9th Australian 1988 Problem B1 A city has 11 bus stops on each route. Any two bus routes have just one stop in common, and any two stops lie on just one route. How many routes are there?
1 or 111 Solution The wording of the question implies that there is at least one route. But there could be just one. That is one possible solution. Suppose there is more than one. Let R, S be any two routes. They must have a stop X in common. Let the other stops on R be R 1 , R 2 , ... , R 10 and the other stops on S be S 1 , S 2 , ... , S 10 . Now there is just one route through R i and S j . Call it (R i , S j ). Note that all these 100 routes must be distinct and must be distinct from R and S (or we would have more than one route through two stops). But any route T must have a stop in common with R and a stop in common with S. If neither of those stops is X, then it must pass through some R i and some S j , so it must be the route (R i , S j ). If it passes through X, then it cannot have any other stops in common with R or S. But it must have a stop in common with Q = (R 1 , S 1 ). Q has 9 other stops, call them Q 1 , Q 2 , ..., Q 9 . So T must pass through X and some Q i . But there is a unique route (X, Q i ) through X and Q i . Thus we have R, S, 10 2 routes (R i , S j ), and 9 routes (X, Q i ), total 111. Solutions are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust. It can be ordered by email. 9th Aus 1988 © John Scholes jscholes@kalva.demon.co.uk 6 Jan 2004 Last corrected/updated 6 Jan 04