearrange and stack the pieces however she likes, so that one cut can pass through several pieces. Without restacking, 3 + 3 + 3 = 9 cuts are needed. What is the smallest number of cuts with restacking allowed, and why can it not be beaten?

Accepted Answer

Six cuts: two in each of the three directions. How to do it. Cut the cube in half (2 slabs of 2 by 4 by 4). Stack the two slabs and cut through both at once, producing 4 slabs of 1 by 4 by 4: two cuts have finished one direction. Now stack all the slabs and cut them in half the second way, restack and cut again: 4 by 4 columns of 1 by 1 by 4 after two more cuts. Finally stack the 16 sticks and cut them in half, restack the halves and cut again: 64 unit cubes after two more cuts. Six in total. Why six cannot be beaten. Focus on one direction, say front to back. The cube is 4 inches deep and th…

Cutting a 4 by 4 by 4 cube when you may restack

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