Here’s a logic puzzle that may have some vague relevance to the topic.
You and two teammates are all going to be taken into separate rooms and have flags put on your heads. Each flag has a 50% chance of being black or being white. None of you can see what color your own flag is, but you will be told what color flags your two teammates are wearing. Before each of you leave your respective rooms, you may make a guess as to what color flag you yourself are wearing. If at least one of you guesses correctly and nobody guesses incorrectly, you all win. If anyone makes an incorrect guess, or if all three of you decide not to guess, you all lose.
If one of you guesses randomly and the other two choose not to guess, you have a 50% chance of winning. Even though it would seem that knowing what color your teammates’ flags are tells you nothing about your own, there is a way for your team to win this game more than half the time. How can it be done?
My attempt at a solution: if you see two flags of the same color, guess the opposite color, otherwise don’t guess. This wins 75% of the time.
Lemma 1: it’s impossible that everyone chooses not to guess. Proof: some two people have the same color, because there are three people and only two colors.
Lemma 2: the chance of losing is 25%. Proof: by lemma 1, the team can only lose if someone guessed wrong, which implies all three colors are the same, which is 2 out of 8 possible assignments.
This leaves open the question of whether this strategy is optimal. I highly suspect it is, but don’t have a proof yet.
Here’s a logic puzzle that may have some vague relevance to the topic.
You and two teammates are all going to be taken into separate rooms and have flags put on your heads. Each flag has a 50% chance of being black or being white. None of you can see what color your own flag is, but you will be told what color flags your two teammates are wearing. Before each of you leave your respective rooms, you may make a guess as to what color flag you yourself are wearing. If at least one of you guesses correctly and nobody guesses incorrectly, you all win. If anyone makes an incorrect guess, or if all three of you decide not to guess, you all lose.
If one of you guesses randomly and the other two choose not to guess, you have a 50% chance of winning. Even though it would seem that knowing what color your teammates’ flags are tells you nothing about your own, there is a way for your team to win this game more than half the time. How can it be done?
My attempt at a solution: if you see two flags of the same color, guess the opposite color, otherwise don’t guess. This wins 75% of the time.
Lemma 1: it’s impossible that everyone chooses not to guess. Proof: some two people have the same color, because there are three people and only two colors.
Lemma 2: the chance of losing is 25%. Proof: by lemma 1, the team can only lose if someone guessed wrong, which implies all three colors are the same, which is 2 out of 8 possible assignments.
This leaves open the question of whether this strategy is optimal. I highly suspect it is, but don’t have a proof yet.
UPDATE: here’s a proof I just found on the Internet, it’s elegant but not easy to come up with. I wonder if there’s a simpler one.