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"?

(In reply to re: not sure if i understood this correctly, but by Charlie)

If identical cards are allowed, the answer can simply be doubled.  If a set includes two identical cards, the third must also be identical.

In my experience with Set (yes, it's a real game!), in order to get identical cards, you'd have to buy another deck.

One of the things about Set, if you want to have two specific cards in a single set, there is only one possibility for the third card.  It's sort of like tic-tac-toe in that manner.  Any two places in tic-tac-toe will only form a row with one other place.  Of course, for this to work, the tic-tac-toe grid would have to wrap around, and be 4-dimensional.  Can you imagine that--some kind of hyperdonut?

In any case, to see if 16 (or 32 if identicals are allowed) is really maximized, I suppose we can think of a wrapping tic-tac-toe board.  If rows can wrap, what is the maximum number of Xs we can put on the board without a row?  In 3-d tic-tac-toe? 4-d?  I think 16 is correct, but I'm not sure.

Posted by Tristan on 2004-05-12 19:49:43