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Ref: https://en.wikipedia.org/wiki/Squaring_the_square

This is a planar version of the question at Cubing the cube - as 'perfectly' as possible.

Question: How does one cut a square into the least number of squares that are all mutually non-congruent - except that 2 of the smaller squares are allowed to be congruent? One would hope that the number of square pieces would less than 21 when a pair of duplicates are allowed (21 being the least number of squares in a perfect squaring of the square, as discovered by Duijvestijn).

Note: It seems one can replace "2" above with "3" or "2 pairs" to generate other questions (but not "4" because that will trivialize the problem).

A possibly related question: Which is the rectangle that can be perfectly squared (cut into mutually non-congruent squares) with the least number of square pieces? This must be known stuff; am not able to locate a reference.

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    $\begingroup$ The fewest pieces in a squared rectangle is nine. It's given at squaring.net/sq/sr/spsr/o9/order9_spsr.html and much info can be found at other parts of the site squaring.net/sq/tws.html $\endgroup$
    – Gerry Myerson
    Commented May 13 at 0:18
  • $\begingroup$ Thanks very much! that settles the 'auxiliary question'. The idea was if a perfectly squared rectangle of dimensions in ratio 2:1 ( 3:2) could be found, one could attach 2 (3) squares both (all) of same size to it to form a perfectly squared square with 2 (3) squares identical. looking through that very long list, i haven't been able to find any such rectangle yet. But then, this need not be the only way to attack the main question! $\endgroup$
    – Nandakumar R
    Commented May 13 at 4:16
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    $\begingroup$ Here's a squared 2:1 rectangle; squaring.net/history_theory/gfx/o22-2x1-136-272.png $\endgroup$
    – Gerry Myerson
    Commented May 13 at 6:00
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    $\begingroup$ It looks like the proprietor of the website is Stuart Anderson; stuart.errol.anderson AT gmail.com – maybe it's worth contacting him with your questions? $\endgroup$
    – Gerry Myerson
    Commented May 13 at 8:49
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    $\begingroup$ Yes I did. No word from there $\endgroup$
    – Nandakumar R
    Commented Jun 17 at 15:23

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