Three logicians

[jfgw]

Three logicians are unable to decide who among them is the most logical so they decide to seek the advice of a wise man.

The wise man explains that he has some black hats and some white hats. He will switch off the light and place one hat upon each logician, then switch the light back on again. The first logician to correctly state the colour of his own hat, and to explain how he knows, will be declared the best logician of the three.

The wise man switches off the light and the room goes dark so that no-one can see. He then places one hat upon each of the three logicians. (He is wise and knows how to work in the dark.) He explains that there are five hats: Three black ones and two white ones, and that he has hidden the two unused ones. He then switches on the light.

One logician demonstrates his superior logic skills by stating the colour of his hat. What colour is it, and how does he know?

No-one can see his or her own hat. There are no mirrors or other reflective objects. No-one asks anyone-else and no-one finds the two hidden hats. It is purely logical deduction.

Julian F. G. W.

[Lootman]

It's obvious, isn't it? There are only two white hats. When the lights come on, one logician sees the other two are wearing white hats, so he knows that his hat is black.

[jfgw]

It's obvious, isn't it? There are only two white hats. When the lights come on, one logician sees the other two are wearing white hats, so he knows that his hat is black.

That is not the correct answer. I will not say at this point whether you are right or wrong about the hat being black but the logic is wrong for a very important reason.

Julian F. G. W.

[Gengulphus]

Three logicians are unable to decide who among them is the most logical so they decide to seek the advice of a wise man.

The wise man explains that he has some black hats and some white hats. He will switch off the light and place one hat upon each logician, then switch the light back on again. The first logician to correctly state the colour of his own hat, and to explain how he knows, will be declared the best logician of the three.

The wise man switches off the light and the room goes dark so that no-one can see. He then places one hat upon each of the three logicians. (He is wise and knows how to work in the dark.) He explains that there are five hats: Three black ones and two white ones, and that he has hidden the two unused ones. He then switches on the light.

One logician demonstrates his superior logic skills by stating the colour of his hat. What colour is it, and how does he know?

Spoiler...

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The easy case for a logical deduction is if both white hats are being worn by logicians. In that case, the third logician should almost immediately deduce that his own hat must be black.

The middling case for a logical deduction is if one white hat is being worn by a logician and one is hidden. In that case, each of the two logicians who is wearing a black hat can see one white hat and one black hat, and should think "Well, if my hat were white, the black hat wearer can see two white hats and should almost immediately deduce that his own hat must be black. So I'll give them just a moment and if they haven't made that deduction, my hat cannot be white and so must be black." The one who judges the length of the momentary wait best wins.

The hard case for a logical deduction is if both white hats are hidden and all three logicians are wearing black hats. In that case, each of them should think "Well, if my hat were white, each of the other two would see one white hat and one black hat, and should be able to make the logical deduction for the middling case. That takes a bit longer, since the chain of thought is more involved and they would have to wait a moment anyway - but if neither of them makes that deduction after a suitably longer pause, I can deduce that my own hat cannot be white and so must be black." The one who judges the length of that longer pause best wins.

Unless of course someone plays psychological games. E.g. logician A sees two white hats when the light comes on, and immediately deduces that his own hat must be black. But that's such an easy deduction that it hardly demonstrates "superior logic skills", so she'll have to instead trick the others into demonstrating inferior logic skills... So she keeps quiet and waits for the other two to 'deduce' that their own hats must be black. If she's lucky, they make a dead heat of it, both demonstrating inferior logic skills; if she's unlucky, one of them does so and she should still be able to announce that her own hat is black before the third logician can work out what must have happened and announce that his own hat is white...

Except of course that all three can play at that game of making deductions but not announcing them, in order to try to fool the others into making incorrect 'deductions'. So ultimately, any deduction based on any other logician not yet having announced a deduction is on rather unsteady ground, which means that the only really safe logical deduction is that of seeing two white hats and deducing that one's own hat is black. But while safe, that deduction is so elementary that it fails to demonstrate "superior logic skills", and it's clear that unsafe deductions can hardly demonstrate them either...

The conclusion is that demonstrating "superior logic skills" is a lot harder than it looks, and it's likely to be futile to try - which was probably the wise man's point! ;-)

Gengulphus

[Lootman]

Except of course that all three can play at that game of making deductions but not announcing them, in order to try to fool the others into making incorrect 'deductions'. So ultimately, any deduction based on any other logician not yet having announced a deduction is on rather unsteady ground, which means that the only really safe logical deduction is that of seeing two white hats and deducing that one's own hat is black. But while safe, that deduction is so elementary that it fails to demonstrate "superior logic skills", and it's clear that unsafe deductions can hardly demonstrate them either...

The conclusion is that demonstrating "superior logic skills" is a lot harder than it looks, and it's likely to be futile to try - which was probably the wise man's point! ;-)

Yes, if this is claimed to be purely a logic problem then speculations based on the tactics of others should not be a factor. That is more akin to the tactics that work in a game of poker, with concepts like bluffing and double bluffing.

