How should the hexagon be coloured?

Question

Given the colours and banding of the first four rectangles, how should you colour the hexagon, and why?

There's a single, unambiguous solution (ignoring symmetry), which will be obvious when found. Nothing is hidden in the image, you could print it out and still solve it with nothing more than your eyes and some lateral thinking.

_{Edit: I intended this to be solved by eye, and so my image wasn't drawn 100% accurately. However, since a few people are approaching things somewhat mathematically (which is admittedly a valid approach), I've checked my calculations and updated the image slightly. It shouldn't make much difference in solving it, but I believe the maths is now correct to the nearest pixel.}

Owing to the lack of responses thus far, I think you've stumped a few people :p — nine9, Jan 20 '16 at 11:58
@nine9 - time will tell if I made it too hard. I don't normally make puzzles like this, so it was hard to judge the difficulty. — Alconja, Jan 20 '16 at 13:13
How much do we care about the relative sizes of the side lengths? Just by eye, if we say the cyan rectangles are 1u (1 unit) wide, then I'd say the red square is 6u6u, cyan rectangles are 1u6u, green square is 6u6u, yellow rectangle is 6u9u, magenta and blue rectangles are 6u*2.5u and the hexagon base is 5u. — SpiritFryer, Jan 20 '16 at 17:20
Separate observation: I assume we are dealing with additive color mixing due to the black background? — SpiritFryer, Jan 20 '16 at 17:23
@SpiritFryer - on relative sizes: yes, that's definitely important (though I haven't necessarily got things pixel perfect, just eyeballing should be enough). — Alconja, Jan 20 '16 at 20:52
@SpiritFryer - on additive colour mixing: excellent observation. — Alconja, Jan 20 '16 at 20:55
Oh, I just noticed that the yellow rectangle is the same height as the hexagon! :D — SpiritFryer, Jan 21 '16 at 04:43
Is the width of the green-cyan rectangle the same as the height of the magenta-blue rectangle, and vice versa? — lirtosiast, Jan 21 '16 at 06:13
@ThomasKwa - The magenta-blue rectangle should be taller than the cyan-green one is wide, but I'm not 100% sure that it actually is in the image... I'll double check when I get the chance and fix it if it's wrong. — Alconja, Jan 21 '16 at 06:25
Am I heading in the correct direction by saying that, the amount of area covered by the colors are also alphabetically in order: B, C, G, M, R, Y? Though it seems that there is more Cyan than Blue...is this coincidence? — nine9, Jan 21 '16 at 11:58
A different observation, there are 6 different colors and 6 sides to the hexagon. — nine9, Jan 21 '16 at 12:31

MathET · Accepted Answer · 2016-01-22T18:35:30.570

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These images are all (orthographic) projections of the same cube.

It has the three basic colors, with opposite faces identical. When one is in front of the other their colors are added in the projection.

The final answer is a variation on these two pictures, depending on how you rotate the cube:

A more detailed explanation of the answer:

First: We just see the two red faces superposed.

Second: Now the cube is rotated, with the green faces in place of red. However, it is slightly tilted so the blue sides mix with the green forming the cyan rectangles.

Third: The cube is at 45 degrees with the ground, so red and green faces superpose completely at top, as well as green and red at the bottom, giving an uniform yellow color.

Fourth: Finally, we see a similar arrangement to the second picture except in a different plane of rotation.

Also, we can notice differences in size between the rectangles. That's no mistake: When a cube rotates along a coordinate axis, its visual size increases from $l$ to $l \sqrt2$, then goes back down to $l$.

The yellow is the biggest rectangle because the diagonal of a square is the longest line segment inside it.

Our last projection is the cube viewed vertex-on, which has a hexagonal envelope.

This picture shows how you combine colors of face pairs in the X, Y and Z axis to get the final answer:

This puzzle was actually pretty easy for me to figure out because I love trying to visualize higher-dimensional spaces, so I was very well familiar with projections, envelopes, cross sections and stuff.

It's a fascinating way to improve your spatial reasoning, so I highly recommend anyone interested to take a look. Or you could just try exploring a 4d maze and see if you can find your way out

