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## Pearls and Girls

Math brain teasers require computations to solve.

 Puzzle ID: #13063 Fun: (2.7) Difficulty: (2.74) Category: Math Submitted By: willymapo Corrected By: Sunrose

Once an old man died, and in his Will he stated that all the pearls he kept in a little bag were to be distributed among his daughters. To make things fair, the distribution must be done in the following way:

The older girl should receive one pearl, and a seventh of the remaining pearls in the bag.

The second girl should receive two pearls, and a seventh of the remaining pearls in the bag.

The third girl should receive three pearls, and a seventh of the remaining pearls in the bag.

And so on until all the pearls are exhausted.
Needless to say, the younger girl grew angry, and went to talk to the Judge because, if distributed that way, she probably would not get a pearl at all.
The wise Judge looked at her and said: "Don't worry, my child. Your father was a wise man. He made everything quite right so that, after all the pearls are given, each one of you will have the same quantity of pearls and there will be no pearls left."

And so it was.
How many pearls were there in the bag?
How many daughters did the old man have?
How many pearls did each girl get?

There were 36 pearls.
There were 6 daughters.
Each daughter got 6 pearls.

The girl number x will have x pearls and a seventh of the rest. Since the last girl leaves no pearls in the bag, then when she takes her pearls the rest in the bag is zero (a seventh of nothing is nothing).
Now, let's say the last girl was number k. Then she took k pearls and left the bag empty. But as all the girls end up with the same number of pearls, then each girl has k pearls in her hands. That is, the total number of pearls is: k pearls * k girls = square of k.
Let's look at the girl number k-1. She takes k-1 pearls and a seventh of the rest, and leaves exactly k pearls for the last girl (remember the last girl found k pearls in the bag and takes them all). But, since each girl ends up with k pearls, then the "seventh of the rest" for the k-1 girl should be 1 pearl (to sum up k pearls at last: k-1+1=k). That means that k pearls of the last girl plus 1 pearl of the k-1 girl divided by 7 results in 1 (a seventh of the rest is 1, (k+1)/7=1 ).
That is, k+1 = 7 and then k = 6.
So, the number of pearls is 6*6 = 36, and there were 6 girls.

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