Party of five couples - Solution

Mr. Smith asked nine persons, who replied him different numbers. The highest number could be 8, because everybody of the 10 participants knows at least himself and his/her partner.

If the highest number of 9 different numbers is 8, this means, that the answers were: 0, 1, 2, …, 8.

Let’s call the nine persons after the answer they gave X0, X1, X2, …, X8.

Let’s look at X8. He/she shook hands with 8 persons of his/her 9 party-mates. Among these 8 persons could be X1, X2, …, only X0 could not be among them, since X0 didn’t shake hands with anybody. That is, X8 shook hands with everybody except X0 (and himself, of course). We could say, X8 didn’t know anybody at the party except X0. From this follows, that X0 must be the wife/husband of X8.

Now let’s look at X7. He/she didn’t shake hands with 2 persons. One of these two must be his/her partner. One of these two is X0, who didn’t shake hands with anybody, the other one is X1, because X1 shook hands with only one person, and this was X8. This means, that only X0 and X1 were known by X7. X0 is the partner of X8, so X1 must be the partner of X7.

Similarly: X6 didn’t shook hands with 3 persons …

Thus X6 and X2 are married couples.

Finally, X5 and X3 are partners too.

One person, X4 remains who has no partner among the nine persons interviewed by Mr. Smith. She is Mrs. Smith.

The answer to the problem: Mrs. Smith answered: 4.

You can make a graphical solution to the problem

Let’s make them take a seat around a table:

              *       *

         *                  *

        *                     *

         *                  *

              *       *

Call one of them X8, and draw straight lines to those persons, with whom he shook hands.

              X8       *

         *                  *

        *                     *

         *                  *

              *       *

Oly one asterisk has no connection to X8, this will be called X0, and they must be partners.

In the second step take another asterisk, call it X7 draw the missing 6 lines to the other asterisks, and the only asterisk that has no connection to X7, must be called X1. They must be partners too.

At the end two asterisks remain, both with 4 connections. One of them must be Mr. Smith, since he has got nine different answers. So the other asterisk stands for Mrs. Smith.

Sent in by da one and only Peter Hosszu.