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Pass the Candy!

Submitted By:eighsse
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Difficulty:*** (2.21)

A group of 9 friends have a package of 40 W&W's chocolate candies to share. They each, one at a time, take a prime number of W&W's to eat. After that, the bag is empty. Exactly four of the friends took a number of W&W's that had previously been taken by someone else. Of the group, the number of people who took exactly 5 is twice the number of people who wear glasses.
Without any regard to the order in which they were taken, what individual quantities of W&W's were taken?


Answer: 2, 2, 2, 3, 3, 5, 5, 7, 11

There is only one way that any number 13 or greater can be included: {2,2,2,3,3,3,5,7,13}. This would be a valid solution, but... We know that an even number of people must have taken 5 W&W's, because it is twice the number that wear glasses.

This leaves only 5 unique primes (2,3,5,7,11) that can be involved, and we know they all are used, because there must be exactly 5 different primes (from the fourth sentence in the teaser). And we know that 5 must be used an even number of times.

So, that gives us {2,3,5,5,7,11} requiring three more primes whose sum is 7. Obviously, the other three must be {2,2,3}.

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