1729 mathematicians are at a party. Some pairs of mathematicians are already friends. Others don't yet know each other.
At the start of the evening, the host, who is not one of the 1729, announces, "if you have an even number of friends at the party, put on a blue hat."
One of the guests, Alex, looks around as all the other mathematicians follow the instruction but realizes he doesn't know how many friends he has at the party. He counts 163 mathematicians wearing blue hats.
Should Alex put o...
