Another party puzzle

More party puzzles! This one is from a thoroughly-recommended book, Puzzle Based Learning¹.

Mr and Mrs Smith invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse or themselves, and nobody shook hands with the same person twice.

After that, Mr Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody.

How many times did Mrs Smith shake someone’s hand?

At first glance it seems that there is not enough information to solve the puzzle—which is why I like it! Once we consider each piece of information, we can put the bits together to find a solution.

Warning: mathematical spoilers (but not the solution) ahead. I’ll post the solution in the comments.

Some prompts:

• How many people are there at the party?
• What is the minimum number of handshakes possible?
• What is the maximum number of handshakes possible?
• Can you draw a diagram to represent the handshakes made by the person who made the most handshakes?
• What can you conclude from this?
• What can you add to your diagram?
• Can you now solve the puzzle?

Good luck!

[1] Michalewicz, Z., Michalewicz, M., Puzzle Based Learning, Hybrid Publishers, 2008. pp 99-102.