Ultimate Tic-Tac-Toe (original post)

See Updated Version Here

Once at a picnic, I saw mathematicians crowding around the last game I would have expected: Tic-tac-toe.

As you may have discovered yourself, tic-tac-toe is terminally dull. There’s no room for creativity or insight. Good players always tie. Games inevitably go something like this:

But the mathematicians at the picnic played a more sophisticated version. In each square of their tic-tac-toe board, they’d drawn a smaller board:

As I watched, the basic rules emerged quickly.

  1. Each turn, you mark one of the small squares.
  2. When you get three in a row on a small board, you’ve won that board.
  3. To win the game, you need to win three small boards in a row.

But it took a while for the most important rule in the game to dawn on me:

You don’t get to pick which of the nine boards to play on. That’s determined by your opponent’s previous move. Whichever square he picks, that’s the board you must play in next. (And whichever square you pick will determine which board he plays on next.) For example, if I go here…

Then your next move must be here…

This lends the game a strategic element. You can’t just focus on the little board. You’ve got to consider where your move will send your opponent, and where his next move will send you, and so on.

The resulting scenarios look bizarre. Players seem to move randomly, missing easy two- and three-in-a-rows. But there’s a method to the madness – they’re thinking ahead to future moves, wary of setting up their opponent on prime real estate. It is, in short, vastly more interesting than regular tic-tac-toe.

A few clarifying rules are necessary:

  1. What if my opponent sends me to a board that’s already been won? Tough luck. If there are open squares, you must pick one. While you can’t really affect that board, you can at least determine where your opponent will go next. [see edit below]
  2. What if my opponent sends me to a board that’s full? In that case, congratulations – you get to go anywhere you like, on any of the other boards. (This means you should avoid sending your opponent to a full board!)

When I see my students playing tic-tac-toe, I resist the urge to roll my eyes, and I teach them this game instead. You could argue that it builds mathematical skills (deductive reasoning, conditional thinking, the geometric concept of similarity), but who cares? It’s a good game in any case.

One final note: I have devised a strategy that I have (im)modestly dubbed THE ORLIN GAMBIT. You begin by taking the very center square.

O will take one of the other squares.

Take the center again.

O claims a two-in-a-row.

Take the center again.

O grabs a three-in-a-row, laughing at your stupidity.

Take the center again.

O begins to see the trick you’re pulling…

Take the center again. O will grudgingly realize that there’s no way to stop you.

When all is said and done, X has sacrificed the center board in exchange for superior position on the other eight. It’s by no means a perfect strategy. It stings to lose that center board without a fight. But forcing so many O’s to cluster uselessly together on one board gives X the upper hand for the rest of the game.

Anyway, that’s Ultimate Tic-Tac-Toe. Go play! Let me know how it goes!

EDIT: RULE CLARIFICATIONS:

  • As many have pointed out, the rules as I’ve described them are not the best. My gambit is too strong, and can be extended into a guaranteed win for X. So I recommend modifying “Clarifying Rule #1” to say: If you are sent to a board that’s already been won, you may go wherever you like.
  • A common question is, “What happens if one of the small boards is a tie?” I recommend that the board counts for neither X nor O. But, if you feel like a crazy variant, you could agree before the game to count a tied board for both X and O.

EDIT: ONLINE VERSIONS

  • Tic Tac Toe Ten is the real deal, with an app for mobile devices, cool graphics, and an adjustable winning condition. You play against the computer. And Ultimate Tic-Tac-Toe is another app (this one with the rule change I recommend above).
  • Some folks at Khan Academy have a nice version they cooked up for fun. You can read and even modify their code. You play against the computer.
  • There are several two-player versions: one at Ken Soft, one at xoxo.GL, an Android holo app from Niek Haarman, and my new favorite from Nikhil Baliga. In each of these, your opponent must be sitting next to you. (No through-the-internet play… yet!)

10 thoughts on “Ultimate Tic-Tac-Toe (original post)

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