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 the dimensions of the five rectangles which can tile the $11 \times 11$ square?
Can you find all the possible different solutions?
Here different means not a reflection or rotation of another solution.
With thanks to Don Steward, whose ideas formed the basis of this problem.