# Hard tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of 1:2.

Select two sets of rectangles from the list:

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 16, 17, 18, 21, 23, 31

Some of the rectangles will be in both squares, no rectangle will be repeated in one square. Every rectangle will be used at least once. No gaps or overlaps, one tiling is unique and the other has a sub-rectangle that can be flipped. Only the short side of each rectangle is listed.

This time I think it's only fair to allow computers. However good hand solvers will probably still be able to hand solve this.

• What do you mean "two sets of rectangles from the list"? are those side lengths of one side of the rectangle? if so, which side? Are we supposed to find matching numbers to fit the 1:2 ratio? – Riker Jan 14 '18 at 20:16
• @Riker I clarified it, just the short sides are listed. – theonetruepath Jan 15 '18 at 0:06
• Do we have to use all the rectangles at least once? – dpwilson Jan 16 '18 at 13:45
• Yes. Good point, added. – theonetruepath Jan 16 '18 at 19:40

I think I have found a solution.

Set 1

$\{1, 3, 4, 6, 7, 10, 11, 12, 14, 17, 31\}$

Square 1

Set 2

$\{2, 3, 4, 5, 7, 10, 13, 16, 18, 21, 23\}$

Square 2

Both squares have side length

$62$