3650.13 – A Solid Cube of Bricks?

Can a solid cube 1 foot on a side be constructed from bricks measuring $2 \ \inch \times 4 \ \inch \times 8 \ \inch$ ? Explain your answer completely.

Solution

The cube has volume $12^3 = 1728$ cubic inches; a single brick has volume $2 \times 4 \times 8 = 64 \ \inch^3$ . As $64$ is a factor of $1728 = 64 \times 27$ , the construction of the proposed cube is theoretically possible. It would use 27 bricks.

And that is as much as we currently know. Can you carry this further? Perhaps this problem calls for making a model!