Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

John and Shane are playing a game, where after each round you get 1 point if you win and lose 1 point if you lose the game. The winner is the one who first reaches 4 points. In how many ways can any one of the players win?

I had this question in the exam yesterday and I was trying to solve it by taking either of the two possibilities game after game and add them all, but I thought it wouldn't work if in general if we have $n$ points for winning.

So can anyone help me show how to solve it?

share|cite|improve this question
I think there are infinite ways of winning. Assume John and Shane win alternately. After 2*n rounds their score would be each 0. Then John wins 4 rounds in a row. And he wins the game. For every natural number n exists 1 solution. (actually a lot more than one) – user38034 Aug 24 '12 at 10:15
Do scores drop below zero? Is it always the case one wins and one loses? – rschwieb Aug 24 '12 at 12:00
@Gerry Myerson I wanted to say at least $4$ times. Of yourse as you said it might be more actually $7$ at most. However the idea is that there is a certain number of differences and among them infinitely many oscillation is allowed. – Seyhmus Güngören Aug 24 '12 at 12:17
As already said by others, there are infinitely countably ways of winning. However, if the game is fair, it takes 16 rounds in the mean before one of the players wins. – Did Aug 24 '12 at 12:22
Could it be that the loser gets $0$ points instead of $-1$ points? – Christian Blatter Aug 24 '12 at 12:42

If player a is awarded a point then player b loses a point and also the other way around. Therefore the final score is always going to be 4,-4 or -4,4. However there are infinite number of ways this can happen. The game can last 4 turns, 6 turns, 8 turns, any even number of turns. Therefore there are infinite ways it can happen but only two possible outcomes.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.