### Brain Teasers

# Pass the Candy!

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?

Without any regard to the order in which they were taken, what individual quantities of W&W's were taken?

### Hint

Even though no information is given about how many people are wearing glasses, the last statement is actually completely relevant, and in fact crucial! What does it tell you about the number of people who took 5 pieces?### Answer

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