We are looking for 4 categories.

32 different flags

Other (independently solvable) puzzles of this type: 2, 3, 4, 5, 6, 7, 8, 9, 10.

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    $\begingroup$ Some of those definitely aren't real flags :-D $\endgroup$ – Rand al'Thor Feb 5 at 9:32
  • $\begingroup$ Is the space after the first column of flags intended? $\endgroup$ – Edivad Feb 5 at 9:41
  • $\begingroup$ The premise is a bit vague >.> are we supposed to identify different vexillological characteristics? $\endgroup$ – Adib Feb 5 at 10:09
  • $\begingroup$ @Evidad Yes it is. $\endgroup$ – sarsaparilla Feb 5 at 10:23
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    $\begingroup$ @Adib I guess part of the puzzle is finding out exactly what to do to solve it (although I have included a hint). $\endgroup$ – sarsaparilla Feb 5 at 10:26

The four groups of flags (and their connections) are as follows...

NB This was solved independently of @Adib's answer, but their working was very useful for corroborating with my own afterwards, as - before a couple of corrections thanks to @Jens in comments - we initially agreed on 86% of the flags (yes, I worked out the percentage!).

Note that in each case:

The OP has provided us with a starting flag in its usual colours. The 7 flag designs that follow it are then a sequence of pairwise mergings of 8 flags, the first of which is the given starting flag, and all of whose countries satisfy a certain theme. I don't profess to understand anything technical behind the merges but I assume like others suggest that there is some algebra going on with the RGB codes. (My key approach to solving was to note that where one merged flag contains white, the other flag's colour shows in the merged image; where both merged flags are white, the merged image shows black.)

Group 1:

Belgium, France, Russia, Croatia, England, Sweden, Uruguay, Brazil.

These are the quarter finalists at the men's football World Cup in 2018.

Group 2:

Mali, Ukraine, Poland, Hungary, Sudan, Iran, Niger, Algeria.

All of these countries border exactly 7 other nations. I shall not list them here, for brevity's sake, but note that Algeria's total includes the disputed territory of Western Sahara as one of the seven.

Group 3:

UK, India, Russia, France, Pakistan, North Korea, China, USA.

These are all countries known to have nuclear weapons.

Group 4:

Uzbekistan, Estonia, Honduras, Taiwan, Georgia, Faroe Islands, Tunisia, Japan.

Each of these nations have a capital city beginning with 'T'. These are (respectively): Tashkent, Tallinn, Tegucigalpa, Taipei, Tblisi, Tórshavn, Tunis and Tokyo.

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  • $\begingroup$ France! Excellent observation! $\endgroup$ – Adib Feb 6 at 8:27
  • $\begingroup$ @Adib That was a surprisingly tricky one! In my case I spotted the connection between the others first and that helped me pinpoint it as the missing one... $\endgroup$ – Stiv Feb 6 at 8:34
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    $\begingroup$ If my understanding of the blending process is correct, the flag next to Mali is that of rot13(Hxenvar). $\endgroup$ – Jens Feb 6 at 18:15
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    $\begingroup$ Also, it is not Palestine, but rot13(Fhqna). $\endgroup$ – Jens Feb 6 at 18:49
  • $\begingroup$ @Jens I'd been toying with that Palestine alternative all afternoon with no success! However, that suggestion you make in your other comment is one I hadn't even considered! I'll have a play... :) $\endgroup$ – Stiv Feb 6 at 19:04

Partial attempt:

Each flag is paired except at the right hand side of the chart where it appears once.
Otherwise each flag appears in adjacent images so that two flags are superimposed.
I have several possibilities for the flags left blank (e.g. outlined cross, three stripes).
So far, the arrangement I have is like this:

enter image description here

I can't figure out how the colour blending works. It looks as if the r-g-b colour components are exclusive-ORed between two flags sharing the same image. But if you look at the first two images on the bottom row, that would mean a flag has two horizontal bands in height ratio of 1:2 and I can't find a flag like that. It looks as if there might be a different rule in various situations.

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  • 1
    $\begingroup$ You've misused Japan where you meant Tunisia. $\endgroup$ – Ian MacDonald Feb 5 at 15:00
  • $\begingroup$ @IanMacDonald thanks, I have corrected that - I had them the wrong way round. $\endgroup$ – Weather Vane Feb 5 at 15:05
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    $\begingroup$ @Adib the fragment of motif near the top left corner seems to be part of a Maltese cross and I can not find any other flag it might be a part of. If you look at a full size download of the image posted, you can detect the outline of the three other crosses of the Georgia flag overlaid on the flag of Rot13(Gnvjna) and on the next flag which I could not identify. $\endgroup$ – Weather Vane Feb 5 at 21:42
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    $\begingroup$ @Adib perhaps the "four categories" are Rot13(naq be abg kbe). $\endgroup$ – Weather Vane Feb 5 at 23:35
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    $\begingroup$ The blending seems to work like this: 1. Take the RGB values of the flag you know. 2. Subtract the RGB values of the blended flag to its right. 3. If the subtraction gives a negative value, add 256. 4. The resulting RGB values give the colors of the flag that was blended with the known flag. As an example, the flag blended with Uzbekistan is the Estonian flag. $\endgroup$ – Jens Feb 6 at 6:03

Partial Attempt:

My flag map:

Row 1: Belgium,Romania,(UNKNOWN),Russia,Croatia,England,Sweden,Uruguay,Brazil
Row 2: Mali,(UNKNOWN),Poland,Hungary,Kuwait/Palestine,Iran,Niger,Algeria
Row 3: UK,India,(UNKNOWN MONO/TRI-COLOR),Pakistan,N.Korea,China,US
Row 4: Uzbekistan,(UNKNOWN TRI-COLOR),Honduras,Taiwan,Georgia,Faroe Island,Tunisia,Japan

I think the first category is..

Color inversion. After inverting the colors, I noticed certain flag colors pop up. For example, Honduras, Hungary, Faroe Island, Georgia, Pakistan, Brazil - they all have their correct color. Inverted

The methods I've tried so far is using OpenCV2 to experiment on different color conversions, and one of the flags that was 'hidden' stood out very well when using color conversion from RGB to HSV (cv2.cvtColor(image, cv2.COLOR_BGR2YUV)).

Here's what I mean...

enter image description here

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