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Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square

Two difficult "Seventeen right isosceles triangles into a square" tilings

V.hard problem, 20 right isosceles triangles into a square

Each tiling has only one solution, the smaller lists will be easier but all should be possible by hand, computers allowed.

The five challenges are to arrange $7, 13, 14, 15, 16$ right isosceles triangles of the listed areas into a square of area $882$ with no gaps or overlaps. The square has a diagonal of length $42$.

$7:$ $9, 18, 36, 72, 144, 162, 441$

$13:$ $16, 18, 25, 32, 36, 50, 64, 72, 81, 98, 100, 128, 162$

$14:$ $1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 64, 98, 128, 441$

$15:$ $1, 2, 4, 8, 16, 25, 32, 36, 50, 64, 72, 98, 121, 128, 225$

$16:$ $1, 2, 4, 8, 16, 18, 25, 32, 49, 50, 64, 81, 98, 128, 144, 162$

The answer tick will be given to whomever posts the greatest number of "placed triangles" in completed puzzles first. In the unlikely event of a tie, the solver that got the highest scoring single puzzle wins.

By way of illustration/clarification, here are the right isosceles triangles of area

$1, 2, 4, 9, 16, 18, 50$

arranged into a $10\times 10$ square:

10x10_7

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  • $\begingroup$ I like these riddles :D $\endgroup$
    – NL628
    Apr 22, 2018 at 3:31
  • $\begingroup$ I like these tiling problems too. I'm trying to think of an approach to these that's better than just tapping them into PolySolver and hoping to be lucky. $\endgroup$
    – nickgard
    Apr 24, 2018 at 12:20

2 Answers 2

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Here are the solutions to the five problems.

enter image description here
I was able to find by hand the first three solutions. The first one, in particular, can be downsized by a factor of 3 in all dimensions to simplify work. The last two I used PolySolver to help. The general methodology is to stack several triangles, often doubling in area, together.

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  • $\begingroup$ Yup all the same as mine, up to R&R. I'll post my pix for completeness. $\endgroup$
    – theonetruepath
    May 22, 2018 at 4:47
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Just for completeness, to see all solutions in my 'style':

7 13 14 15 16

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