Fifty-three bricks in a six-by-six-by-six box

Question

A cubical box has inside dimensions 6 by 6 by 6. You have bricks measuring 1 by 1 by 4, to be placed with their faces parallel to the box's faces, on the unit grid. The box has volume 216 and each brick has volume 4, so 54 bricks would exactly fill it if volume were the only issue. Can you fit 53 bricks into the box? If not, what is the maximum, and why?

Accepted Answer

No. At most 52 bricks fit. The colouring. Cut the 6 by 6 by 6 box into 27 blocks of size 2 by 2 by 2, arranged 3 by 3 by 3. Colour the blocks in a checkerboard pattern: a block is white if the sum of its three block-coordinates (each 0, 1 or 2) is even, and black if odd. There are 14 white blocks and 13 black blocks, so 14 × 8 = 112 white cells and 13 × 8 = 104 black cells. What a brick covers. A 1 by 1 by 4 brick runs along one axis and occupies four consecutive unit cells in that direction. Along any axis, the blocks alternate in colour every two cells, so four consecutive cells always cons…

Fifty-three bricks in a six-by-six-by-six box

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