No. of players |
2 |
3 |
4 |
5 |
n |
1234 |
Mo. of games |
1 |
2 |
3 |
4 |
n-1 |
1234 - 1 = 1233 |
On the other hand let n be the number of games, played and r be the number of players remaining in the tournament.
After every game, r will be reduced by 1.
r → no. of players remaining
n → no. of games played
If r = 2 then n = 1
As n increases, r decreases
n, r : = n + 1, r – 1
n + r = (n + 1) + (r – 1)
= n + 1 + r – 1
= n + r
Therefore n + r is invariant. n + r = 1234 (No. of players initially)
The winner of the tournament is the player that is left after all other players have been knocked out. After all the games, only one player (winner) is left out.
i. e. n = 1
Put n = 1 in (1)
n + r = 1234 …. (1)
1 + r = 1234
r = 1234 – 1 = 1233
No. of games played = 1233