The problem of squaring the circle is the archetype of a hard mathematical problem, and we know it is impossible. But what about what is commonly called squaring the square, that is, the dissection of a square into smaller squares of different sizes? If the condition on different sizes is not required, the answer is immediate: yes, it is always possible to cut a square into squares whose side is a fraction of the side of the original square, as in a grid.
A trivial dissection of a square into squares.
Two conjectures
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In 1934, Paul Erdős was in residence at the University of Manchester. During a visit to Cambridge, he put forward two conjectures:
