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"?

To further my analogy of the wrapping 4-d tic-tac-toe game, I'm going to find the number of rows if all the spaces were occupied.  Using a bit of logic, there are 15 rows that go through any one location.  Since there are 81 spaces, and each row goes through three of them, there are 405 rows total.

It would seem that to block all those rows, you would need to leave only 27 spaces empty.  This is incorrect, because every pair of spaces left out will end up blocking one row twice.  The number of rows blocked is 15x-xC2, where x is the number of spaces left empty.  Actually this is wrong as well because it overcompensates for the spaces that block the same rows.  Three empty spaces can all block the same row so the same row is counted three times instead of three rows each being counted twice.

I leave off here because I ran out of time.  I hope to finish later.

Posted by Tristan on 2004-05-13 20:26:46