Cube2

[cinelli]

A carpenter has a wooden cube 3 inches on each side, which he wishes to cut into 27 one-inch cubes. He can do it by making six cuts through the cube, keeping the pieces together in the cube shape. Can he reduce the number of necessary cuts by rearranging the pieces after each cut? If so how, if not why not?

Cinelli

[kiloran]

No, can't be done, unless he has a saw which is infinitesimally thin.

--kiloran (smart alec since he was born)

[Gengulphus]

A carpenter has a wooden cube 3 inches on each side, which he wishes to cut into 27 one-inch cubes. He can do it by making six cuts through the cube, keeping the pieces together in the cube shape. Can he reduce the number of necessary cuts by rearranging the pieces after each cut? If so how, if not why not?

Spoiler, assuming an infinitesimally thin saw...

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

No, he cannot. To show that's the case, look at the three dimensions of the largest piece remaining after each saw cut. Each of them is in the range 1-3, only one of them can be reduced by the next saw cut, and it can only be reduced by 1. So the sum of those three dimensions starts at 3+3+3 = 9 after no saw cuts, can be reduced by at most 1 by each saw cut, and must end at 1+1+1 = 3. Therefore at least 9-3 = 6 saw cuts are needed.

If it were a 4x4x4 cube, he would be able to reduce the obvious 9 saw cuts to 6 by first sawing the cube in half, then stacking the two resulting half-cubes to saw them both in half with the same cut, and repeating for the other two dimensions, and something similar goes for all bigger cubes. But not for the 3x3x3 cube.

Gengulphus

[argoal]

I came to the answer by a different line of thought.

One of the 27 cubed that will result is the one in the very center of the 3 x 3 cube and it needs all 6 of its faces to be cut to expose them as none are currently exposed.

There is physically no possible way for a single cut to expose more than one of this cube's faces.

Therefore 6 cuts is the minimum required.

[Gengulphus]

Yes, a more elegant solution than mine!

Gengulphus

[cinelli]

There is some interesting analysis from Gengulphus but I agree with him that argoal's explanation is more elegant.

Cinelli