Mathematical problem 1: 29 rounds

Now and again I shall offer some interesting mathematical problems that my blog readers might like to solve – and perhaps use in the classroom, if they are teaching.

Here's a little mathematical problem that could be used with bright Year 6 pupils (10 to 11 years).

I was watching a snooker tournament final. This consisted potentially of 29 'frames' (rounds). But when the score was 15 to player A and 9 to player B then A was declared the winner, even though only 24 frames had been played. Why?

That's easy, of course. But I wondered how it would work if there was some kind of game of 29 rounds between 3 players (A, B and C), in which only one player can win each round. When would the match be over and one of the players declared the winner? Let's use a, b and c for the numbers of rounds won so far by A, B and C.

If a = 12, b = 9 and c = 6, then A has won and the match is over. Why?

But if a = 11, b = 9 and c = 6, then no player is certain yet to win and the match must continue. Why?

Can you find a general rule for when we can say that A has won? (Assume that A has the highest score and C the lowest).

You know what's coming next ... what about a rule for 4 players playing 29 rounds? 5 players?

Can you find a generalization for any number of players playing a game consisting of n rounds?