The Couples Handshake Problem

The problem
5 couples = 10 people · no one shakes their own spouseme♥AABBCCDDhost + spousehost asks 9 others · gets 9 distinct counts · spouse's count?

You and your spouse host a dinner party with four other couples — 10 people in total. At various points people shake hands, but nobody shakes their own spouse's hand.

After dinner you ask each of the other 9 people (including your spouse) how many hands they shook. Each gives a different number: 9 people, 9 distinct counts.

How many hands did your spouse shake?

Tempting (but wrong)
"is it 0? 4? 8? totally unclear"9 distinct answers ⇒ from {0, 1, 2, ..., 8}spouse is one of those 9 peoplebut which value? feels under-determined9 unknowns, 9 numbers — seems like guesswork

It feels underdetermined. Nine unknowns, one constraint ("they're all different"). People typically conclude:

  • "Not enough information."
  • "Could be any value from 0 to 8."

Both wrong. The "distinct counts" condition is far more restrictive than it looks.