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 player’s tally was equal to another’s, and no player’s 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 number of primes less than 45 is 14, thus three of the primes less than 45 are not represented by the team players, with one of the primes not represented by the team players represented as the team average.

The smallest 11 primes averages a fraction less than 15 and the largest 11 primes less than 45 averages a fraction greater than 24. The team average is given as a prime number and the primes between 15 and 24 are 17, 19, and 23, thus the team average is one of those three primes.

The sum of primes less than 45 is 281. Subtracting the possible team average prime of 17 gives a total of 264. 264 mod 11 is 0, thus, if 17 is the team average, the two other primes will sum to a multiple of 11.

Following is a table of primes (less than 45) mod 11:
Prime : 2,3,5,7,11,13,17,19,23,29,31,37,41,43
Mod 11: 2,3,5,7, 0, 2, 6, 8, 1, 7, 9, 4, 8,10
As the two additional missing primes must have a digit sum equal to 11, thus, the possible prime pairs not represented with the corresponding average of included primes are:
 2 and 31: 21
 3 and 19: 22
 3 and 41: 18
 7 and 37: 20
13 and 31: 20
23 and 43: 18
29 and 37: 18

Subtracting the possible team average prime of 19 gives a total of 262. 262 mod 11 is 9, thus, if 19 is the team average, the two other primes plus (11 - (11 - 9)) will sum to a multiple of 11.

Again, as the two additional missing primes must have a digit sum equal to 11, the possible prime pairs not represented with the corresponding average of included primes are:
 2 and  7: 23
 2 and 29: 21
 3 and 17: 22
 5 and 37: 20
 7 and 13: 22
13 and 29: 20
23 and 41: 18

Subtracting the possible team average prime of 23 gives a total of 258. 258 mod 11 is 5, thus, if 23 is the team average, the two other primes plus (11 - (11 - 5)) will sum to a multiple of 11.

Again, as the two additional missing primes must have a digit sum equal to 11, the possible prime pairs not represented with the corresponding average of included primes are:
 2 and  3: 23
 3 and 13: 22
 5 and 11: 22
 5 and 37: 20
 7 and 31: 20
17 and 43: 18
19 and 41: 18

The only prime average among the possible pairs is 23. As this average is not represented by an individual players score, the set of player scores are:
{5,7,11,13,17,19,29,31,37,41,43}
and the team's season average is 23.

Edited on April 17, 2013, 1:26 pm

Posted by Dej Mar on 2013-04-17 13:21:32