# Unlucky tiling: Arrange thirteen right isosceles triangles into a square

Link to next puzzle in this series: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

This problem has only one solution, and will be challenging by hand, but probably more satisfying. Computers allowed, probably challenging too. Not for the computer, but for the programmer.

The challenge is to arrange thirteen right isosceles triangles of the following areas into a $36\times 36$ square with no gaps or overlaps.

$2, 4, 8, 18, 32, 64, 72, 98, 128, 144, 196, 242, 288$

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: If you can solve this with scalene right triangles of the correct area... I'll accept that too.

• Do the triangles have to be isosceles? – NL628 Apr 21 '18 at 0:48
• Yep... I'll add that. Possibly it could be done with non-isosceles right triangles... much harder though. I'll accept an answer with non-isosceles right triangles! – theonetruepath Apr 21 '18 at 3:20
• It's my puzzle. I found it with my tiling program, there's only one solution, the proof is "brute force". – theonetruepath Apr 21 '18 at 7:57

## 2 Answers

Solution:

Based on the size of the pieces and the board, I saw that the area 288, 144 and 72 pieces would fit together nicely along one side, so I assumed this would be a correct placement. A solver filled in the rest to confirm my assumption.

• May I ask what solver? – NL628 Apr 21 '18 at 20:48
• Well done, that's it. I'll post my pic just for completeness. – theonetruepath Apr 22 '18 at 0:53

Answered by @nickgard, this is my pic

• +1 for a fun puzzle, but I confess I don't see the point of you posting a pic which is just a reflection of the answer @nickgard found. – IanF1 Apr 24 '18 at 20:17
• Not much point, but as I'm posting a series of related puzzles I thought it would be good to have the solutions all posted in the same style. Doesn't do any harm does it? – theonetruepath Apr 25 '18 at 2:09