Question

Brain Teaser:

two brothers, John and Matthew, were recently invited to a party attended by four other pairs of siblings, a total of ten people.

During the party various handshakes took place, but no person shook their own hand or the hand of their sibling.

At the end of the party John asked each person, including Matthew, how many different people they shook hands with, and was surprised to note that every number was different!

How many hands did Matthew shake?

Answer 1

4 !

Answer 2

Since each person shook a different number of hands, and they didn't shake their own hand or their sibling's, the most handshakes anyone could have reported was 8.

The person who shook 8 hands must have shaken the hand of everyone except themselves and their sibling. Therefore everyone, except their sibling, has shaken hands, so their sibling shook 0 hands.

Similarly it follows that the persons who shook 7 and 1 hands must be siblings, the persons who shook 6 and 2 hands must be siblings, and the persons who shook 5 and 3 hands must be siblings.

The only number left is 4. As there were 10 people, there must have been 10 answers to the question, so two people must have shaken 4 pairs of hands (as we've already had 0, 1 , 2 , 3 , 5 , 6, 7, 8), and as no two people gave the same number, one of these people must have been John, and the other was therefore his brother. So both John and Matthew shook 4 hands.