Blue, and the Two Sentences That Contain No Answer

Five hats sit in a box: three blue and two red. Three men each take one and put it on without looking, so nobody sees his own colour. The first two can see the other two heads. The third cannot see at all.

The first man is asked his colour and says he does not know. The second is asked and says the same. The third, who has seen nothing, names his own colour correctly. It is blue, and the interesting question is not what he says but why it is a deduction rather than a good guess.

Seven configurations, not eight

Call the men first, second and third in speaking order. A configuration is a colour for each, so there are $2^3 = 8$ of them on paper. One is impossible: three red hats cannot be drawn from a pool holding two. That leaves

feasible configurations. The two hats left in the box play no further role. What the pool constrains is how many red hats can be worn, and that is all any of the three men uses.

What the first silence deletes

The first man sees the other two heads. Suppose he saw two red hats. Both red hats in the pool would then be accounted for, so his own could only be blue, and he would have said so immediately. He did not. Therefore

This is straight modus tollens. If the antecedent held he would have spoken, he did not speak, so the antecedent fails. Exactly one feasible configuration is removed, the one where the first man wears blue and the other two wear red, taking the count from 7 to 6.

What the second silence deletes

The second man now knows everything we do, plus he can see the first and third heads. Two situations would let him answer.

If he saw two red hats, the pool argument settles it and his own is blue. If he saw a red hat on the third man, he can use the first man's silence. The first man's silence says the second and third are not both red. The third is red. So the second is not red, and his own hat is blue.

He said nothing, so neither situation holds. In particular the third man is not wearing red. Two more configurations fall away, the ones with a red hat on the third man that had survived so far, and the count goes from 6 to 4.

Why the third man can speak

The four survivors are the four configurations in which the third man wears blue and the first two wear anything at all, which is $2 \times 2 = 4$ cases:

His own colour is the same in every one of them. That constancy is what upgrades his answer from a guess to a deduction. He does not know which of the four configurations is real, and he does not need to, because they agree about the only thing he was asked.

His blindness costs him nothing

The instinct is that the man who cannot see must be the worst placed of the three, and it is backwards. Sight would have let him rule out configurations, but the two announcements already ruled out all the ones that mattered. What he needs is the knowledge that two perfect reasoners looked at the scene and found it inconclusive.

This is why the puzzle is really about a chain of statements rather than about hats. Each "I do not know" is an announcement about a whole set of configurations, and an announcement of that kind carries information precisely because the speaker is known to be reasoning correctly. Everyone knowing that everyone reasons correctly is the assumption doing the work, and it is worth stating out loud because the puzzle is false without it.

Both silences are load-bearing

A tempting shortcut is that the first silence alone settles it. It does not. After only the first man has spoken, six configurations remain and two of them put a red hat on the third man. He would have nothing to say. The second silence is what removes those two, and stripping it out breaks the puzzle.

Change the pool and the puzzle breaks

Three blue and two red is the exact pool that makes the chain work, and no other small pool does. Running the same knowledge model over other pools shows how narrow the window is.

• Two blue and two red. No configuration survives both silences, so the scenario as described cannot happen.
• Three blue and three red. Nobody can ever answer. All eight configurations are feasible and all eight survive both silences, because with three red hats available the first man learns nothing from seeing two of them.
• Three blue and one red. The premise is impossible for the opposite reason. Every configuration in which the first man cannot answer is one in which the second man can, so the story never reaches the third man.

The general shape is that the red hats must be scarce enough for their absence to be informative and plentiful enough for two silences to be possible. Two out of five is the smallest pool where both hold.

What the argument does not survive

The order of speaking matters. Ask the second man first and the eliminations happen in a different order, with a different result. The model above fixes the order as stated.

Perfect reasoning matters more. If the first man might simply have failed to notice that two red hats force his own to be blue, his silence carries no information and the chain collapses at the first link. Puzzles of this kind are sometimes described as being about logic, which undersells them: they are about reasoning over other people's reasoning, and every step assumes the others are as careful as you are.

Sources and further reading

• The assumption named in the section on blindness — Common knowledge
• The formal setting for reasoning about what others do not know — Epistemic modal logic
• The rule of inference behind equation (2) — Modus tollens

The knowledge model was written so the conclusion falls out rather than being encoded. The second man's reasoning is implemented as a filter over his candidate own-colours, keeping only those that would have left the first man unable to answer, and asking whether one candidate remains. Running it over all seven feasible configurations gives four survivors, all with a blue hat on the third man. The pool sweep above came from the same code.