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Twelve business associates meet for lunch. As they leave to return to their offices a couple of hours later, one of them conducts a small mathematical experiment, asking each one in the group how many times he or she shook hands with someone else in the group. The twelve reported values were 3, 5, 6, 4, 7, 5, 4, 6, 5, 8, 4, and 6. Is this data believable?

Summing these numbers, I got 63. Shouldn't the sum be even? Starting with a simple graph with four vertices and four edges (a rectangle), each person shakes hands with two others, making the sum 8. If one person wanted another handshake, the only way this is possible is by increasing the sum by two.

If this line of thinking is not correct, can someone guide me in the right direction?

Alex
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    You're correct. In general, the sum of the degrees of a graph is twice the number of edges. – Benjamin Wang Jan 03 '24 at 02:54
  • Yes, it should be even: every single handshake involves two people, so the sum of all reported handshakes should be exactly twice the number of handshakes that took place. – Bram28 Jan 03 '24 at 02:59
  • That's what I was thinking, since everything was double-counted. Since my reasoning was correct, what should I do with the status of my question? – Alex Jan 03 '24 at 03:02
  • Without having to even sum the handshakes, the number of vertices of odd degree must be even, i.e. the number of people who report an odd number of handshakes must be even. It isn't. – Sammy Black Jan 03 '24 at 03:06
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    Regarding your process question: It's good practice to write an answer yourself (possibly mentioning the comments), and mark it answered. – Sammy Black Jan 03 '24 at 03:08
  • Sammy Black's advice is especially true in this case, since you gave the answer yourself, and all you needed was confirmation that you had understood the matter correctly. – Paul Sinclair Jan 04 '24 at 12:50
  • Thank you to Sammy and Paul for help with what to do, as far as an answer is concerned. – Alex Jan 08 '24 at 00:16

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This data is not believable. Any time one person shakes hands, their sum count goes up, but someone else's sum count must also go up by one. Therefore, the total sum of handshakes must be exactly twice the number of interactions (edges on a graph). Specifically, the sum must be even, which it is not.

Alex
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