Puzzle: perfectly dissecting a cube

This puzzle comes from the first chapter of Littlewood’s Miscellany (J. E. Littlewood, ed. B. Bolobás, CUP, 1986), which is entitled “Mathematics with minimum ‘raw materials'”. Littlewood quotes a very neat solution which is worth trying to discover for yourself…

Finding a perfect dissection of a square means dividing it into a finite number of smaller squares, all of different sizes. (Perhaps surprisingly, this problem can be solved in an infinite number of ways.) Analogously, finding a perfect dissection of a cube means dividing it into a finite number of smaller cubes, all of different sizes. It turns out that there is no way to solve this problem: the puzzle is to prove this fact!

A hint is given below.

We’ll suppose that a solution does exist, and try to deduce a contradiction. The trick is to start by considering what the bottom face of the dissected cube must look like, and then to think about the smallest cube in that face…

(DP)

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One Response to Puzzle: perfectly dissecting a cube

  1. Pingback: Solution to the cube dissection puzzle | Degree of Freedom

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