When I was sick recently, my boss e-mailed me a math problem he said I might try to solve to pass the time. Now, I had thought that this was a problem he had read somewhere, but no: this is an actual thing that happened to him. It happens to lend itself to a cool problem, and here it is:

You have ten players and five two-player games at a party. There will be five rounds of games, and each round will include all five games played simultaneously. You want every player to play each game once, and never play against the same player more than once. Is there a solution?

Once you’ve worked that out, consider:
(a) Is there a solution if you have N rounds of games and 2N people, where N is an arbitrary odd number?
(b) Is there a solution to (a) if N is even?

I know the answer to (a). I suspect I know it for (b) but I can’t prove it. I’d like to see what you all can come up with.

UPDATE 11/14/2011 This problem has been completely solved in the comments! Check them out.

UPDATE 2: 11/15/2011 I’ve also put the C++ code I used to brute-force a solution in the N=6 case online here, in case anyone’s interested.