Bricks in a box
What is the largest number of bricks that can be packed into a box, with the faces of the bricks parallel to the faces of the box?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The volume allows 54 bricks. A colouring argument rules some of them out.
Cut the box into 27 blocks of and colour the blocks like a three-dimensional chessboard.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Colour the box so that every brick covers the same number of cells of each colour, whatever its position. Then the number of bricks is limited by the smaller colour class.
Step 1: the colouring. Cut the box into blocks of size , and colour them black and white alternately, like a chessboard in three dimensions. There are 14 blocks of one colour and 13 of the other, so cells of one colour and of the other.
Step 2: each brick is balanced. A brick runs along one direction across 4 consecutive cells. Along that direction, the block colour changes every 2 cells, so the brick covers 2 cells of each colour, wherever it lies.
Step 3: the bound. All bricks together cover at most 104 cells of the smaller colour, 2 per brick, so there are at most bricks.
Step 4: 52 is reached. A packing of 52 bricks exists (one can be found by a short computer search, and it leaves 8 cells empty). So the answer is 52.
Usually asked as “Can you fit 53 bricks?” The volume says yes (it allows 54); the colouring says no.