## Friday, January 12, 2007

### Five Daughters

Mr. Reader's five daughters each gave books for Christmas to one or more of her sisters. Each presented four books and each received four books, but no two girls allocated her books in the same way. That is, only one gave two books to one sister and two to another. Beth gave all her books to Alice; Christy gave three to Edith. Which sisters gave the four books to Deborah?

1. Let's see if I got it right this time:

Dorothy received books from Edith, Christy, and Alice.

2. OK, I am going to go out on a limb here and say that Deborah gave all the books to herself (she kept her own).

In order to meet the premise that each daughter distributed the books in a different way there must be five ways to distribute the books: 0, 1-3, 2-2, 3-1, and 4. Now, I do have an issue with the 1-3/3-1 as being different, but that is the only way I can come up with 5 different ways to distribute.

So, if Alice, Beth, Christy, and Edith have already done some sort of exchange, then I think that Deborah is the '0' exchanger.

3. Nope, I take all that back...

4. OK, I am sticking with my orginal answer, but I have a variation on why... (Am I perseverating over this too much?!?)

Gave -
A - 1 to B
- 2 to C
- 1 to E

B - 4 to A

C - 3 to E
- 1 to B

D - 4 to D

E - 2 to B
- 2 to C

The 1-3/3-1 issue became 1-3 and 1-1-2 distributions. OK, so I have had too much coffee this morning...

5. Indeed you have, Mr. Don. Gill's original answer was correct. She got two from Edith, one from Alice, and one from Christy.

I did this by making a chart that looks like this (based on info given), where the rows represent books given and the columns represent books received. Each row and each column have to add to 4.

A B C D E
A X
B 4 X X X X
C X X 3
D X X
E X X

Also, each row must be unique and be one of the 5 different possibilities (4, 3-1, 2-2, 1-2-1, 1-1-1-1).

My final chart looks like this

A B C D E
A X 1 1 1 1
B 4 X X X X
C X X X 1 3
D X 2 2 X X
E X 1 1 2 X

6. Ok, so the formatting didn't come out like I had hoped. On the first chart there should be Xs down the diagonal (since you can't give books to yourself), and Xs all down the row and column containing the 4 (row 2, column 1). On both charts, the top row of "A B C D E" should be a little to the right so it actually sits over the five columns correctly. That should be easy to reconstruct.

7. Nice work! This is such a great puzzle because you can come up with so much with so little.

Deborah received one book from Alice and Christy, and two from Edith.

I just switched over to the new version of blogger, so look for labels on the new posts!

8. Duh! I missed the 1-1-1-1 possibility. Boy, are youpeople lucky I don't work at NASA!

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