Quote by banlwales
On the theme of hat colours, here's a puzzle that was on Radio 4 a couple of years ago:
It's Bilbo Baggins's birthday and he decides to throw a huge party which is going to go on for weeks and weeks - but he does like to sleep so the guests go home each evening and return the next day.
The trouble is there is not enough room at Bag End so when they all arrive Bilbo gives everyone a hat. A white hat means that that guest is invited and may stay, a black hat means that he or she may not.
Everyone knows that there is a least one black hat. The guests cannot see their own hats but can see everyone else's. They don't even try to look at their own hats. Even in the evening when they go home they are too proud to peek.
Pride also prevents them from returning to the party once they work out whether they have a black hat as they know they haven't really been invited. It turns out that there are ten black hats although only Bilbo knows this. How long is it before only white hatted people turn up to the party?
That'll be the 11th night surely.....or is that one of Shakespeare's plays...?
Sorry, I had to pop out - Well done, banlwales.
Here's the reasoning for anyone interested:
Suppose there had been just one black hat. The partygoer wearing it would see no other black hat, and knowing that there must be at least one black hat, would deduce that he himself must be wearing it. He would not turn up on the second night.
Suppose there where two black hats. Each black-hatted one would see one black hat only and would be unable to deduce anything about his own hat. On the next day they would therefore both turn up. Seeing one black-hatted one turn up enables the other black-hatted one to deduce his own hat is black, otherwise the other black-hatted one would not have turned up again on the second night. So both black-hatted ones deduce on the second night that they are both wearing black hats and so don't turn up on the third night.
Similarly, if there are three black hats, these three will not turn up on the fourth night. If there were 10 black hats they would realise this on the 10th night and not turn up on the 11th night.
