Colourful Hats

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This is a true classic. There are three logicians and five hats (three red ones and two blue ones). The hats are shuffled and one hat placed on each person’s head (the other two hats are hidden). As it turns out, each person is wearing a red hat. The hats are placed such that each person cannot see the colour of their hat, only the colour of the other people’s hats. They are asked if they can determine the colour of there hats.

After a brief pause, the smartest person is able to do so. How?

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… er, He asks his neighbour?

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… er, He asks his neighbour?

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OMG, I’m in knots over this. Best start with the situation where A, the smartest, is actually wearing blue.
If this was the case, B, the next smartest would be watching C. B knows that if he is wearing blue, C would have declared his own hat red. But C doesn’t speak, so B would know he was not wearing blue and must therefore be wearing red, and duely announce it. But A hears none of this from either of the others, and therefore concludes that his own hat can’t be blue, and must therefore be red.

Phew! That’s worse than my spaghetti code!

Alan

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Yeah – that is the answer but I’ve always had a problem with it. It assumes that person B is playing with a straight bat. What happens if person B really hates person A and, more that being first to annouce what colour hat they are wearing, they would perfer that person A got it wrong. In that case, it could go like this … person A is wearing a blue hat. Person B looks at A (sees blue), looks at C (sees red), waits until C doesn’t say anything, therefore concludes that they must be wearing red. THEN thinks, now if I pause, then person A will go though the logic you mentioned and annouce RED – oh how sweet that would be when they take off their hat and see blue. So … person B pauses and pauses … and BINGO!

But wait, it gets worse, what if person C really hates person A and B!

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Ah yes, but that would mean that person B was the best logician, which is what the question asked. In fact, such a ploy could be regarded as tactical genius, forcing A to answer wrongly but using A’s answer, rather than silence, to determine that his own hat is, in fact, red. Love/ hate aside, this might be a classic case of conning the conman.

great problem

Alan

BTW, I think I stumbled on a (counter)revolutionary solution to this sexist / political correctness rubbish that’s forced us to write “if he/she gives his/her hat to him/her ….” I coined what I believe is a revolutionary term the other day – after being corrected on this rubbish, I simply qualified myself by stating that I was using the generic “he/ his/ him”.

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It’s simple…. he took his hat off to wipe his brow, and checked out the colour before putting it back on.

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Not as simple as my answer above, but as pointed out maybe his neighbour would lie.

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Not as simple as my answer above, but as pointed out maybe his neighbour would lie.

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Even simpler – the one facing the mirror wins every time.

Alan

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Even simpler – the one facing the mirror wins every time.

Alan

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It’s simple…. he took his hat off to wipe his brow, and checked out the colour before putting it back on.

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Ah yes, but that would mean that person B was the best logician, which is what the question asked. In fact, such a ploy could be regarded as tactical genius, forcing A to answer wrongly but using A’s answer, rather than silence, to determine that his own hat is, in fact, red. Love/ hate aside, this might be a classic case of conning the conman.

great problem

Alan

BTW, I think I stumbled on a (counter)revolutionary solution to this sexist / political correctness rubbish that’s forced us to write “if he/she gives his/her hat to him/her ….” I coined what I believe is a revolutionary term the other day – after being corrected on this rubbish, I simply qualified myself by stating that I was using the generic “he/ his/ him”.

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Yeah – that is the answer but I’ve always had a problem with it. It assumes that person B is playing with a straight bat. What happens if person B really hates person A and, more that being first to annouce what colour hat they are wearing, they would perfer that person A got it wrong. In that case, it could go like this … person A is wearing a blue hat. Person B looks at A (sees blue), looks at C (sees red), waits until C doesn’t say anything, therefore concludes that they must be wearing red. THEN thinks, now if I pause, then person A will go though the logic you mentioned and annouce RED – oh how sweet that would be when they take off their hat and see blue. So … person B pauses and pauses … and BINGO!

But wait, it gets worse, what if person C really hates person A and B!

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OMG, I’m in knots over this. Best start with the situation where A, the smartest, is actually wearing blue.
If this was the case, B, the next smartest would be watching C. B knows that if he is wearing blue, C would have declared his own hat red. But C doesn’t speak, so B would know he was not wearing blue and must therefore be wearing red, and duely announce it. But A hears none of this from either of the others, and therefore concludes that his own hat can’t be blue, and must therefore be red.

Phew! That’s worse than my spaghetti code!

Alan

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