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Miraculous Handshakes

2022-07-03T04:52:25.980Z

At an evening reception, people greet each other warmly – and shake hands in the classic way. A mathematician makes a startling observation.



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THE MIRROR

The World Association of Combinatorists meets for its annual congress.

On the first evening, each person present should shake hands with each other once.

A nice idea if there weren't various animosities: some like variations without repetition, others combinations with repetition.

And some just want to talk about permutations.

The various camps therefore largely keep to themselves in the evening.

When the reception is over, Italian combinatorist Lotta Facultati makes a bold claim: "The number of those present who shook hands with an odd number of people is even."

Is she right?

Note: There was at least one handshake at the evening reception.

And there were at least two people present who didn't shake hands.

Lotta Facultati's claim is true.

There are different ways to prove that.

For example with the method of complete induction: If there was only a single handshake, Facultati's statement is correct.

And if the number of people with an odd number of handshakes is even, that doesn't change when a new contact comes along.

Martin Gardner describes another solution in his book »My best mathematical and logicial puzzles«, which I find very elegant: The sum of the handshakes counted by everyone present is an even number, because every handshake always involves two people and therefore a handshake is always counted twice.

From this number we subtract all handshakes from people with an even number of handshakes.

The result is an even number and at the same time the sum of the handshakes of all people with an odd number of handshakes.

However, a sum of odd numbers is only even if the number of odd numbers added is even.

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Source: spiegel

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