# Q-Puzzle on $\mathbb F_7$

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 $$\mathbb F_{17}$$?

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.

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

I get the picture

A toucan

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.

• 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 ;) – JKHA May 12 '20 at 18:35
• I decided to change the puzzle's aim tanks to your answer ;) – JKHA May 12 '20 at 18:37
• 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). – trolley813 May 12 '20 at 18:53
• Ah thanks. I was trying to be stealthy. :) – MacGyver88 May 12 '20 at 18:56
• Alright, I understand the issue. I've decided to select your answer as first answer to original puzzle :) – 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:

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.

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