@LeppyR64 Are you still pondering this or have you given up on?

Still looking. Haven't had a good session to sit down and really look deep.

Ok, I'm on to something.

@BmyGuest I have begun :)

No one's said this explicitly; I don't know whether because it's obvious or because no one's noticed. If each U/V represents a relation between three groups-of-circles (which might represent numbers or something), then I think P2 depicts a bunch of "composite relations".

I agree with that assessment. All of the "U" relations all seem to be 1-to-1. Since we're operating on the colours being prime factors, this function could represent multiplication.

That is, if e.g. we say that V(a,b,c) means it's legit to have a V with pattern a at top, pattern b at lower left, pattern C at lower right and likewise for U(a,b,c) (but note, I think the U needs to be thought of the other way up -- the "top" is the bit by the "nipple" on the curvy bit) then ...

... then e.g., the pattern at lower left on P2 says: there are some X and Y such that V(empty+beige, X, Y) and U(X,blue,lilac) and U(Y,blue+yellow,blue+yellow+red).

Where of course my use of symbols like + isn't meant to beg any questions, I'm just gesturing towards the actual arrangements of dots on the image.

Yes, the Us are telling you how to split up a given bunch of dots.

(maybe the dots should be thought of as prime factors, indeed)

I hope the Us don't just represent multiplication, because in that case they convey no information at all.

The one at lower right of P2 is a bit strange; it's oddly spaced compared to the others. I wonder whether it means V(empty,black+X,red) where also V(X,Y,Z) and V(Y,...,...) and U(Z,...,...). That will be impossible to make sense of without comparing against the actual picture :-).

Actually, I notice that I'm really confused about this, because some of these things I'm claiming are composite relations have a bunch of dots in the "shared" place and some don't. Maaaaybe the latter specifically have "no dots" in that place, but that doesn't seem likely. The only other way I see to fit this with my theory is that sometimes the aliens have kindly indicated what X and Y are, and sometimes not. That's a bit unsatisfactory.

So maybe I'm just entirely wrong.

I think it is V(V({Blue, Blue, Yellow}, {Cyan}), U({Red},{Yellow})) = Black

Hi, I'm deliberately not going to comment too much here, because the two of you are clearly on the right track now, so eventually you'll get it.

I'm confused about this "Empty" colour though.

returning to P1, here's a boring but kinda obvious guess: U really does just mean multiplication of prime factors and V means addition. In that case, look at the V towards the northwest with red and yellow dots under it. Let red,yellow,blue be p,q,r; then we have pq(p+q) =pqr^3 so p+q=r^3. If p,q are both odd then this means {p,q}={3,5} and r=2. If not then we have prime+2 = prime^3, which I think has lots of solutions but none with small primes. So I bet {red,yellow}={3,5} and blue=2.

This is all conditional on V = addition, of course.

Then near the northeast we have 3^3+5^3=2^3.cyan or 27+125=8.cyan or 152=8.cyan or cyan=19. That is in fact a small prime, which is promising.

I guess you mean P4? (i.stack.imgur.com/iwl32.png)

sorry, yes, P4

Now if my conjecture about P2 is also right then at the southeast of that we have 15 from the U and either (3x4+19) or (5x4+19) from the lower-left V; that is, either 31 or 39. [will continue in a moment]

@BmyGuest yes, that's the one that I think tells us {red,yellow}={3,5} and blue=2 unless there are surprisingly large prime numbers involved.

so then I guess the "intermediate V" means to add the 15 to the 31-or-39, yielding 46 or 54 respectively.

I agree (That V seems to be addition)

But then I'm kinda stuck because (1) I'm not sure how to interpret the topmost V -- I think probably as "(black x thing-we-just-found) + light-red = empty" -- and (2) there are too many colours here that I don't have any information about.

oh, but wait

there's a problem with the interpretation I just described

... oh, sorry, no there isn't. My brain had a parity error :-).

How does the image fit to "pq(p+q) =pqr^3" ? (Not saying its right or wrong, just not seeing your derivation)

Ah, c now.

oh, there's another useful V on P4, also towards the northeast, which says that red.green = yellow^3+blue^3

which means, depending on which way around red and yellow are, either 3.green = 5^3+2^3 or 5.green = 3^3+2^3. That is, either 3.green=125+8=133 or 5.green=27+8=35.

Obviously only the latter is possible.

