Mathematical Problem 1: Solution

Here's a solution to Mathematical Problem 1, posted on 13 March 2010.

With 2 players, A wins a match of 29 rounds when A has won 15 rounds. Even if A loses all the remaining rounds B will only have won 14.

The principle is that A would still win even if all the remaining games were to be won by B (A's closest rival).

With 3 players, A wins when a is greater than b added to the number of rounds still to be played (assuming B has the second highest number of wins so far). The number of rounds still to be played is 29 – (a + b + c). So, algebraically:

a > b + [29 – (a + b + c)] which simplifies to 2a > 29 – c.

With 4 players, A wins when

a > b + [29 – (a + b + c + d)] which simplifies to 2a > 29 – (c + d).

The general solution , with any number of players and n rounds, is that A wins when

a > b + [n – (a + b + c + d + e + ...)] which simplifies to 2a > n – (c + d + e + ...)

The surprise in this result is that b does not appear in the simplified rule! Provided B has won more rounds than C, D, E and so on, the actual number that B has won does not make any difference to when A is declared the winner.

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?