Like Plato's Meno, this small structure illustrates a very clear geometric concept: doubling an area does not mean doubling a side.
The double square
Riddle statement
You have a square drawn on a sheet. Someone challenges you to draw another square that has exactly twice the area, but with one condition: you cannot measure lengths or use formulas.
How can you build it using only the square itself as a guide?
Show solution
Solution
Answer: use the diagonal of the original square as the side of the new square.
Draw a diagonal on the initial square, joining two opposite vertices. Then build a new square taking that diagonal as one of its sides.
Why does it work? The diagonal divides the original square into two equal right triangles. The square constructed on that diagonal can be divided, in turn, into four triangles equal to those.
The original square contains two of those triangles; the new one contains four. Therefore, the new square has exactly twice the area.