So blue=2, yellow=3, red=5, green=7.

yes, that's the one I'm talking about.

now near the southwest is another one that will tell us what light-red is

...I see a puzzle crumbling into it's solution :c)

it says blue.red + blue.yellow.green = blue.blue.lightred or 2.5 + 2.3.7 = 2.2.lightred or (losing a common factor of 2) 26 = 2.lightred so lightred = 13.

currently known primes are

blue=2, yellow=3, red=5, green=7, lightred=13, cyan=19.

now at the east

we have 2.5.5 + 2.3.7.7.13 = 2^5.lilac^2

or, again losing a pointless factor of 2, 25+1911 = 16.lilac^2 so lilac=11

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, cyan=19.

there's a V slightly north of centre which uses ... oh, only these colours so it won't tell us anything new, but it will give us some confirmation

ignoring visible common factors of green,red,lightred it says red + green = blue.blue.yellow or 12 = 2.2.3 which is in fact correct, so that's encouraging

Guess, you mixed up east & west ;c)

where?

oh yes

I did indeed say east when I meant west

sorry about that

(is in the west (left hand side) of the image)

Took me a minute to find it too :)

so I guess now we want the V slightly SE of centre

oh, no, that's got two new colours

BTW, I believe using the cutouts in the final, reasoned solution to the puzzle would be a good...

well, never mind

it says blue.yellow.lightred = pink+grey or 2.3.13 = pink+grey or 78 = pink+grey

{17,61} {31,47} {37,41}

oh, the V just east of north looks like it will tell us more

it'll tell us pink.grey

Maybe useful "reference" to fill:

I have the current list starred...

which means the slightly SE one in fact gave us no information this won't give us better

common factors to ignore: yellow^2. Then we have blue.red^2.lightred + green^2.lilac = pink.grey or 2.25.13+49.11 = pink.grey or 1189 which is, er, 29x41

so it looks like I made a mistake on the other one

or failed to distinguish two different colours

or something

(that's the one that allegedly gave us pink+grey=78)

Yes, your' too fast for me! :c)

but apparently also too incorrect since I seem to have proved both pink+grey=78 and pink.grey=29.41 which can't both be true

Never ignore the possibility of me messing up :c) Hence I'm joining in to try to cross-check. But you're too fast...

perhaps I mistranscribed a prime/colour correspondence earlier

D@rn. I need to be AFK for a while. (Or rather, other K needed.)

everything looks OK to me aside from having arrived at a contradiction :-)

I'm just getting caught up to the grey/pink thing.

Are you looking at the one just East of North?

yes

I think that's a hot pink.

oh crap, there are two pinks

you're right

but that's good news

because now we know grey = 41, the prime common to those two things we looked at, and dull-pink = 37 (the other one to make up 78) and hot-pink = 29 (the other one to make up 1189)

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, cyan=19, hotpink=29, dullpink=37, grey=41.

this would be much easier with those better-discriminating alien eyes

hmm, maybe we can now get somewhere with the north side of P2

You can find white just west of north

red.grey + dullpink+blue = yellow^2.white

so its "outer" V has green.green.green.lilac + cyan.lightred.lilac which is 3773 + 2717 = 6490. Its "outer" U is red.red.red.blue.emptyempty = 250.empty.

And its "inner" V has sum blue^3.yellow.red.green.lilac = 9240

which would work if empty=11

so maybe empty means "unspecified prime"??

CAreful, there's two greens as well.

anyway, sorry, you were saying: just west of north. That's red.grey + blue.dullpink = yellow.yellow.white or 5.41+2.37=9.white or white=31

two greens? where?

Two shades of green

See in P4 - Center.

true but have we encountered a light green yet?

or were you just giving me advance warning so I don't screw up like I did earlier?

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, cyan=19, hotpink=29, white=31, dullpink=37, grey=41.

I wasn't looking at the same one you were on P2, my confusion, but yes, also for advance warning.

can we do far northeast of P4 now? Let's see.

Common factor of yellow.green.red.blue.

Other than that we have blue.red.green^3 + lilac.lightred.cyan.brown = hotpink^2.orange

Looking good so far...

or 3430+2717.brown = 841.orange

( This is the solution as state by you guys, not saying that is MY solution (yet))

Can't do that in my head...

Can get purple in southeast

so brown = -3430/2717 mod 841 which means brown=17 (unless it's very big) and then orange = 59

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, brown=17, cyan=19, hotpink=29, white=31, dullpink=37, grey=41, orange=59.

(thanks to Mathematica for being better at modular arithmetic than I am)

They seem to be following the pattern of the first 17 primes so far.

hardly a surprise

lightgreen in center

so now, as you say, in the southeast, if I am seeing these colours correctly we have red.lilac+hotpink.maroon = blue.yellow.green.white or 5.11+29.maroon = 2.3.7.31 or 55+29.maroon = 1302 so maroon=43

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, brown=17, cyan=19, hotpink=29, white=31, dullpink=37, grey=41, maroon=43, orange=59.

black on east

?

Wrong cropping :)c

and yes, in the centre we have cyan.grey + orange.lightgreen = blue.yellow^2.green.white or 19.41 + 59.lightgreen = 2.9.31 which unfortunately means lightgreen=-221/59 so I have done something wrong :-)

ah, missed a factor on the RHS

19.41 + 59.lightgreen = 2.9.7.31 so lightgreen=53 which is much much more believable

( Breaking in cold sweat. So far I have not made a mistake... Can it really be? A BeMyGuest-puzzle without mistake?!? )

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, brown=17, cyan=19, hotpink=29, white=31, dullpink=37, grey=41, maroon=43, lightgreen=53, orange=59.

we'd better hope black=47

or 23

oh, damn, there's another one spare. yeah, OK

so, (17.19 + 2.3.5.13) / 31 = 23

blue=2, yellow=3, red=5, green=7, lilac=11, lightred=13, brown=17, cyan=19, black=23, hotpink=29, white=31, dullpink=37, grey=41, maroon=43, lightgreen=53, orange=59.

