This stalemate continues for a few minutes, at which point one of them realises it has to be a 3-y situation, not a 2-y, because nobody’s realised they have to be yellow.
It is important to remember that all three know that each man is smart. So they know that all three, on seeing three hands raised, know that there must be at least two yellow dots.
If there was one blue and two yellows, the two with the yellow dots would each see a blue and a yellow. They know there must be two yellows, so they’d know immediately that their personal dot was yellow. Only the one woth the blue dot would be confused, because he knows there are either two yellows or three yellows, and seeing two yellows doesn’t tell him which is the case.
So the important fact is that these smart guys don’t immediately know what’s what. That means that they must all be seeing two yellows (the only ambiguous case). Thus, each must have a yellow dot. There can be no blues.
Why do you say that one of them realises that it HAS to be a 3-y situation? I don’t see it as it having to be a 3-y situation at all.
X sees Y and Z with yellow dots. He raises his hand. He sees Y and Z raising their hands as well, which indicates that Y sees at least Z’s yellow dot, and Z sees at least Y’s yellow dot. He cannot tell if he has a blue or yellow dot, because the two others could be raising their hands because Y sees yellow on Z and Z sees yellow on Y. X has no way to know from this if he has a blue or yellow dot. This same problem exists for all three of them; I do not see any reason why it HAS to be a 3-y situation. Please explain.
OK, I get it now. Saltire’s explanation helped.
Because if it’s 2y, Both yellow people would have seen a blue dot. Upon seeing a blue dot, and everyone having raised their hands, it’s just a matter of which one realises he can’t have a blue dot first. After a while, once nobody’s figured that out, it becomes clear that nobody’s seeing a blue dot.
Now, the original puzzle doesn’t specify that the person has seen 2 yellows, so the situation in the puzzle could be 2y - what’s important is that he’s yellow, and the only people who can know what colour they are in this scenario are yellow, but anyone who is yellow can know what they are.
There are four variations that exist:
- 0 yellow/3 blue - 0y scenario
- 1 yellow/2 blue - 1y scenario
- 2 yellow/1 blue - 2y scenario
- 3 yellow/0 blue - 3y scenario
The 0y and 1y scenario are out as soon as all of them raise their hands; if there were 2 or 3 blues, at least one of them would leave their hand lowered. This leaves the smart men with the 2y and 3y scenario.
If it was the 2y scenario with 1 blue dot to be seen, 2 of the men would see the blue dot and a yellow dot and, reconciling the sight of the blue dot with the above knowledge of a 2y or 3y scenario, a smart man would realize this must be a 2y scenario and therefore they must have the second yellow dot.
It is because this DOESN’T happen after a few moments that the 2y scenario can then be rejected and, by elimination, it must be a 3y scenario, and so the man is able to deduce that he has a yellow dot.
Heck, one of my favourite “thee men” puzzles is the one where three intelligent men have to divide up a stash of 30 gold pieces. The deal is that any man can propose a distribution, and if it is approved two-votes-to-one, it passes, though counteroffers can be made before the vote is taken. How is the loot ultimately divided?
Answer: equally.
I don’t get it. Why don’t man A and man B just agree to split the loot 15 each, adn screw man C over? Man C could propose and counterpropose all he wants, but presuming each man is out to get as many as he wants, surely it’d end up with some kind of 15-15-0 distribution. There’d be no way to pick who gets the 15s and who gets the 0 though.
Because the negotiations take the following sequence:
A: Hey, B. Let’s each take 15 and screw C over.
C: Wait a sec! B, you can have 16 and I’ll take 14, and we’ll screw A over.
A: Oh, yeah? Listen, B buddy, you take 17 and I’ll take 13…
…eventually, A will offer B 29 while he takes 1. Somewhere along the way, C wises up and says:
C: Wait a sec, why are we dealing with this fickle schmuck? Hey, A ol’ salt, why don’t you and I each take 15 and screw B over?
B: Hey! Uhm, A old pal, why don’t you take 16 and I’ll get by with 14?
C: Oh, no you don’t! Listen, A-whose-life-I save-in-'Nam, you take 17 and I’ll take 13…
And so on indefinitely. Since all three men are intelligent, they’ll see that trying to negotiate is a pointless waste of time, so they split the loot equally and go get drunk.
I’m not sure I agree with this. If the three men are all intelligent, they all understand that if any one of them attempts to be too greedy, the other two can collude against him. I would envision the negotiations going something like this:
A: Hey B, lets split things 15-15 and screw over C.
B: Sounds good to me.
C: Wait B, I’ll give you 16 and take 14 for myself.
B: Nah, I’ll take my 15, thanks. If I take 16, A may decide to get together with you and split 15-15 to screw me over. 15 sure beats going around and around until I only get stuck with 10.
OK, let me see if I can explain the yellow dot thing to those who are still confused.
If there were no yellow dots, no hands would be raised
If there were one yellow dot, the guy who had it wouldn’t see any, so his hand would stay down.
Thus, the fact that all three raised their hands proves that there are at least two yellow dots.
The man who solves the puzzle (I’ll call him the wisest man) assumes that all three of them are smart enough to figure this out. In other words, he assumes that all three men know there are at least two yellow dots, because all three hands went up.
Let’s pause there for a moment. Everyone with me so far? All three men have figured out that there are at least two yellow dots.
Now, the wisest man knows that if any of the men saw a blue dot, then that man would know his own dot was yellow. This is because there are only three men, but by this point everyone knows there are at least two yellow dots. So the wisest man assumes that anyone who sees a blue dot would have been smart enough to figure out that his own dot was yellow.
Thus, the wisest man knows that no one saw a blue dot, because otherwise that person would have known the answer.
Thus, the wisest man knows he must not have a blue dot, because he knows the other men didn’t see a blue dot.
Thus, the wisest man knows that he has a yellow dot.
That’s as simple as I can make it. It’s not a simple puzzle, so I’m not surprised people are having trouble with it. Anyway, I hope I helped.
hmm . . . I just realized that it’s possible that no one was still confused. The person who said “I don’t get it” was questioning the men’s honesty, not the wisest man’s reasoning.
Oh well. Maybe some other confused person will wonder into this thread and my post will help them.
Darn you for getting here first 
I had been tossing this problem round my head last night and come to the same conclusion. It must be 15,15,0 since that bid can beat any other. So the quickest witted will offer 15,15,0 (one 15 for her self obviously) and the other one who is offered 15 will realise that any ‘better’ offer to him from the poor schmuck offered 0 will be countered by a 15.15.0 offer between the quick witted one and the schmuck. So he must accept the 15,15,0 as his best possible payoff.