I have (1) a math problem to tell you about (though I suspect many readers already know it), (2) a story about it, and (3) a point to make. TODAY I'll just do the math problem. Feel free to leave comment with solutions--- so if you haven't seen it before and want to try it, then don't look at the comments. Tommorow or later I will tell you the story and make my points.
PROBLEM 1: There are n people sitting on chairs in a row. Call them p1,...,pn. They will soon have HATS put on their heads, RED or BLUE. Nobody can see their own hat color. pn can see p(n-1),...,p1. More generally, pi can see all pj j < i. They CAN meet ahead of time to discuss strategy.
Here is the game and the goal: Mr. Bad will put hats on people any way he likes (could be RBRBRB..., could be RRRBBB, could be ALL R's - like when a teacher has a T/F test where they are all FALSE.) Then pn says R or B, p(n-1) says R or B, etc. They want to MAXIMIZE how many of them say THEIR hat color. Assume that Mr. Bad knows the strategy the people will use.
What is the best they can do?
Here is a strategy: pn says R if the MAJORITY are R, and B if the MAJORITY are B, and then everyone says what pn says. They are guaranteed around n/2 correct.
Here is a strategy: Assume n is even. pn says the color of p(n-1). p(n-1) then says what pn said and gets it right. then p(n-2) says what p(n-3) has. Then p(n-3) gets it right. You are guaranteed to get around n/2 right.
GEE- can we do better than n/2? Or can one prove (perhaps using Ramsey Theory, perhaps something I learned at Erdos 100 over dinner) that you can't beat n/2 (or perhaps something like n/2 + log(log(n))).
PROBLEM 2: Same as Problem 2 but now there are c colors of hats.
NOTE- there are MANY hat problems and MANY variants of this scenario--- some where you want to maximize prob of getting them all right, some where everyone sees everyones hat but their own. These are all fine problems, but I am just talking about (1) people are in a row, (2) Want to maximize how many they get right in the worst case.
ADDED LATER- WARNING- THE ANSWER TO PROBLEM 1 IS IN THE COMMENTS NOW.
SO IF YOU WANT TO SOLVE IT YOURSELF DO NOT LOOK AT THE COMMENTS.