I have produced a new kindle ebook containing many of the previous Friday Puzzles! It is called PUZZLED (available in the UK here and USA here) and contains 101 puzzles and the solutions. If you enjoy the Friday Puzzles and the blog, feel free to show your appreciation by buying one of the books.
This week’s puzzle was kindly sent to me by Milan V.
Imagine a monastery in which ten of the monks may have a disease which causes them to have blue spots on their foreheads but has no other symptoms. All the monks have taken a vow of silence, they meet just once a day, and there are no mirrors in the monastery, so nobody knows whether he has a blue spot on his forehead or not.
If a monk discovers that he has a blue spot on his forehead, he will have to leave the monastery by the end of the day. All the monks are perfect logicians – that is, they can instantly infer all the logical consequences of any statement made to them – and they all know that all the other monks are perfect logicians.
One day, the Guru, who is known to be truthful, gathers all the monks together and announces “At least one monk in this monastery has a blue spot on his forehead.” Nothing happens for nine days, but on the tenth day, all the monks with blue spots leave.
How many monks left and why?
If you have not tried to solve it, have a go now. For everyone else, the answer is after the break.
This is how Milan presented the explanation:
“In order to solve it you first have to consider a scenario, where there’s only single ill monk. He looks around, fails to see anyone with a dot on his forehead and so concludes that he himself has to be the one with the disease. He packs
his things and leaves by the end of the first day. Now, consider there are two ill monks, say you and me. You see that I (and no-one else) have a dot on my forehead, so you naturally expect me to leave by the end of the first day. But I don’t because I see you with the dot and expect the same from you. The only possible reason is that we both have the dots, so we both pack our things and leave by the end of second day.
Now, there is you, me and John. We both clearly see that John is sick and each one of us see one additional spotty monk. John, on the other hand, sees the two of us. So each one presumes that the other two will have left by the end of the day and the only reason why they have not is that there are actually three of us who have the dots. So we all just leave together by the end of the third day.
The rest is just extrapolation. Each additional ill monk makes the inference process a day longer. Ten days, ten monks.”
Did you solve it? Any other explanations?