That leaves 47 in the wind so far.

is there a colour we haven't used yet?

There were only 16 colours represented in that panel.

There's a deeper violet in the other panels.

Do you mean the one I've been calling "empty"?

I don't think so.

because conditional on my interpretation of the "composite" things in P2 it seemed like in at least one case we had empty=11

ah, OK, where's the deeper violet?

Panel 3 has a very distinct violet, but it's not represented on Panel 2.

@BmyGuest, please verify.

The legend of P3 is given above. Convenience pic in a sec

I think that violet is also present on P2, at the west

Yes it is, and it's distinct from our empty.

Side-note: Why do you call it empty ? (And if for reason, that might be hint!)

Lack of a better term :)

It's the same colour as the background.

I call it empty because its colour appears to match the background in your pictures. I don't know whether it did before the "enhancement".

..so your brain has given you a hint (unconsciously)

It might actually be "dark blue" or something

well, I already conjectured that it might stand for arbitrary primes

Or it might, indeed, be empty.

More of a point that when all else is eliminated, it will present itself.

Why guess at hints when we can prove it.

@GarethMcCaughan I've seen that. Did not want to comment.

of course if it means an unspecified prime then maybe at some point we need to fill it in with (the colour of) the prime it represents

@LeppyR64 Correct. Husaa, onward!

are the empty circles arranged in a pattern that might hint at something useful to do with them after filling them in?

@GarethMcCaughan I don't see the pattern yet, let's identify them, then maybe it makes sense.

And @GarethMcCaughan Well done! And @LeppyR64: You're good at giving my puzzles the "get it rolling" kick! You did it again!.

OK, maybe we should look again at the southeast of P2

@BmyGuest: I'm proud of that. Thanks.

To avoid endless scrolling: P2

so the lower-left V is now 12+19=31, the U is 5.3=15, so the intermediate V is 46. Then the top V seems to say 46.23+29=empty so empty=1087 which is certainly prime but not one of the ones we've seen so far.

P2 with legend (so far):

What you've done with the black seems awkward to me, but seems to be the only logical method.

P2 southwest looks like it has a colour we haven't seen before

at its "top" it has one empty and one light grey (R=G=B=153)

18 colours, interesting.

I should really get some work done.

Legend = helpful.

Me too

Still, it seems like we've done a big chunk of this.

Leave it for now @GarethMcCaughan. Real life comes first. We'll crack this nut soon enough.

Indeed. Glad you've picked it up though, guys.

Was already afraid it's not a good puzzle....

I think it's a great puzzle, so far. It fits with your other themes of diving deep into images and then pulling out small details. If I had been around at the start (life gets in the way) then we would be further along already.

So southwest is 2.11 + 2^2.3^2.5 = 22 + 180 = 202 = x.y = 2.101, so the new greyish is 101 and the unknown is blue (2).

Seems odd, but ok

Unknown in the North is 11.

Southeast is not correct. The lightred is 13, not 29.

Something is wrong with the stacking of the operations. Going for a walk.

Not sure what you mean by SW is not correct... ?

Gareth calculated that the "blank" was 1087. I'm not certain that's correct.

Not sure that we're handling the black being inserted in the stream correctly.

(2^2.3+19) + (3*5) = 46

The "empty" (blank) is indeed NOT 1087.

But of course, the image is obviously missing stuff...

46.23 + 13 = 1071 which is not prime.

I'm not sure how to interpret "missing", so I'm going to ignore it for now.

Why 46.23 ?

Don't see any multiplication (U) exectp the very first one

Gogin to leave now for today.

Exactly my conundrum, unless there should be another blue in that spot, then why is the black there and not the blue? Not sure how to handle that black. It's odd.

And if that's the case, then what is the indication of what is missing? I can tell when something is missing because it won't balance, but when we get to multiple unknowns, it's awkward. (I'm looking at you P2-East)

For P2-SE, (2^2.3+19) + (3*5) = 46 + 13 = 59 (Orange)

Assume purple = 47, since all other constellation colours are used in the first 17 primes. West = 13 (lightred)

So southwest is 2.11 + 2^2.3^2.5 = 22 + 180 = 202 = x.y = 2.101, so the new greyish is 101 and the unknown is 2 (blue)

North unknown is 11 (lilac)

Unknown from P2-East is 7 (Green)

Pairing those colours with the constellations in P3 and then finding the combination of them in the stargate, we need to push the right most white one from Panel 1.

That is assuming that the orientation of the star gate matches the orientation of Panel 1.

(Which I have no clue yet)

#4 from 2012rcampion's answer