It is a social and logical piece at the same time: each answer fits with the others until the party is secretly ordered. The solution doesn't count one-to-one squeezes; discover the structure that forces them.
The marriage handshake
Riddle statement
At a party there are several married couples. No one shakes hands with themselves or their own spouse.
You ask everyone else how many hands they have shaken and you get different answers.
How many hands has your spouse shaken?
Show solution
Solution
Answer: Your spouse has shaken $n - 1$ hands, where $n$ is the total number of couples at the party.
Explanation:
If there are $n$ couples, then there are $2n$ people. You ask everyone except yourself, so you receive $2n - 1$ answers.
Each person can have shaken between $0$ and $2n - 2$ hands: no one can shake hands with themselves or with their own spouse.
Since all the answers you receive are different, they must be exactly:
Now we pair off the extremes.
The person who shook $0$ hands must be married to the person who shook $2n - 2$ hands. The latter shook hands with everyone except their spouse; therefore, their spouse is precisely the person who shook no hands at all.
Once that pair is removed, the same reasoning repeats: among the remaining people, the person who shook $1$ hand must be married to the person who shook $2n - 3$ hands; the person who shook $2$ hands, to the person who shook $2n - 4$ hands; and so on.
All the answers are paired off by extremes:
But one answer remains exactly in the middle:
That answer cannot belong to any of the other paired couples. Therefore, it must be your spouse’s answer.
So your spouse has shaken exactly:
hands.