# Welcome to the club

A financial manager of a bowling club, whose members are all couples (a man and a woman), creates three different membership lists.
In the first list the couples are sorted by increasing age of the men.
In the second list the couples are sorted by increasing age of the women.
In the third list the couples are sorted by increasing age of the couple (sum of the age of the couple).

In the first list the couple A is at position 7 and couple B at position 8.
In the second list the couple B is at position 7 and couple A at position 8.
In the third list couple A is at position 1 and couple B is at the last position.

How many couples are members of the club?

## 1 Answer

The total number of couples is

14.

Proof: let $$A_1$$ and $$A_2$$ be the ages of the man and woman (respectively) in couple A, and similarly $$B_1$$ and $$B_2$$ in couple B. We know three facts:

• $$A_1+A_2$$ is minimal and $$B_1+B_2$$ is maximal among all couples;
• $$A_1$$ is seventh largest and $$B_1$$ is eighth largest;
• $$B_2$$ is seventh largest and $$A_2$$ is eighth largest.

That means there are six men younger than $$A_1$$. All of their wives must be older than $$A_2$$, otherwise we would contradict minimality of $$A_1+A_2$$. So there are exactly six couples with a man younger than $$A_1,B_1$$ and a woman older than $$A_2,B_2$$.

Similarly,

for every man older than $$B_1$$, his wife must be younger than $$B_2$$ otherwise we contradict maximality of $$B_1+B_2$$. So there are exactly six couples with a man older than $$A_1,B_1$$ and a woman younger than $$A_2,B_2$$.

That makes an exact number of couples in all, and we have solved the puzzle.

An example to show that this is possible:

Men's ages: 24, 25, 26, 27, 28, 29, 30, 50, 51, 52, 53, 54, 55, 56
Women's ages: 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53
Couples' ages: 71, 72, 73, 74, 75, 76, 77, 83, 84, 85, 86, 87, 88, 89

• ... what about if all ages are equal? Doesn't that mean arbitrary order (and a minimum of 8 couples)?
– Avi
Feb 11 '20 at 14:25
• @Avi I guess we're assuming that there actually is an ordering (i.e. 7th place means 7th place, not possibly equal 7th place with some others). Good point though; I thought of the same thing. Feb 11 '20 at 15:48
• Yup, fair enough
– Avi
Feb 11 '20 at 15:57
• I don't understand which woman and man end up in which couple in your final example. Could you edit it to label the men and women as A, B etc as well as their ages? Feb 11 '20 at 17:22
• @Vicky can't you figure it out based on the position of the ages, since that's given in the puzzle?
– Kat
Feb 11 '20 at 22:41