Mike and Dominique go to a party with four other couples. Each person there shakes hands with everyone he or she doesn't know.

Question

Later, Mike does a survey and discovers that every one of the nine other attendees shook hands with a different number of people.  How many people did Dominique shake hands with?

Answer 1

For “every one of the nine other attendees” to “shake hands with a different number of people” we need nine different values representing the number that is “everyone he or she doesn’t know.”  Each of those numbers must be nine or less, since there are only ten people present.

We must assume that each of the “four other couples” know each other, so that limits the number of people each of these people doesn't know to 8.

O.K., let’s assign the digits 0-8  (yes, remember to include 0 as a possibility):

     (1) None of the 8 people in the 4 couples gets a “9,” because they each know at least 1 other person (10 – themselves – other person in couple <= 8).

     (2) Now, we must distribute 0-8.  If all 9 others -- 8 of the people in the “four other couples” and Dominique -- must have a unique number of people they shook hands with, that means that 8, 7, 6, 5, 4, 3, 2, 1, and 9 all get taken very quickly.

Wait a minute !!  How can you use 9 ?  Well, the explanation has to be that Dominique doesn't know Mike and that "other couples" was was meant to mislead us into thinking that Mike and Dominique knew each other.  Dominique gets the 9.