There are numerous squared squares known to exist, and since there is no limit to how many squares one can use in constructing one, it seems likely that there are infinitely many solutions, although finding them seems rather daunting. The proof that there are infinitely many solutions follows:

Construct any known squared square. Find its smallest sub-square. Replace just the smallest sub-square with a tiny squared square of the same pattern. You have created a new squared square composed of 2n-1 squares (where n is the number of squares in the first. Repeat recursively ad infinitum always replacing the tiniest square. Physicists might balk about the Planck length, but mathematicians have never been limited by reality.

The degree of infinity is startling to consider too. One need not replace the smallest square to generate a new unique squared square. Any of the squares will probably do. To be successful, one must not accidentally create a second instance of some square size. If the dimension of the replaced square is not a multiple of any of the other squares, then the resulting squares cannot be repeats.

One need not limit oneself to iterations of the same original pattern either. You could substitute a second pattern, or choose 10 different squared squares to represent each digit, then choose the corresponding pattern for the new iteration by the next digit of your favorite irrational number! Ow! My brain is sore.

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