Every two players play each other once. The outcome of each game is a win for one of the players. Player n wins an games and loses bn games. Prove that ∑ an2 = ∑ bn2.
Solution
Suppose there are N players in total. Each player plays N-1 games, so bn = N - 1 - an. Hence ∑ bn2 = ∑ (N - 1)2 - 2(N - 1) ∑ an + ∑ an2 = N(N - 1)2 - 2(N - 1) ∑ an + ∑ an2.
Each game is won by just one player, so ∑ an = no. of games = N(N - 1)/2. Hence ∑ bn2 = N(N - 1)2 - 2(N - 1)N(N - 1)/2 + ∑an2 = ∑an2.
© John Scholes
jscholes@kalva.demon.co.uk
25 Jan 2002