# 20 right isosceles triangles into a square

Five graded difficulty isosceles right triangle into square tilings

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

The challenge is to fit the smallest $20$ integer-sided scaled right isosceles triangles into a square with diagonal $46$

For convenience, I list the areas of the triangles:

$1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, 121, 128, 144$

There are four ways of doing it (not two as originally posted). A brave person it would be who tackled this by hand.

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: ## 2 Answers

Here are at least two solutions (up to reflection and rotation) The trapezoid outlined in red can be flipped. This is presumably the other "substantially different" solution (not counting the above trapezoid flip as different).

• Your second solution is the same as one of mine. Your first is not... which means my software butchered something. I'm re-running it finding all 32 solutions, ie I turned off the code that rejects R&R solutions. Originally it found 4 solutions, ie rejected 7 out of every 8. But two of them were reflections. There should be four solutions, I'll have to ID them by hand. Meanwhile I'm awarding your answer... – theonetruepath May 23 '18 at 8:49
• OK my R&R code needs fixing for triangle solving... there are 4 solutions, my two and your three (with an overlap of 1). I'll post the full set in my graphic style for completeness. – theonetruepath May 23 '18 at 10:52

Here are the four solutions. In the first diagram, there is a flippable area bottom centre so two solutions from that, as noted by @Phenomist: