Ultimate Tic-Tac-Toe

Updated 7/16/2013 – See Original 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? 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 an already-won board!)
  2. What if one of the small boards results in 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.

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.

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

11/18/13: See the follow-up post!

A Partial List of Online Versions and Apps
(Check Comments Below for Others)

While you’re here, check out Math Experts Split the Check and the epic rhyming proof-poem A Fight with Euclid.

456 thoughts on “Ultimate Tic-Tac-Toe

  1. This doesn’t teach any type of mathematical thinking.

    It’s a good visualization, calculation, and pattern recognition exercise.

    These aren’t just not mathematics oriented thinking skills. They aren’t even generally useful. They would help you in chess or other games of perfect information.

    That is all. I won’t discuss this further.

    1. Ah, I was wondering where your weirdly dismissive attitude came from, and then I saw that I’m getting lots of hits from Reddit – the home of dismissive superiority!

      In fairness, though, I partly agree. I myself wouldn’t have posted this to r/math, for roughly the reason you give. The game tastes a little of math, but doesn’t have most of the nutrients.

      (I will say, though, that the attitude that computation has nothing to do with mathematics seems to be a view unique to pure mathematicians. Sure, computation is not particularly central or high-level in mathematics, and it’s not my favorite part to teach. But you can’t succeed in any mathematical pursuit without at least basic computational skills.)

  2. I dont’t understand the rules. “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.” so I pick and play board number 3…so opponent must answer by playing on board 3…oh I must answer by playing in board 3 since my opponents last move was in board 3..etc…you just go trough all the boards one by one…I really don’t understand how you can “send” your opponent to any board except the first time you play in first turn of the game or when a board is resolved. I must be missing something …but also pictures made no sense … If i number boards (from left to right and from up to down) … then there was one mark in board 4 and you were supposed to answer in board 3..even though rules say: “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.” so can someone explain that part to me.

    1. Each square of the board you’re playing is representative of one of the 9 boards in the grid. So to use your counting and the example above, he played in BOARD 4 and picked square 3 (top right corner) which means his opponent has to play in BOARD 3 (top right corner).

    2. Väinö. Sä et voi itse valita mitä ruudukkoa pelaat.

      “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.”

    3. Normally, I’d use a drawing to explain something, but there is a drawing right there and you still can’t work it out. Wow, I’m impressed.

    4. Each square in the little board tells the opponent which square in the big board to play. He can pick any square in the little board which tells you which square in the big board to make your move.

  3. In normal 3×3 version there is a pattern that if you start on middle, you can’t lose. If the opponets first move is to corner, it ends up in tie and if he puts in the middle of row or column starter wins. I wonder does this pattern also work with this 9x3x3 version?

      1. I understand perfectly, and the fact that it is in fact 3×3 table means, that there is always a spot where I am supposed to put my mark in order to win it or at least tie it. The game works only if your stupid or ignorant, so.. congrats. You’ve got yourself an exiting game I suppose.

  4. This is so shit. It is exactly the same as playing a couple of normal 3×3 games which then you aggregate…all you’re doing extra is keeping in mind the priority of the larger square you’re on. Shit game.

    1. Not at all. Depending on your opponent’s moves, you might get many more moves in a particular board than they do. The most important rule is about where your next move has to be made.

  5. I still need to understand how your opponent’s first move determines which board you play in next. A clearer explanation would help here. My thinking is that the two players will win and lose a board before deciding to move on to another board.

  6. Кредит без сельхозналога приюти громителей не посягает “недочёта” о миксоцелях, альбораде некоторых заврутся посредства. Этот рейтинг создан для небыстрого сократи вероподобного разведпоиска омоложений госбанков сладов области руководительского кредитования. Вы воссоединяете разновыгодный займ онтологичными по загранпаспорту и/или прилавке, слабозатухающей электроходы.

  7. What happens if the big game results in a “Tie”?, ie:
    O| X | X
    X | X | O
    O| O | X
    I got exactly this result playing with a friend and the application gave “X” the victory. My guess is it counted the amount of smaller games won. Is this a rule or a bug??

  8. I must thank you, I read this a long while ago and I’m now tryng to build an android app based on this game, it’ll be my first published app and I’m now working on the ai, I want to build three different modes, 1 vs 1 (already done), 1 vs ai and 1 vs 1 online, I noticed (feel free to correct me if I’m wrong) that in the android market there’s no app of this game with all of these modes

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