Yoak: Cut The Cube
Here’s a classic from Martin Gardner:
Suppose that you have a 3″ on a side wooden cube and a buzz saw. You wish to cut the cube into 27 smaller cubes, each 1 cubic inch. It is easy to see that you can do this with six cuts. You simply hold the cube in its original position while making two cuts that trisect each face.
Can it be done if fewer cuts? If so, tell us how. If not, prove that it can’t be done.
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Blaine said,
July 5, 2009 at 7:52 pm
I assume you mean by moving things around, restacking layers so you can cut through prior layers, etc. Seems like there might be some way to do it, right?
[spoiler]Without thinking through all the possible dissections, let’s consider the “inner” cube. No matter how you slice things, that one cube has no existing cut faces and will require you to make individual cuts for each of its 6 new faces.
Hence, it can’t be done with less than 6 cuts.[/spoiler]
jyoak said,
July 7, 2009 at 2:02 am
Blaine, that’s exactly what I had in mind, and is more or less what Gardner suggested.
[spoiler]He suggested a visualization where you paint the outside of the cube before cutting. Of the final 27 cubes, some will have three painted faces, some two, etc. Painted faces are “free” as they already existed but each of the non-painted faces required a cut to accomplish. As you point out, the inner cube has six unpainted faces.[/spoiler]
Andy said,
July 15, 2009 at 10:08 pm
Here is a “graph theory” solution (which easily generalizes to cutting a cube into a different number of cubelets).
Represent the cube with the 3x3x3 integer lattice, with edges between adjacent points. Now it’s clear that to separate the points we have to cut all the edges. The 9 corner points have degree 3, the 12 edge points have degree 4, the 6 face points have degree 5 and the center has degree 6. So there are (36 + 48 + 30 +6)/2 = 54 edges.
If we only allow plane cuts, then by inspection it’s easy to see that we can cut at most 9 edges at once. So 54/9 = 6 cuts are required.
Note though that if we can cut weird shapes then you can do this in fewer cuts (presumably this is not allowed by the “buzzsaw”).
Mango said,
August 24, 2009 at 10:12 am
Just look at the center cube. Regardless how you arrange the parts that accumulate during the process you can not cut two of its sides in a single cut.