My high school math student had to solve this puzzle for his homework. I made it harder by not telling you what is the definition of $\mathbb{F}_7$. Neither why letter Q is used and which of his chapter this is.

Your aim is to reconstitute the original image and discover the pattern. Would you use your technique for any random picture and $\mathbb F_{17}$?

enter image description here

Note: the LaTeX mathbb font and color (blue) are different from the ones in puzzling.se, but that's not part of the puzzle (it was a URL link in the image).

Note bis: while it could be easier to solve it thanks to computers, I will favor no computers solutions and math computations. That's why no-computers tag for this puzzle.

  • $\begingroup$ While many image stitching algorithms require partial overlap, I suspect numerical methods could get you most if not all the way to the reconstruction. $\endgroup$ May 12 '20 at 17:18
  • $\begingroup$ @Galen You made me add the no-computers tag :) $\endgroup$
    – JKHA
    May 12 '20 at 17:22

I get the picture

A toucan

enter image description here

I think the maths connection relates to the Farey sequence of order $7$.

Some sequence can be see in the skip-1 vertical sequence in the original arrangement which translates to horizontal placings in the solved picture.

Thanks to @MacGyver88 for pointing the way.

  • $\begingroup$ Well! That's a correct no-computers answer. However, would you use your technique if it was $\mathbb F_{17}$? You still get my +1 ;) $\endgroup$
    – JKHA
    May 12 '20 at 18:35
  • $\begingroup$ I decided to change the puzzle's aim tanks to your answer ;) $\endgroup$
    – JKHA
    May 12 '20 at 18:37
  • $\begingroup$ Of course, F7 is rot13(Tnybvf svyrq bs beqre 7 (v.r. svryq bs vagrtref zbqhyb 7)). So, the table may be rot13(n xvaq bs nqqvgvba/zhygvcyvpngvba gnoyr bire S7). $\endgroup$
    – trolley813
    May 12 '20 at 18:53
  • $\begingroup$ Ah thanks. I was trying to be stealthy. :) $\endgroup$
    – MacGyver88
    May 12 '20 at 18:56
  • 1
    $\begingroup$ Alright, I understand the issue. I've decided to select your answer as first answer to original puzzle :) $\endgroup$
    – JKHA
    May 12 '20 at 20:06

The subimages are permuted using an affine map in $\mathbb F_7^2$. I wrote a quick program that lets me adjust the affine map using keypresses until the image is correct, and this is the result:
enter image description here
You can read off the exact affine map by hand if you are so inclined, by comparing the positions of tiles that are supposed to be adjacent, but I figured this was more fun and also provided a more visual solution.

  • $\begingroup$ Congrats! That is the answer I wanted to be found ;) if I could give two selected answers, I would for yours $\endgroup$
    – JKHA
    May 13 '20 at 13:31
  • $\begingroup$ You can change your selection if you really want to do that. $\endgroup$
    – Magma
    May 13 '20 at 13:54
  • 1
    $\begingroup$ That would no be fair according to the comments on @WeatherVane's answer, that's partially my fault, I admit $\endgroup$
    – JKHA
    May 13 '20 at 16:23

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