from Peter Cave’s Do Llamas Fall In Love: 33 Perplexing Philosophical Puzzles
Suppose ten philosophers, attending an international convention, are sitting in a semi-circle at the dining table, having just finished some gorgeous creamy dessert in which they all delighted. They can see everyone’s face, except their own. No mirrors, no squinting allowed; and they are not permitted to confer. You are the draconian waiter, gazing at them all.
No philosopher knows that he has cream on his lips; he cannot see himself. Just two of them have, though, been messy eaters this time, their lips all creamy; the others, surprisingly, are sparklingly clean. For ease, let us call those with cream on their lips ‘creamy’.
Naturally, all of them, being philosophers, are perfectly rational reasoners – despite the flow of wine – ready to obey instructions. (Well, true, there is something a little 'out of this world' re this puzzle.)
You say to them, ‘Hands up all those who know that they are creamy.’
They can all see at least one creamy philosopher. That there is at least one creamy philosopher is universally known, mutually known. Not one of them, though, knows that he is creamy. So, no hands go up. Importantly, no hands go up, if you ask the question again – and again. No philosopher can work out by reason alone that he must be a creamy one.
BUT NOW.... Suppose the same set-up, except that, before any instruction, you announce, ‘At least one of you is creamy.’
Of course, they knew that in the first scenario: they could all see at least one creamy philosopher. Your announcement, though, magically it seems, alters what will happen.
You now instruct, ‘Hands up all those who know they are creamy.’
No hand is raised.
You make the same demand again, ‘Hands up all those who know they are creamy.’
Amazingly, the two creamy philosophers, and only those two, do now put their hands up. Your announcement, yet of what they each already knew, made all the difference.
Why did the creamy philosophers now raise their hands?
[This puzzle has a long history from unfaithful wives to muddy children.]