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"?

I beleive the answer to be 14. I arrived at this after reading Axorion's solution. I was following a similar line but Axorion put it very well. But as it was still unsolved I set about trying to see what was wrong with his logic.....I couldnt.

There is symetry with this puzzle with the numer of possible cards (81 or 3^4) and sets (3 cards) , sets into cards = 27. I felt that at the end of his solution number of cards held out should exactly equal 81 and it does not so therefore it is wrong. But its not.

Then I realised that the puzzle is not exactly symetrical because there are 4 charactoristics and only three cards to a set.

Axorion's solution I beleive is correct he just did not finish it off where he said 12 - 66 - 78 must be it because its the last one under 81 posible cards. I beleive he should of considered that the last 3 cards would form a perfect set allowing you to add 2 of these without forming a set thus making the number of posible card up to 14.