# 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 – NL628 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. – nickgard 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. – theonetruepath May 22 '18 at 4:47