Fun with representations I – Nine numbers

Here’s a two-player game for you to try out:

You need nine cards, numbered 1 to 9. You and your opponent take turns picking cards -each card can only be picked once. The first player with three cards that add up to 15 wins the game.

Nine Numbers

Before you keep reading, it’s best if you give it a try, as a thought experiment. You’ll notice how, although the game is not really very complex, a winning strategy is a bit hard to find.

OK, moving on. Now imagine those same nine cards, but arranged in a 3×3 grid with a ‘magic square’ pattern (where each row, column, and diagonal adds to the same number -in this case, 15). You get something like this:


If you try it out, you’ll notice the game becomes much easier to grasp. Every possible three-number combination adding up to 15 is a row, column, or diagonal in this square! You’ll surely also notice that the game becomes strangely familiar:

Tic Tac Toe

So there you go -it turns out the nine numbers game is, in fact, exactly the same as the tic-tac-toe game. The difference is that tic-tac-toe is a much simpler version of it, but the two games map perfectly to each other, however non-intuitive that might be.

Snap quiz: You’ve picked the numbers 3, 5, and 8. I’ve picked 1, 4, and 7. It’s your turn. What number should you pick? Well, mapping to tic-tac-toe makes it easy -number 2 of course!

Tic Tac 2

The interesting point here is that problems can be represented in such ways that their solutions become clearer or more obscure. The two games above are isomorphic, but that doesn’t mean that they require the same amount of cognitive effort from their players. They don’t even require the same type of knowledge. The first game requires that we know how to add numbers, and forces us to perform additions constantly. A player without math skills would have a hard time with it. The second game, on the other hand, requires no math -only that we know how to detect lines visually, something that comes almost effortlessly to us.

The problem of adequate representations extends to many problems beyond tic-tac-toe. A diagram of pulleys in a Physics problem is more helpful than its equivalent description in English. Arabic numerals are easier to handle than Roman numerals. Code in Python is clearer than its Assembly counterpart -you get the idea. Herbert Simon (from whom I got the nine numbers game) used to say that problem solving consists of re-representing a problem until its solution becomes trivial.

The key, then, is to be able to define why is a representation better than another one, in a fairly predictable manner. Finding the reason would allow us to design representations that make our cognitive tasks easier. However, this analysis turns out to be a little complicated: there are actually plenty of reasons that could make one representation preferrable to another, and they are not always evident. Over the next few entries I will be covering some of them.


About Jorge Aranda

I'm currently a Postdoctoral Fellow at the SEGAL and CHISEL labs in the Department of Computer Science of the University of Victoria.
This entry was posted in Cognition, External cognition, XCog. Bookmark the permalink.

6 Responses to Fun with representations I – Nine numbers

  1. Greg says:

    Scheme programmers are fond of saying that the right way to solve a problem is to define a mini-language in which the solution to the problem is trivial 😉

  2. Ricc says:


    Does anyone had try this game?

    I’ve, but just for myself.. any comments?

  3. Pingback: Catenary » Fun with representations II - Where is the train going?

  4. juan says:

    Wow… (I just saw the future)

    We need to stop trying to solve complex problems. All we need to do is figure out how to explain they are not real problems… “There is no spoon” makes a lot of sense all of the sudden.

  5. Pingback: Fun with representations VI - Sharing the load « Catenary

  6. number games says:

    wow the future!

    I think we should start solving the little problems in life!

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