The intelligent footballer

I first came across this mindbender in a great puzzle book called "The Moscow Puzzles" by Boris A. Kordemsky. This is the most popular puzzle book ever published in the Soviet Union and it contains a collection of 359 brain teasers ranging from simple catch riddles up to incredibly difficult problems. I have adapted one from the "Difficult Problems" chapter to include here.

The puzzle

The manager of Fulchester United is looking for a player with brains as well as brawn to join his squad. He think he has found his man, so he sets him a problem to solve to check that he is quick witted. In a training session he circles the player with 12 members of the squad along with himself. He tells the player the rules of the problem. The triallist is to pass the ball to one of the players encircling him. That player then passes the ball back to him in the middle and drops out of the circle. The triallist must then pass the ball to the player who is thirteen places around the circle amongst the remaining players, that player then passes the ball back to him in the middle and drops out of the circle. This continues until one player is left. That player must be the manager. With which player should he start the sequence to ensure that the last player left is the manager?

The solution

In the diagram above, if the starting player is denoted by an X, then if the puzzle procedes in a clockwise direction around the circle then the players that drop out are, in order, X, 1, 3, 6, 10, 5, 2, 4, 9, 11, 12, 7 & 8. So the player in position 8 is the last one left. The player in position 8 is five places anti-clockwise from the starting position. If this is to be the manager, then the game must be started five places clockwise from the manager.

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