# Five graded difficulty isosceles right triangle into square tilings

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:

• I like these riddles :D Apr 22 '18 at 3:31
• 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. Apr 24 '18 at 12:20

## 2 Answers

Here are the solutions to the five problems.

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.

• Yup all the same as mine, up to R&R. I'll post my pix for completeness. May 22 '18 at 4:47

Just for completeness, to see all solutions in my 'style':