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The problem is: A group of people met and some of them (NOT all of them) shook each other hands. Prove that the number of people who shook others' hands an odd number of times is even.

My attempt:

  1. I have already shown the cardinality of the group of people must be finite, since saying infinity is even doesn't make sense.

  2. I tried to use the method of graph theory. But I encountered a critical problem that "the member of people who shook others' hands an odd number of times" can shook hands with "the member of people who shook others' hands an even number of times".

I thought the condition was "all of them shook each other hands", which is a easy case. But this one seems harder. I cannot figure it out. Any help will be appreciated.

Especially Lime
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Sam Wong
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    Add up the number of times each person shakes hands. Relate this to the total number of handshakes. – Gerry Myerson Sep 04 '18 at 07:19
  • In graph theory language: how does the sum of vertex degrees relate to the number of edges in a graph? – Christoph Sep 04 '18 at 07:20
  • See also https://math.stackexchange.com/questions/560394/understanding-the-proof-of-even-odd-handshake-problem and https://math.stackexchange.com/questions/1418658/must-the-number-of-people-at-a-party-who-do-not-know-an-odd-number-of-other-peop and https://math.stackexchange.com/questions/2099740/in-a-party-people-shake-hands-with-one-another and https://math.stackexchange.com/questions/2246392/people-shaking-hands-in-a-party and probably others. – Gerry Myerson Sep 04 '18 at 07:25
  • Thank you Myerson – Sam Wong Sep 04 '18 at 07:58

1 Answers1

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You mentioned using the methods of graph theory so I'll assume you are somewhat familiar with graph theory terminology. This question is equivalent to the question: Prove that the number of odd degree vertices of a finite graph is even. Now there is a close relationship between the sum of the degrees of the vertices of a graph and the number of edges of that graph. Do you see what it is? What does it tell you about the number of odd degree vertices?

hendreyth
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  • The sum of the degrees of the vertices of a graph is 2 times of the number of edges of that graph. But I don't know what does it tell me about the number of odd degree vertices... Could you explain it more? Thank you so much... – Sam Wong Sep 04 '18 at 07:35
  • Does it help if I remind you that that the sum of 2 odd numbers is even, and the sum of an even and an odd number is odd? – hendreyth Sep 04 '18 at 07:44
  • I understand it now. We need to argue by contradiction. But all of the answers quoted by Gerry Myerson seems also arguing by contradiction. I wonder if there is a straight proof without using contradiction. Anyway thank you so much hendreyth. :) – Sam Wong Sep 04 '18 at 07:57