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.
- Each turn, you mark one of the small squares.
- When you get three in a row on a small board, you’ve won that board.
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:
- 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!)
- 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)
- Super Tic Tac Toe
- Tic Tac Toe Ten
- Ultimate Tic-Tac-Toe
- XO9
- Nikhil Baliga
- Niek Haarman
- Kongregate
- Inception Tic Tac Toe
- Khan Academy (open-source online version)
- Live-coding of an online version (French, with English subtitles)
- C code for a strong artificial intelligence
While you’re here, check out Math Experts Split the Check and the epic rhyming proof-poem A Fight with Euclid.
This is the greatest.
Someone showed me a cool variation. instead of eliminating boards that have been won from future play, you must still go on that board if forced. However, your move doesn’t affect the outcome of that board. For example, if X wins one of the small boards, and then forces O to move on that board, instead of O getting to choose where to go, O still must go on that board. But no matter which move O makes, X remains the winner of that board.
Be warned: this variation makes the game take much longer, and it makes winning the first board critical.
See the original version:
https://mathwithbaddrawings.com/ultimate-tic-tac-toe-original-post/
This is also a fun variant, but is sadly solved, as explained here:
https://www.joachim-breitner.de/blog/604-Ultimate_Tic_Tac_Toe_is_always_won_by_X
The rule that if you get sent to a board which is already won, you get to choose a board was introduced to break this strategy.
If you like this, give this game a go it’s fantastic https://play.google.com/store/apps/details?id=com.MonWiz.Nexo
Bossed it!! Great game!
Reblogged this on Shaman's Antics and commented:
O.M.G.
YES.
Reblogged this on nuknoe and commented:
Check out this different variant on #TicTacToe!!!
NÜK
That looks like a lot of fun! Normally tic tac toe does get boring. Will definitely play this one with my friends. Thank you for posting it:)
coding it!
I made a super-fast and super-strong computer opponent for this game here: http://www.theofekfoundation.org/games/UltimateTicTacToe/
This also includes 3 new game modes:
Anti Tic Tac Toe: The goal of the game is to make your opponent win 😛
Tie Tic Tac Toe: X still wants to win, but O wants to tie 😀
And of course, Anti Tie Tic Tac Toe, where O wants to win and X wants to tie XD
Hey Bud, I was wondering if i could get the Github of this code!? it’s really interesting.
very good game
Mind=Blown! Definitely gonna give this a try! 😀
Here is another variant. Suppose a small board is a tie. Then it counts as a win for neither player… but only at first! Now next time someone is directed to play there they become the temporary winner of that board (instead of making a move). In other words, tied boards become wins for the last player to have been directed there.
You can also find this game on Kongregate:
http://www.kongregate.com/games/GeneralVimes/strategic-tic-tac-toe
There’s a mistake, dunno how nobody has caught it. So you say “…the player must then go to that same square” but then show a picture where you circled a square that wasn’t where the last play was made, it was a new square. So the only thing I can think of is you meant to say “…the player must then go to the PREVIOUS square”
Perhaps it would have been better stated: “The SMALL square chosen by the first player represents the LARGE square the next player must play. That player selects any available SMALL square, which represents the LARGE square the next player must play…. and so on.”
In that case, the image is still seemingly contradictory. Why did you circle (in blue) a “large” square that wasn’t “in play”? Could you clarify the meaning of the image as it relates to the rules?
or, well, I suppose it wasn’t you that made the image (this isn’t your blog), but that’s beside my point
if anyone interested:
https://github.com/lkiril/big-tic-tac-toe.git
Ok, this is really cool!
Will try as soon as possible 😀
Love the game a lot. Made an implementation with React this week: https://github.com/maracuja-juice/ultimate-tic-tac-react
This is the best Ultimate Tic Tac Toe write-up on the net! Check out my android Ultimate Tic Tac Toe app – it has a pretty strong AI! https://play.google.com/store/apps/details?id=com.paperandpencilgames.ultimatetictactoe
I don’t think these drawings are bad at all!
Ah, you are too kind I think.
Like the idea of this. There is a variation of tic tac toe I’ve just downloaded using a 10×10 grid which is similar to this game but you score points for odd numbers 3s 5s 7s and 9s. With checking out, great game
https://play.google.com/store/apps/details?id=com.MonWiz.Nexo
Sounds cool – thanks!
you can chunk small board to smaller boards
The first time I played, I had only heard part of the rules so one of my students and I invented the rule on what to do if one player had already won a board. If you send me to a board you have already won, then I have to play there. If I send you to a board you have already won, then you get to play wherever you like. If the board is full, then anyone sent there gets to play wherever they like.
I’ve never taken the time for a full analysis, but it does avoid the immediate ‘X wins’ solution of being sent to a won board whilst adding some advantages for winning a board.