24th Australian 2003 Problem B1 After 10 minutes A, B, C were in that order in a cycle race. Thereafter the leader changed 19 times and the rider in third place changed 17 times. At no time were all three riders abreast. B finished third. Who finished first?
C Solution The initial order is ABC. The final order must be ACB or CAB. But ACB differs from ABC by 1 transposition and CAB differs from ABC by two transpositions. Each change in the first place involves one transposition and each change in the last place involves one transposition. So we know that there were at total of 36 transpositions between the initial and final order. An even number of transpositions can give the identity, but an odd number cannot. So the final order must be CAB. 24th Aus 2003 © John Scholes jscholes@kalva.demon.co.uk 15 June 2003 Last updated/corrected 15 Jun 03