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(Similar to the recent 3:1 and my 3:2 rectangle question)

Tile a square completely with rectangles which have aspect ratio 1:2, integral side lengths and all different sizes. In other words selected from 1x2, 2x4, 3x6 etc. Usual tiling rules apply: No gaps, no overlaps.

  1. Find the smallest area tiling.
  2. Find the smallest area tiling with with the fewest rectangles.

The answer to (1) is not the same as the answer to (2).

Tagged as no-computers, but probably should be tagged computer-puzzle as well. Also tagged as logical-deduction.

I found these by computer, but a Google search turned up a paper that answers both parts and proves (2) minimal.

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  • $\begingroup$ I don't get it. What's stopping me from applying 2 of these rectangles in the same orientation to make a square? $\endgroup$ – tuskiomi Jan 13 '18 at 17:48
  • $\begingroup$ 'The 'all different size' part plus the bit I left out, 'excluding the trivial case"... I'll add it to the question. $\endgroup$ – theonetruepath Jan 13 '18 at 19:12
  • $\begingroup$ Scratch that.... you only need to exclude that trivial case when tiling rectangles with these... you can't make a square that way. So just the 'all different sizes' part is the answer to your question. $\endgroup$ – theonetruepath Jan 13 '18 at 19:51
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Using the program from my previous answer, I found the following solution for part 1:

solution

Size: 34; Rectangles: 10 (again)

The best I found for part 2 is so far:

solution

Size: 36; Rectangles: 9

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  • $\begingroup$ Coulda sworn I tagged it "no-computer", maybe someone edited it, I changed it...Ah it was me. Ah well. I should tag it both. $\endgroup$ – theonetruepath Jan 13 '18 at 9:50
  • $\begingroup$ Your part 2 is minimal, too. Here's the proof, by Jepsen: ac.els-cdn.com/0012365X9400267M/… $\endgroup$ – theonetruepath Jan 13 '18 at 9:54

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