There's an interesting trick you can do with an $11 \times 11$ square...
It's possible to make five rectangles, each with different widths and lengths, using each of the following dimensions once only: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, that can be used to tile the $11 \times 11$ square!
Convince yourself that only one of the arrangements below could satisfy these conditions.
Can you find two different solutions (not just reflections or rotations)?
With thanks to Don Steward, whose ideas formed the basis of this problem.