Handshakes with guests
Mr. Smith and his wife Mrs. Smith invite 4 married couples for dinner. Upon arrival, they shake hands, but in a pretty random way. Not all of the people greet all others. It is possible that someone never shakes hands with anybody. Nevertheless, they keep two rules: nobody shakes hands with his/her spouse, and nobody shakes hands with the same person twice or more.
During the dinner, Mr. Smith asks everyone to say how many times they shook hands. The eight guests and Mrs. Smith answer. Mr. Smith notices with surprise that every number he hears is different from the others.
How many times did Mrs. Smith shake hands?
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1 solution
Mr and Mrs Smith had handshakes with 4 guests.
The minimum number of handshakes is 0, the maximum is 8 (because excluding self and spouse leaves only 8 guests). That offers 9 choices. Since there are 10 persons in the company, at least one of the numbers of handshakes has to occur twice. That must be the number of handshakes Mr Smith had, since he was asking the question, but not answering it. Our goal is to determine how many handshakes Mr. Smith had, and deduce the number of handshakes of Mrs Smith.
The attached spreadsheet has one possible distribution of handshakes. The numbers 1 to 10 represent the guests, the same color represents a couple. The "x" means that handshake is forbidden; the "1" means that handshake happens.
The table was obtained by first filling in guest #10 with 8 "1"s in the last row and the last column. Filling in guest #8 followed, so that the total number of handshakes is 7. After that we did guests #6 and #4 with totals of 6 and 5. Once all the corresponding rows and columns were filled in we obtained the required number of handshakes without adding any more handshakes to the even-numbered guests. If the table is filled this way we have to identify #1 and #2 with Mr and Mrs Smith, because of the argument we outlined in the first paragraph.
There are many different ways to fill in the table, but they will all share a few common features:
-- the total number of handshakes for husband/wife couple must always add up to 8
-- there is a total of 40 handshakes
-- the number of handshakes that appears twice is 4, and those numbers must belong to Mr and Mrs Smith, as discussed earlier.