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 all different answers. How many hands has your spouse shaken?
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The marriage handshake
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.
Hints
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- The possible answers range from 0 to a maximum that depends on the number of couples.
- If all the answers are different, they necessarily form a complete consecutive list.
- Extreme responses are paired as spouses: 0 with the highest, 1 with the next, and so on.
Solution
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Answer: Your spouse has shaken the middle number of hands. If there are \(n\) couples in total at the party, then you have shaken exactly \(n-1\) hands. Explanation: You ask everyone else, so you get \(2n-1\) different answers. Since no one can shake hands with themselves or their spouse, each person may have shaken between 0 and \(2n-2\) hands. Since they are all different, the answers must be exactly: \[
0,1,2,\dots,2n-2.
\] Now observe the following: - The person who gave 0 hands must be married to the person who gave \(2n-2\), because that person greeted everyone except their spouse, who is precisely the one who did not greet anyone;
- similarly, the one who gave 1 hand is married to the one who gave \(2n-3\);
- who gave 2 hands, with whom he gave \(2n-4\);
- and so on. Couples are matched by complementary sums. Everyone is paired except for one answer: the one in the middle. That intermediate response is necessarily that of your spouse, and it is worth \[
n-1.
\] That's why your spouse has exactly shaken \(n-1\) hands.
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