# Circling the rectangle: A strange IQ problem

I'm sure you're all familiar with pattern recognition puzzles where you have to find the next picture in a sequence. Well, I’m completely stumped on this one… Can you help?

There is a sequence of pictures in the top row of this image. Which of the four options in the row below completes the sequence by filling the box labelled with a question mark?

Which of the four options should replace the question mark?

Hint:

The graph in each square represents an object. What might this object be, and how might the colours be important?

This is my own problem, with a well-defined solution. It is an inherently visual puzzle; the colours of the dots are listed in a pastebin. Note the precise details of the lines (length/straight vs. curved) and dots (precise coordinates, RGB values) are not required; some of the presentation is aesthetic.

• Does your clarification mean that whether a vertex is on the circle or not is irrelevant to the answer? Commented Oct 13, 2021 at 13:17
• @GeorgeMenoutis Yes, you are correct in that circular and straight edges mean the same thing. A circular structure just turns out to be an efficient and aesthetically pleasing way of encoding graphs of the size and structure you see in these images. Once you realize the method by which the images are constructed (which will be a bit of an "aha!" moment for some, or at a glance perhaps for others), the images will make sense.
– Anon
Commented Oct 13, 2021 at 13:22

The sequence can be completed using image:

B - despite the fact this is the only one of the options that does not exclusively use red, white and blue nodes!

How come? Because each of these graphs represents:

the coloured sections of a national flag! The three already in the sequence are the flags of Faroe Islands, Liberia and Australia, where each individual block of colour is represented by a coloured node, with adjacent colour blocks connected by lines:

How should we complete the sequence?

First we need to identify each of the candidate flags. These are: Cuba, Guyana, Russia and Samoa...

Next, notice that the initial letters of the original images appear to be spelling out the thematic word FLAG - all that is missing is a flag beginning with the final letter, G. This corresponds to option B, Guyana!

Of course, this makes good sense of the title too, since:

Each of these flags is rectangular, and we are converting them into these artistically curved graphs!

• Awesome!!! Somehow I guessed you might be the one to get this!
– Anon
Commented Oct 15, 2021 at 8:13
• @Anon Haha, it is one of my favourite subjects :) I'd had an idea it might be something like this for a while but the first one had stumped me. Once I extended my search parameters I found it and then the pattern fell into place! (Beautiful images too, by the way - something very visually pleasing about them!)
– Stiv
Commented Oct 15, 2021 at 8:14
• Yes the first one was slightly cheeky... Senapr would have been the most obvious choice, but it is topologically indistinguishable from Argureynaqf. I'm glad you found the images visually pleasing too - I wanted to make them aesthetically nice and do justice to the beautiful nature of the original objects (which I also love like you), and they took me some time!
– Anon
Commented Oct 15, 2021 at 8:17