# Game theory problem

1. Oct 10, 2004

### galois427

i need help with solving tihs problem. i'm not really sure how to prove it.

several people started with $300 each, and played a game with the following strange rules. each player pays$10 to the house at the beginning of each round. during each round, one active player is declared the loser, and he distributes all of his money in equal amounts to the remaining players. the loser must then leave, but all of the other players go on to the next round. the game is over when only one player remains. at the end of the game, the surviving olayer was surprised to discover that he had exactly \$300, equaling his starting amount. how many players were there at the beginning?

2. Oct 10, 2004

### Gokul43201

Staff Emeritus
This looks more like an algebra/number theory problem to me....

Assume there are p players.

The total initial amount of money is 300p

The money going to the house is as follows :

1st round : 10p
2nd round : 10(p-1)
3rd round : 10(p-2)
.
.
(p-2)th round : 20

Sum these to find the total money given to the house.

The difference is what the winner has.

You have a simple quadratic equation in p. Solve it.

3. Oct 10, 2004

### galois427

can you explain that a little more. how does (p-2)th round : 20?
i found out, by guess and check, that p=58, but how do you go about proving it?

Last edited: Oct 10, 2004
4. Oct 11, 2004

### galois427

nvm, i just figured it out. thanks.

5. Oct 11, 2004

### Gokul43201

Staff Emeritus
Oops, sorry. The last round should be the (p-1)th round.

Use the formula for the sum of p consecutive natural numbers.