Prove that, at a recent convention of biophysicists, the number of scientists in attendance who shook hands an odd number of times is even. The same problem can be expressed graphically as follows: Put as many dots (biophysicists) as you wish on a sheet of paper. Draw as many lines (handshakes) as you wish from any dot to any other dot. A dot can “shake hands” as often as you please or not at all. Prove that the number of dots with an odd number of lines that connect them is even.
Because two people are involved in every handshake, if we add up the number of handshakes for every scientist at the convention, we count each handshake twice. The total must be evenly divisible by 2 and therefore even. The total number of handshakes made by the scientists who shook hands an even number of times is also even. If we subtract this even number from the even total number of handshakes, we get an even total for those who shook hands an odd number of times. Only an even number of odd numbers will total an even number, so we can conclude that an even number of people shook hands an odd number of times.
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A version of this puzzle originally appeared in the August 1958 issue of Scientific American.