A problem from Cambridge mathematician J.E. Littlewood’s Miscellany (1953):
Is it possible to pack a cube with a finite number of smaller cubes, no two of which are the same size?
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No. Think of the large cube as a square well with a flat bottom. If the task is possible, then this floor is tiled perfectly with cubes of different sizes. Consider the smallest of these cubes. This can’t adjoin an edge of the floor, because in that position it would be impossible to surround it with cubes larger than itself. So it lies somewhere on the expanse of the floor, surrounded by larger cubes.
But now we’re back where we started. The top of this small cube forms the floor of a square well, and all the reasoning above applies again. And so on forever: At each stage we’ll be forced to supply smaller cubes, and our task will never be finished.
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