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Math Help - football tournament

  1. #1
    Member billym's Avatar
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    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)?
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Pigeon-hole principle

    Hello billym
    Quote Originally Posted by billym View Post
    "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 = ?

    Grandad
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