At the end of a soccer season, every player on a team scored a prime number of goals, and the average for the team as a whole was also a prime number. No playerís tally was equal to anotherís, and no playerís tally was the same as the average.
Given that nobody scored more than 45 goals, how many goals did each player score?
*** There are 11 players in a soccer team.
The team average must be an odd number, so the team total must be an odd number, so 2 cannot be one of the scores (because that would make the total even).
That leaves 13 primes less than 45: 3,5,7,11,13,17,19,23,29,31,37,41,43, which sum to 279.
The average of the 11 scores is unchanged if we treat the average as a 12th score. So, in exploring the range of possible averages, all we need to do is exclude one prime:
The minimum possible average is (279 - 43)/12 = 19.7
The maximum possible average is (279 - 3)/12 = 23
The only prime between 19.7 and 23 is 23, so that is the team average.
The team total is 23 * 11 = 253.
The other score to be removed = 279 - 253 - 23 = 3.
So the team scores are all the primes under 45 except for 2, 3 and 23.
Edited on April 19, 2013, 9:32 am