In a version of the game of set, cards with shapes on them are dealt out and each has four characteristics:

Type of shape (Circle, Square, or Triangle) Color of the shape (Red, Blue, or green) Fill type (Empty, Half filled, or Completely filled) Number of the shape on the card (1, 2 or 3)

A "set" is defined as a three card subgroup of the cards "in play" such that for each of these four individual characteristics are either all the same, or all different. (The cards could be all different on one characteristic and be same on another.)

What is the greatest number of different cards that can be "in play" such that there is no subgroup that can be designated a "set"?

For any pair of cards there is only one card that will make a set. As you add to the cards in play you must take one card out of play for every possible pair. When the number of cards in play plus the number of cards out of play is greater than the total number of cards back up a step and there's your answer.

in, out, total

2, 1, 3

3, 3, 6

4, 6, 10

5, 10, 15

6, 15, 21

7, 21, 28

8, 28, 36

9, 36, 45

10, 45, 55

11, 55, 66

12, 66, 78

13, 78, 91

There are only 81 cards and 13 would need 91 so the answer is 12.