# Number of candies

A few candies were distributed to three people Alice, Bob, and Charlie. The sum of the reciprocals of the number of each person's candies was equal to 1. (Alice, Bob, and Charlie knew their number of candies and they knew that the sum of the reciprocals was 1, but they did not know the number of candies the other two persons had received.)

I asked Alice whether she knows the number of each person's candies and Alice said she does not know it. Also, I asked her if she knows the total number of candies and she said she does not know it.

I asked the same questions to Bob and he said that he does not know the number of each person's candies but he knows the total number of candies.

Finally, Charlie answered to the same questions that he knows not only the total number of candies but also the number of each person's candies.

What is the total number of candies?

There are only three different ways of distributing the candies according to the puzzle statement:

        3, 3, 3,
or   2, 4, 4,
or   2, 3, 6


Alice cannot have 4 or 6, and hence must have 2 or 3.

If Bob had 2, he'd known that Alice has 3 and Charlie has 6.
If Bob had 4, he would have deduced Charlie has 4 and Alice 2.
If Bob had 3, he wouldn't be able to deduce the total number of candies.
Hence Bob has 6.

Charlie can figure out everything by elimination.
The total number of candies is 11.

• You had said that there are only 3 ways of distributing the candies according to the puzzle statement : (3,3,3) , (2,4,4) and (2,3,6) . Can you prove that there are no other ways to distribute the candies ? Jul 5 at 16:36
• @HemantAgarwal None of the reciprocals can be 1/1. The average of the reciprocals is 1/3. If none of them is above it, then it's (3,3,3). If at least one of them is above average, only 1/2 can meet this condition while leaving another 1/2 as the sum of the remaining two reciprocals, so it's either (2,4,4) for a more equal distribution or (2,3,6) if one of them is above 1/4. Jul 7 at 6:17

There are only a few possibilities: You can have $2+4+4=10$, or $2+3+6=11$, or $3+3+3=9$. Alice must have gotten $2$ or $3$, as otherwise she would at least know the total, and Bob knows this. Bob must have gotten $6$, as otherwise he would know how many Alice got. Charlie knows what he and Alice got, but we do not.