# A Handshake Problem From the MathsJam Shout

One of the puzzles in the MathsJam Shout looked impossible, so obviously I sat down with Mr Miller and had a go at it.

I don't have it in front of me, but it went something like:

A couple hosts a party to which five other couples are invited. At some point in the evening, hands are shaken. Afterwards, one of the hosts asks everyone else how many hands they shook and discovers:

• Nobody shook hands with everyone else.
• Nobody shook hands with their partner (or with themselves).
• Everyone he asked shook a different number of hands.

How many hands did the host's partner shake?

Take a moment to have a crack at it, if you like. Don't read beyond the horizontal line, unless you like spoilers.

We drew pictures. We frowned a lot. Mr Miller may have sworn a little bit, but not me.

We figured it out as follows:

• If all eleven people apart from the host shook a different number of hands, and nobody shook eleven hands, the eleven people must have shaken each of the numbers from 0 to 10.
• The person who shook ten hands (call them A) shook hands with everyone except their partner (B) - who is therefore the only person who could have shaken no hands.
• The person who shook nine hands (C) shook hands with everyone except their partner (D) and B. D shook hands with A, but is the only person left who could have shaken just one hand.
• This pattern continues - E shook eight hands and their partner, F, 2; G and H shook 7 and 3; I and J shook 6 and 4 each - so the host and their partner must each have shaken five hands.

A lovely puzzle. Did you solve it a different way? Let me know!

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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