# A chic party puzzler

(Based on this puzzle by Gamow. The answer there might give you a hint (so don't look yet!))

At a fancy party in Café la Tour, everybody is friends with exactly 14 of the other people present.

• Whenever two people at the party are friends, they have exactly 6 friends (present at the party) in common.

• Whenever two people at the party are not friends, they have exactly 2 friends (present at the party) in common.

(Oh, and half of the people at the party came dressed in white, and the other half was dressed in black. How stylish!)

Question: How many persons are at the party?

(Note: Friendships are always mutual. Nobody is their own friend. There is at least one person at the party.)

• Why have you used chess ? Nov 25, 2015 at 14:36
• Although the accepted anwer obviously is a solution, could there possibly be other solutions as well?
– Ivo
Nov 25, 2015 at 15:04
• @IvoBeckers: (hover spoiler)
– Lynn
Nov 25, 2015 at 15:27
• @Mauris thanks. actually the 4x4 still has one solution. It has different graphs yes but both have 16 vertices.
– Ivo
Nov 25, 2015 at 16:15

The chess tag gave it away.

There are exactly 64 people at the party.

To show this:

Imagine every person standing alone on a field on a big chessboard. 64 Fields and therefore 64 people.
Now every person is friends with those 14 other people which stand in the same row or column. (7 in the same row + 7 in the same column)

With this every two people which are friends and therefore are within the same row/column will have the other 6 people in the same row/column as common friends.

In a similar way every two people which are not friends and therefore neither within the same row nor the same column will have those two people as common friends that are standing on the intersections of the respective rows and column.

• Correct! (Did you spot the hint in the first sentence?)
– Lynn
Nov 25, 2015 at 14:46
• @Mauris Nope sorry. And I still can't seem to find the hint after looking again. Nov 25, 2015 at 14:54