The only solution that is purely a matter of logic does not solve in all cases, but rather the subset where the two white hats are distributed and the non wearer of the white hat recognises that. The other solutions all rely on various forms of second guessing, speculation and bluffing.

Incidentally, there are other variations of this puzzle that genuinely rely upon logic. Typically they are based on prisoners rather than logicians, and the prisoners are lined up such that each prisoner can see the hat of the prisoner in front of them. so one prisoner can see two hats, one can see one hat and one can see no hats. It's a cleaner puzzle, in my opinion. This one is messy.

[jfgw]

I will clarify a couple of points:

The purpose of the procedure with the hats was to determine the best logician.

All three logicians were aware of this.

Julian F. G. W.

[Lootman]

The purpose of the procedure with the hats was to determine the best logician. All three logicians were aware of this.

I think that both Gengulphus and I understood that when giving our solutions. Our responses indicated more that this wasn't solely a logic problem but also embraced concepts like game theory, bluffing, speculation, gambling and predicting probability. I'd argue that none of those are matters of pure logic, but rather involve empirical and psychological factors as well. They test a broader range of skills and entail a greater degree of risk and uncertainty.

Except in the case where the two white hats are distributed, as we both noted.

So I think it comes down to how you define "logic", narrowly or broadly. Logicians don't always make the best poker players.

[redsturgeon]

If the object of the exercise was to discover the best logician then that can only logically be done by giving all three black hats...ie. a level playing field.

John

[UncleEbenezer]

If the object of the exercise was to discover the best logician then that can only logically be done by giving all three black hats...ie. a level playing field.

John

That was my thought, too. It is the only permutation that fits the exact question, given:
(a) The wisdom of the wise man.
(b) The need to determine in a single move who is the best logician (i.e. NOT who has been presented the easier problem, or who is quickest to reach a deduction from his fellows' hesitation).
(c) Not least, the supporting evidence of the formulation of the exact question:

What colour is it, and how does he know?

If it were white, then at least one logician has a black hat, and therefore no chance of winning. So the problem has failed (a) and (b).

Sadly, the problem fails anyway. Because it can be deduced without the exercise being carried out, we can infer that none of the parties is actually very bright if they have to wait for the lights to go out (let alone turn back on) before answering. King Solomon did a better job.

[jfgw]

If the object of the exercise was to discover the best logician then that can only logically be done by giving all three black hats...ie. a level playing field.

John

That is the correct answer. In the version of the puzzle that I based this one upon, the winner correctly stated the colour of his hat before the light was switched back on. I didn't want to make it too easy though.

The key point is that the contest with the hats must be fair for the winner to be declared the best logician of the three. A contest with two white hats proves very little as one logician can solve it very easily but the others cannot. Likewise, a contest with one white hat can be solved by two of the logicians more easily than the third. The only fair and, therefore, conclusive contest is one where all three logicians are given black hats.

Julian F. G. W.

[jfgw]

Sadly, the problem fails anyway. Because it can be deduced without the exercise being carried out, we can infer that none of the parties is actually very bright if they have to wait for the lights to go out (let alone turn back on) before answering. King Solomon did a better job.

From the original question, they were not told how many of each hat there were until after the light was switched off and the hats were placed upon their heads.

It is immaterial whether the correct answer was given before or after the light was switched on.

Julian F. G. W.

[UncleEbenezer]

It is immaterial whether the correct answer was given before or after the light was switched on.

Julian F. G. W.

But it is not immaterial that the victory lies in speed rather than in logic. So the well-mannered logician whose reaction on starting to speak simultaneously with his peers is one of "sorry, after you", is a loser.

Just like real life.

Perhaps we could formulate these using the national stereotypes with which we (brits) flatter ourselves. If the subjects are well-brought-up Englishmen and one of them is Stephen Hawking, the others are too polite to reply before he's had a chance with his device. If they're brash Americans, they answer before Prof. Hawking has typed more than his first bit.

[Lootman]

But it is not immaterial that the victory lies in speed rather than in logic. So the well-mannered logician whose reaction on starting to speak simultaneously with his peers is one of "sorry, after you", is a loser.

Yes, it involves speed, which doesn't have much to do with logic. It also relies on making assumptions about others and how logical they are. That was the point of making the participants logicians, presumably, but it rather begs the question we are trying to answer. It also presupposes the notion of wisdom, since an inference is also required about the motives and methods of a "wise man".

Maybe the three participants should instead have been described as three wise quick thinkers of equal politeness, but that's not as laconic.

[jfgw]

Maybe the three participants should instead have been described as three wise quick thinkers of equal politeness, but that's not as laconic.

I can see this board getting rather tedious

Julian F. G. W.

[StepOne]

Maybe the three participants should instead have been described as three wise quick thinkers of equal politeness, but that's not as laconic.

I can see this board getting rather tedious

Julian F. G. W.

Julian, I enjoyed the puzzle

[redsturgeon]

Maybe the three participants should instead have been described as three wise quick thinkers of equal politeness, but that's not as laconic.

I can see this board getting rather tedious

Julian F. G. W.

Julian, I enjoyed the puzzle

As did I of course. I thought it well constructed, giving exactly the right level of information.

John