edited Jan 22 '16 at 18:35

answered Jan 21 '16 at 20:00

MathET

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Nice solution. If this is correct, though, the colors should be magenta, cyan, and yellow (hex codes #FF00FF, #00FFFF, and #FFFF00.) Your purple and light blue aren't quite right. – Michael Seifert Jan 21 '16 at 20:32
Ok. Will fix and add more pictures detailing stuff, wait a minute – MathET Jan 21 '16 at 20:36
This is a cool answer, but if I'm looking at this right (I'm using a color picker, because I'm the tiniest bit colorblind), the original image uses six colors, and this only uses three - how to account for the other three colors? – question_asker Jan 21 '16 at 21:30
When two colors are superposed, their values are added. So, for example, Red+Green = Yellow or R+G+B = White – MathET Jan 21 '16 at 21:31
I made a picture explaining more or less how the secondary colors are the only ones to appear. Basically, only different colors can combine because faces of the same color are opposite to each other – MathET Jan 21 '16 at 22:03
Yeah, this makes perfect sense, thank you. I was hoping it would be 'lateral thinking', as was suggested, but oh well, still a neat puzzle. – question_asker Jan 21 '16 at 22:05
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@question_asker I think it is a pun, since lateral means "of, at, towards, or from the side or sides." :-) – Carl Löndahl Jan 21 '16 at 22:09
@carllöndahl sigh, I get it, it's still disappointing, though. It's a bit like saying "I got you a surprise! The surprise is: I didn't get you anything!" – question_asker Jan 21 '16 at 22:11
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@question_asker - sorry to have frustrated you. I was torn whether to use the tag or not... I guess I felt like making the leap from 2d to 3d required somewhat lateral thinking, and... ok, I admit it, I couldn't resist the pun. – Alconja Jan 21 '16 at 23:12
@IvanBarreto - Well done! Perfect. Thanks for writing such a detailed solution/explanation (I'm going to give you a small bounty in gratitude), and for including the extra reading links. – Alconja Jan 21 '16 at 23:21
@alconja Haha it's not a problem, just getting my hopes up – question_asker Jan 21 '16 at 23:29
simply brilliant. Well done to the @Alconja for creating the puzzle, and Ivan for solving it – nine9 Jan 22 '16 at 06:56
Good job, I believe the word "orthogonal" is typo. I tried this idea with no success, since I would never consider that the cube can be projected at 4 random angles... – klm123 Jan 22 '16 at 08:46
@Alconja Thanks for writing such an inspiring puzzle! It got me thinking for a while and I can't keep things succinct when I find them genuinely interesting – MathET Jan 22 '16 at 19:14

score 2 · Answer 2 · answered Jan 20 '16 at 21:09

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Either Black or White, depending on how you look at them. If they are basic light colors - then seen together they would be seen as white light, but if they are paint pigments - then black. However, because of the background I would bet on white.

answered Jan 20 '16 at 21:09

superefka

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White is definitely better than black, but neither is right. The solution is multi-coloured and has a specific pattern to it. – Alconja Jan 20 '16 at 21:58

score 0 · Answer 3 · edited Jan 21 '16 at 04:49

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The color of the hexagon should be black because all the other colors are a part of CMYK color model and the black is the only color that is left out.

And there should be no pattern in the color because alternate shapes have been colored.

edited Jan 21 '16 at 04:49

SpiritFryer

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answered Jan 21 '16 at 04:16

Sol Infinus

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He already confirmed (unless I misunderstood his comment) that it's additive color mixing that should be applied, and not subtractive, so this can't be right. (So with your argument, the hexagon should actually be white.) – SpiritFryer Jan 21 '16 at 04:31
Sorry, not black. @SpiritFryer is right about the colour mixing being additive. But as I commented on superefka's answer, the solution is multicoloured and has a specific pattern to it. – Alconja Jan 21 '16 at 04:58
@SpiritFryer Oh yeah sorry. Didn't see that. – Sol Infinus Jan 21 '16 at 05:30
@Alconja Oh OK. I will try it again. :) – Sol Infinus Jan 21 '16 at 05:30
Isn't black the colour of the background of the image? – Ian MacDonald Jan 21 '16 at 17:32
@IanMacDonald Yes it is. But that was the color that matched my logic so I thought that would be the answer. Foolish conclusion. '-.- – Sol Infinus Jan 22 '16 at 01:14

score 0 · Answer 4 · answered Jan 21 '16 at 07:34

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I spotted something interesting, not sure if it's right, but here's a go:

I drew overlapping selections by hand onto a layer mask, so not particularly accurate.

Explanation:

If we look at the relative areas of the four rectangles:
5x5 = 25 (Red)
6x5 = 30 (Cyan/green/cyan)
5x7 = 35 (Yellow)
5x6 = 30 (Magenta/Blue/Magenta)
And the total of these is 120 - the internal angle in degrees of a hexagon. So dividing the hexagon from one vertex using these angles gives us:
25 degrees of red;
30 degrees comprised of 5 cyan + 20 green + 5 cyan;
35 degrees of yellow;
30 degrees of magenta/blue/magenta (sub-divided horizontally, not drawn to scale!).

answered Jan 21 '16 at 07:34

jhabbott

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I wouldn't call this solution obvious, but it's definitely comprehensive. – nine9 Jan 21 '16 at 08:01
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If you treat areas as angles, then why subareas (magenta/blue/magenta within one rectangle) is treated differently? It is hard to call "a single, unambiguous solution (ignoring symmetry), which will be obvious when found." At least, there are many ways to apply the same idea. – klm123 Jan 21 '16 at 09:55
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And finally, there is author statement "The magenta-blue rectangle should be taller than the cyan-green one is wide". Thereby downvoted, sorry. – klm123 Jan 21 '16 at 09:59
Interesting theory. But unfortunately wrong. If you want to get mathematical about things, I've now updated my original image to be accurate to the nearest pixel. You'll notice the green/cyan is slightly narrower and magenta/blue block is slightly taller than previously. – Alconja Jan 21 '16 at 12:39
@klm123 fair enough. Although I hadn't seen that comment, since it was posted around the same time I was devising this solution. Regarding the blue/magenta - well the relative areas should still be the same (I drew it by eye, so not to scale as mentioned) and the reason it's different is that they go horizontal. Hey - it's a hypothesis to get feedback and inspire thinking in among not much other activity, I thought that was the whole point. – jhabbott Jan 21 '16 at 13:05

How should the hexagon be coloured?

4 Answers4