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The Amman chair is an interesting shape that can be dissected in two pieces that are smaller copies of the original. The sizes of the two pieces are different. The ratio between the areas of the pieces is the golden ratio.

Because of this ratio, if you repeat the process of dissecting all the larger pieces, you always get a dissection with pieces of only two different sizes. The number of pieces follows the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, ...

Below is the result of level 0 to level 7 of the process of repeating the dissection.

Set of eight dissections

It would be cool to laser-cut for instance the last picture (level 7) and make a puzzle out of it. Right? It would be even more cool if the solution is in fact unique for the given set of pieces. Right again?

So the question for you is: is the solution unique for the examples shown above? If yes, is it unique for larger levels? How far?

If the solution is always unique, provide a convincing explanation why it should be so. If the solution is not always unique, show the smallest counter-example, an alternate solution to the solutions given by the process above.

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    $\begingroup$ It's not 100% clear which of two different questions you're asking. (1) "Is there a unique way to divide the shape into 13 larger and 21 smaller pieces the same shape as the original, their sizes being in golden-ratio proportions?" (2) "Once you've got those pieces as in the dissection here, is there a unique way to fit them together to make the original shape?" (Equivalently: is the puzzle unique? and: Is the solution unique?) $\endgroup$
    – Gareth McCaughan
    Mar 14, 2021 at 20:27
  • $\begingroup$ I added "puzzle-creation" because indeed you could laser-cut it and make a puzzle. $\endgroup$
    – Florian F
    Mar 14, 2021 at 20:28
  • $\begingroup$ If you start with a rectangle of $1:\sqrt{2}$ side length ratio (say, an A4 sheet of paper) and keep halving the largest pieces, you can also make pieces that are the same shape as the original, and you always get pieces of only one size :-) $\endgroup$
    – Bass
    Mar 14, 2021 at 20:28
  • $\begingroup$ Yes, the standard european paper sizes have this property. That is why I added that here the sizes are different. Btw, I edited the problem to clarify Gareth's question. $\endgroup$
    – Florian F
    Mar 14, 2021 at 20:31
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    $\begingroup$ Each dissection of a level 0 chair into two Level 1 chairs is unique. This is easily observed by actually cutting out either of the the smaller chairs and trying to fit it into the larger in any way that leaves the other chair. Therefore every subsequent dissection is unique. However once you have reached some level, the 'other' question becomes germane: Can you rearrange say the 34 chairs of level 7 in a different way. I would normally launch my solver at this after 'squeezing' the chairs in one dimension to make them rational (and integer) sided, I haven't worked out a way of doing that. $\endgroup$ Mar 16, 2021 at 5:00

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I stumbled over this old question.

So here is the answer.

Do all levels have a unique solution?

All levels shown have a unique solution. It is not possible to rearrange the pieces differently.

This is the result of a computer search.

What about the next levels?

Up to 55 pieces the solution is unique. But with 89 pieces you have a whooping 14962 solutions.
Here is one. You can see it is not the standard solution that you can split in 2 smaller chairs.

enter image description here

Note that in the solver the powers of phi have been approximated with Fibonacci numbers. It works because Fibonacci numbers obey the same addition rules as the powers of phi: $\phi^{n} = \phi^{n-1} + \phi^{n-2}$.

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