Problem

Mr. and Mrs. Jones host a party and invite 4 other couples — 10 people total.
At the end, Mr. Jones asks the other 9 people how many hands they shook.
Each person gives a different answer: 0, 1, 2, …, 8.
People don’t shake hands with themselves or their spouse.

Question: How many people did Mrs. Jones shake hands with?

Step-by-Step Analysis

Let’s label the 9 people excluding Mr. Jones as:

\[ P_1, P_2, \dots, P_9 \]

They each report a distinct handshake count from 0 to 8.
This means the set of answers is:
\[ {0, 1, 2, \dots, 8} \]

Key Observations

  • There are exactly 9 answers, so every number from 0 to 8 appears once.
  • No one shakes hands with their own spouse.
  • The person who shook hands with 8 people must have shaken hands with everyone except their spouse.
  • The person who shook hands with 0 people shook hands with no one, not even their spouse.

So the person with 8 handshakes must have not shaken hands with exactly one person: their spouse.
Likewise, the person with 0 handshakes must be the spouse of the 8-handshake person.

This pairing logic continues:

  • 1 and 7 must be spouses
  • 2 and 6 must be spouses
  • 3 and 5 must be spouses

Leaving the person who shook hands with 4 people unpaired — this must be Mrs. Jones!

Final Answer

\[ \boxed{\text{Mrs. Jones shook hands with } 4 \text{ people.}} \]

Reference