Providing that you must use all twelve squares in each rectangle, and you must show the entire surface of each square, you can try the brute force method.

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For 12 squares:

1 x 12

2 x 6

3 x 4

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1 x 7 is unique

1 x 11 is unique

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The others have several possibilities, as in the 12 squares problem.

If you have a better plan, I would like to see it.

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The challenge now is to find an algorithm by which you can determine the number of rectangles possible given the number of squares, but without actually having to list each possibility.

Even more interesting would be writing a simple proof that such an algorithm is true, again without listing all the possible configurations for each problem.

For example, how many rectangles can be made from 4456 squares?

I would not want to go one by one on this, and it almost certainly is amenable to a reasonable algorithmic solution without the aid of software.

Please let me know if you have an answer.

Thank you.

Bye.