Henry Ernest Dudeney/Modern Puzzles/145 - A Cube Paradox/Solution

by : $145$

 * A Cube Paradox
 * I had two solid cubes of lead, one very slightly larger than the other.
 * Through one of them I cut a hole (without destroying the continuity of the four sides)
 * so that the other cube could be passed right through it.
 * On weighing them afterwards it was found that the larger cube was still the heavier of the two.
 * How was this possible?

Solution
The smaller cube is the one which has had the hole made in it.

Proof
From Prince Rupert's Cube, the largest tunnel that can be made in a cube has a side of length $1.06+$ the length of the edge of the cube.

So the larger cube has an edge whose length greater than $1$ and less than $1.06$ times the length of the edge of the smaller cube.

Then the smaller cube has the hole made in it, through which the larger passes.

And, of course, the larger cube is still the heavier of the two.

Also see

 * Prince Rupert's Cube