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Problem 388

Combinatorics Difficulty 2.6 Multiple choice CEMC Pascal · Canada · 2012

The entire exterior of a solid $6\times 6\times 3$ rectangular prism is painted. Then, the prism is cut into $1\times 1\times 1$ cubes. How many of these cubes have no painted faces?

Pick one

Official solution

We visualize the solid as a rectangular prism with length $6$ , width $6$ and height $3$ . In other words, we can picture the solid as three $6 \times 6$ squares stacked on top of each other.

Since the entire exterior of the solid is painted, then each cube in the top layer and each cube in the bottom layer has paint on it, so we can remove these.

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This leaves the middle $6 \times 6$ layer of cubes.

Each cube around the perimeter of this square has paint on it, so it is only the “middle" cubes from this layer that have no paint on them.

These middle cubes form a $4 \times 4$ square, and so there are 16 cubes with no paint on them.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.