EL PA?S Mathematical Challenge presented by Juan Mata

Juan Mata, footballer for Spain and Chelsea, introduces the 27? mathematical challenge of EL PA?S to celebrate the centenary of the Real Sociedad Matem¨¢tica Espa?ola. Please send your solution to problemamatematicas@gmail.com for the chance to win a selection of maths books.

Two high school students, who are goalkeepers, decide to organize a football match. Each of them must choose 10 players out of 20 fellow students. To do so, the 20 candidates line up and each goalkeeper makes his selection, alternately, but they can only choose from the two players who are at either end of the line.

The players have played in a previous tournament, and the goalkeepers know how many goals each of the players scored. The aim of the goalkeepers is choose a team that scored more goals in the previous tournament than the one their rival chooses. The challenge is to find the strategy that the first goalkeeper can use in order to choose a team that will always have scored at least as many goals as their rivals, no matter where the players are in the line, nor how many goals they scored.

The second part of the challenge is as follows. Is there a similar strategy that either the first or second goalkeeper can use if they have to choose from a group of 21 players? (It is understood that one player will end up not being picked and will not get to play.)

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