Stubborn Wooden Block

Probability Level 3

Consider a $3 \times 3 \times 3$ wooden cube. We are using a knife to cut it straight into twenty seven $1 \times 1 \times 1$ cubes. Clearly, if we can use the knife only on one piece of wood at a time, regardless of its dimensions, then we need 27 cuts.

Now, If we are allowed to rearrange and stack up the pieces already cut so that a cut can go through multiple layers, what is the minimum number of cuts needed?

Bonus: How can we generalize this to an $n \times n \times n$ cube?

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1 solution

Jon Haussmann
Mar 27, 2017

It's easy to see that six cuts are sufficient.

Consider the unit cube at the center. No matter how the pieces are rearranged at any time, all six faces of the center unit cube must be cut separately. So six cuts is the minimum.

My thoughts regarding the general case,

If $n$ is prime then the number of cuts requires will be $3\times (n-1)$

If $n$ is composite the number of cuts will be $\sum 3(p-1)$ where, each $p$ is a prime factor of $n$

eg: if $n=6,(6=2\times 3)$ the minimum number of cuts will be $3(2-1)+3(3-1)=9$

Anirudh Sreekumar - 4 years, 2 months ago

Stubborn Wooden Block

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