If I pick a card at random, how many possible sets can I make? Let's say I pick one green oval card as my initial seed. Here are all the possible sets:
It turns out that there are exactly 13 sets. And I noticed that every other card in the deck was used once and only once. And 13 happens to be a very intriguing number: it is the number of tic-tac-toes possible that pass through the middle square in a 3-dimensional game. Visualize it and count it up.
It is possible to arrange the 27 cards into a 3-by-3-by-3 cube arrangement, with the one green oval card in the very middle, and have every line through it yield a set:
Every one of the 13 sets is there, through the G1O card.
But I happened to notice that this arrangement also yielded all sorts of other sets not involving the G1O card. Every horizontal line, every vertical line, and every diagonal line is a set. In fact, if you took off the lower level of the cube and moved it on top of the upper level, every horizontal, vertical, and diagonal (following tic-tac-toe rules) line would be a set. And this also holds true if you took off the left face of the cube and moved it to the right of the right face. And so on. In fact, if this cube is tessellated by translation to fill up a 3-dimensional space, every horizontal, vertical, and diagonal line in that whole space will be a set! (This took quite a while to corroborate, but I am 99% sure I did not make any mistakes when I checked this.)
Every line in the space is a set. So does that mean the space contains ALL possible sets? How many sets are possible in the deck? That is easy: 27 cards to pick as initial seed times 13 sets with each card divided by 3 because each unique set is counted three times (once per each one of its cards as the initial seed). There are therefore 117 sets. How many lines in the space? This was a lot harder to figure out. Here are possible orientations of the lines:
- Left-Right (x-component only)
- Front-Back (y-component only)
- Top-Down (z-component only)
- FL-BR (no z-component)
- FR-BL (no z-component)
- TL-DR (no y-component)
- TR-DL (no y-component)
- TF-DB (no x-component)
- TB-DF (no x-component)
(The diagonal lines have to follow tic-tac-toe rules.) For each orientation of line, there are nine possible parallel lines. I tried to come up with a neat visual of this, but it is really hard to do. Anyway, 9 x 13 = 117 lines in the set. Yes, I think the space contains all possible sets.
What about the regular Set game and a 4-dimensional space?