# Unfair tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of $1:2$.

Select two sets of rectangles from the list:

1, 2, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24

Some of the rectangles will be in both squares, no rectangle will be repeated in one square. Every rectangle will be used at least once. No gaps or overlaps, one tiling is unique and the other has a sub-rectangle that can be flipped. Only the short side of each rectangle is listed.

This one earns the difficulty rating 'unfair'. Additional clue: One square uses $11$ rectangles, one uses $13$. Not the same as "Hard tiling puzzle"

• Possible duplicate of Hard tiling puzzle Commented Jan 16, 2018 at 6:01
• No, it's not the same. It's significantly different, different sized squares, different rectangle lists, different tilings. Commented Jan 16, 2018 at 8:28

After a couple weeks of free-time programming I've found a solution! Essentially my method was:

1. Find a valid square size.
2. Pick a subset A
3. Find all possible side subsets of A
4. Find all possible corners (2 sides) from those sides
5. Find all possible borders from those corners
6. For each border, try to fill the rest in with what's left

I'd be more than happy to upload my code somewhere if anyone is interested and I'd also be interested in checking out anyone else's programs to see how they solved it! It was a pretty fun puzzle to solve, thanks theonetruepath!

Square Size:

64

Size 11 set:

2, 4, 8, 12, 13, 14, 15, 16, 17, 18, 19

Top (left to right): 15, 17
Right (top to bottom): 17, 19, 14
Bottom (left to right): 18, 16, 14
Left (top to bottom): 15, 13, 18
Inner (top most, left most first): 2, 4, 8, 12

Size 13 set:

1, 2, 4, 6, 7, 8, 9, 12, 14, 15, 16, 20, 24

Top (left to right): 15, 8, 9, 16
Right (top to bottom): 16, 24
Bottom (left to right): 20, 24
Left (top to bottom): 15, 14, 20
Inner (top most, left most first): 7, 1, 4, 6, 2, 12

• Good job. My program is probably less efficient than yours, I treat each rectangle as a list of coordinates which works well for small polyominoes but gets out of control with large rectangles. I just throw a bigger CPU at it... Commented Feb 1, 2018 at 2:55