Set: An Introduction To The Idea Of Geometry In Four Dimensions

Believe it or not, the simple card game of set is actually based on four dimensional geometry. No really. And you thought it was just a waste of time...

For those of you who haven't played it, set is played with 81 cards, as described by /dev/joe above. Now 81 = 34 = 3x3x3x3, which is no coincidence - but you'll see why this is later, if you haven't already.

To play, you lay 12 cards out on the table, and the players have to find a set, which is a collection of three cards from the 12, where for each category (colour, number, shape or pattern), either they are all the same (for example, all three are red) or they are all different (one is diamonds, one ovals and one squiggles). And to make a set, you have to do this for all categories at once.

So the triplets that aren't sets are the ones where two are red and one is blue, or two are half-shaded and one is solid, or two have one symbol and one has three - you get the idea.

When you find a set, you yell "Set!", show the set to the other players to check that it actually is a set, and then keep those cards. Then the dealer deals three more cards to bring the total back to 12 on the table. If nobody can find a set from the 12 on the table, another 3 cards are dealt out to help, giving 15 to choose from (although even then it's not always possible to find a set - I'll come back to this).

The idea is very simple - the person with the most sets at the end of the game wins.

Now believe it or not, you can think of the cards in Set as points on a four-dimensional grid, and the sets as lines in the grid.

I'll demonstrate with a simplified game.

Suppose you just had two categories: colour and shape. Your cards are either red, blue or green, and they either have ovals, diamonds or squiggles on - you have 9 cards. Now, suppose you draw a graph with shape down one axis, and colour down the other; then you could put each card into its place in on the graph, and so arrange them in a 3x3 grid.

Here's where it gets nice: the sets in the game of Set are just the lines on the grid - the places where you have to get three in a row to win at tic-tac-toe. There are eight possible sets: three lines of same shape different colour, three lines of same colour different shape, and two diagonals which correspond to sets which share neither colour nor shape.

Now we can add in the possibility of different numbers, and so add an extra axis on our graph. We now get a 3x3x3 grid, like a three-dimensional tic-tac-toe board; the sets are still the lines on the grid, although now there are a lot more possibilities.

Lastly, we add a pattern axis into a hypothetical fourth dimension, making a 3x3x3x3 grid. Now here's where it starts to get difficult to visualize, but that doesn't really matter, because everything works basically the same as the three-dimensional case.

I'll expand on that idea. Suppose you have just one category; let's say shape. Your cards are all red, and have one symbol and are solid shaded, but one's oval, one's squiggly and one's a diamond. Then you'd have three cards arranged in a line. To add colour, you do the following: make two copies of the line of cards, one in blue and one in green. Then lay both copies alongside the first one, and voila! A 3x3 grid with both categories.

Likewise, to add shading, you make two copies of the 3x3 grid, one half-shaded and one unfilled, and lay them next to the first one to make a 3x3x3 grid.

So when you get to a 3x3x3x3 grid, it's not really too hard to think of - you just make two copies of the 3x3x3 grid, with different numbers of symbols on each card, and lay them alongside the first one. You can think of this happening in three dimensions without too much difficulty, although it's not so easy to work with.

You've probably heard of Einstein's theories that the world is four dimensional and time is the fourth dimension, but don't worry about that for now - just think of the fourth dimension as being the same as the first three; as well as up/down, left/right, forwards/backward, you get another direction to go to.

This is how mathematicians and physicists deal with spaces that have more than three dimensions - start small, and work up.

And what's more, even with four dimensions, the sets are still the lines where you would win at tic-tac-toe (although you wouldn't play tic-tac-toe on a 3d or 4d grid, because the first player would always win - but that's another story). It's not entirely obvious why this is, but if you think about it you should be able to convince yourself.

One interesting fact that comes out of all this is that it's possible to have 16 cards on the table without there being a set there - just take any 2x2x2x2 block from the 3x3x3x3 grid, and it can't have any full lines. On the other hand, if there are 17 or more cards on the table, then there has to be a set in there somewhere. I don't think I've ever seen more than 15 cards dealt, but in theory it could happen.

So far, my geometric ideas about the game of set haven't helped me win (:-(), but I'm gonna keep trying - this game is just way to addictive.