Mathcounts Cutting A Larger Cube into Smaller Cubes

Question

What is the smallest number of cuts required to create 64 unit cubes from a 4 by 4 by 4 unit block of wood?

I thought that maybe we could make 3 cuts in the x, y, and z direction, but that would be the wrong answer of 9. The correct answer is 6. Does anyone have a solution for the answer?

Answer 1

HINT

Notice that the Volume of the cube is $4^3=64$ so your cube is essentially comprised of $64$ cubes of volume $1$.

Assume that we have made the required partition. Focus on one side of the cube-a square with sides equal to $4$. Observe that any such partition must consist of an equal number of cubes in each row and each column.

Now picture a tic-tac-toe table and the least number of pencil strokes required to create it. Finally say we add a couple of lines to the table, one vertical one horizontal-now how many pencil strokes (cuts) are needed?

Answer 2

Think about cutting the cube in half in any dimension ($x$ in my example). Now you have $2$ blocks of dimension $2\times4\times 4$.

The trick is to rearrange the cubes so we can cut both of them in half at the same time, now in a different direction ($y$ for instance)

Now we have $4$ blocks of dimension $2\times 2 \times 4$. Rearrange the cubes again to cut all at the same time to knock down the $z$ dimension.

Now we have used three cuts and have $8$ cubes of side length $2$. Keep this pattern going with three more cuts and we see that $6$ cuts can be used to create $64$ unit cubes.