1. ## football tournament

"Ten football teams play in a tournament, and every two of them have to play eachother.

Show that, at any given time, there are at least two teams which have played the same number of games."

I could draw out tedious diagrams and I understand the logic of this (in my head - i think), but how do you go about explaining it mathematically (and simply)?

2. ## Pigeon-hole principle

Hello billym
Originally Posted by billym
"Ten football teams play in a tournament, and every two of them have to play eachother.

Show that, at any given time, there are at least two teams which have played the same number of games."

I could draw out tedious diagrams and I understand the logic of this (in my head - i think), but how do you go about explaining it mathematically (and simply)?
By the end of the tournament, each team has played 9 matches. So at any given time, the number of matches that each team has played is between 0 and 9 inclusive. There are 10 teams, and the only way they can all have played different numbers of matches is if all 10 numbers from 0 to 9 are used.

But, since each match is between 2 teams, if we add up the number of matches played by each team, we must get an even number. And 0 + 1 + 2 + ... + 9 = ?