Enigma 1073: Cross-country match
26 February 2018
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From New Scientist #2229, 11th March 2000 [link]
In cross-country matches, teams consist of six runners. The team scores are decided by adding together the finishing positions of the first four runners to finish in each team. The team with the lowest score is the winner. Individuals never tie for any position and neither do teams because if two teams have the same score the winner is the team with the better last scoring runner.
The fifth and sixth runners to finish in each team do not score. However if they finish ahead of scoring runners in another team they make they make the scoring positions of those scoring runners, and the corresponding team score, that much worse.
In a recent match between two teams, I was a non-scorer in the winning team. Each team’s score was a prime number, and if I told you what each team’s score was you could deduce with certainty the individual positions of the runners in each team. I won’t tell you those scores, but if you knew my position you could, with the information given above, again deduce with certainty the individual positions of the runners in each team.
(1) What was my position?
(2) What were the positions of the scoring runners in my team?