1. ## Chirstmas doll puzzle

Father Christmas had a problem. He was making up a batch of dolls but could not remember exactly how many of each kind he needed. He knew that he needed 57 in total, that 27 had to have blue eyes, and 29 had to have fair hair. His assistant gnome pointed out that some had to have both features and remembered that 3 dolls were needed having blue eyes and fair hair, but which were not able to say 'Mama'. This prompted the old man to remember that he needed a total of 34 dolls able to say 'Mama', of whom 17 needed to have fair hair. The assistant was going around in circles juggling all this information but couldn't work it out. He pressed for any other bits of information that Father Christmas just might be able to manage, no matter how trivial. The only items he got were: "Every doll had at least 1 of the features named"; "That there was one combination of features not asked for at all" and "Several children had asked for all 3 features in the one doll". That was enough!

How many fair-haired, blue-eyed dolls saying 'Mama' were needed?

2. 13. Here's why:

There are eight possible combinations, which I will refer to as A - H, as follows:

A. Blue, Fair, Mama
B. Blue, Fair, NotMama
C. Blue, NotFair, Mama
D. Blue, NotFair, NotMama
E. NotBlue, Fair, Mama
F. NotBlue, Fair, NotMama
G. NotBlue, NotFair, Mama
H. NotBlue, NotFair, NotMama

Here are the clues:

1. He knew that he needed 57 in total,
2. that 27 had to have blue eyes [2a. so 30 do not have blue eyes]
3. and 29 had to have fair hair.
4. His assistant gnome pointed out that some had to have both features
5. and remembered that 3 dolls were needed having blue eyes and fair hair, but which were not able to say 'Mama'.
6. he needed a total of 34 dolls able to say 'Mama',
7. of whom 17 needed to have fair hair [7a which means that 17 did not have fir hair].
8. "Every doll had at least 1 of the features named";
9. "That there was one combination of features not asked for at all"
10. "Several children had asked for all 3 features in the one doll".

From clue 5 you know B = 3
From clue 8 you know H = 0
From clue 3, A+B+E+F = 29. We already know B = 3, and from clue 7 A+E = 17. So F=9.
From clue 9 you know one of the remaining is 0. Let's guess that C=0 (you can try others, but this seems to work out)
Then from c7a, since C=0 that means G = 17.
From 2a, since E+F+G = 30 and already know F=9 and G=17, so E=4.
From 7, given E=4 then A = 13.
Finally from 2, since A+B+C+D = 30, then D=11.

So:
A = 13 = Blue, Fair, Mama