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.
I made another 2 players side by side version in Javascript with the AngularJS framework. I also have a screencast (in French) of me live-coding it in 45 minutes. Take a look : http://www.youtube.com/watch?v=zrawFEawKkc
Ok but I have to read it again 🙂
Nice work!
Ironically me and a friend made this game up nearly 10 years ago in middle school. but we never found a better way to fix the problem of when you get sent to a full square. im glad someone else had discovered the joy of a good game of tic tac toe
Probably would have been better had you studied your grammer!
If you’re going to correct somebody, then you better make sure your own house is in order there Donnie. Nothing makes you look dumber when you misspell something while correcting a post regarding the English language.
Haha! Nice one! But what were the grammar mistakes anyway? 😀
s/you better make sure/you had better make sure/
HA! and you should probably study your spelling! Bravo on the correction…
I don’t think you can name a move after yourself when it’s already been considered an “instant win” and it was the first move I thought of after reading the “you determine what board your opponent plays on, even if that board has been won.” That would be similar to someone at the dawn of baseball naming a bunt “The Johnson Swing” because the rules allow you to do so.
Fair complaint! I’d been used to a readership of a few hundred per post, and didn’t anticipate the 500,000 who have seen this one. I probably would’ve written it differently if I knew how many people would be coming. (In particular, I’d have chosen the other rule variation, which doesn’t allow for that winning strategy.) But I find it better to add edits at the bottom and to leave mistakes intact.
Reblogged this on smile..
It gets interesting when it’s forbidden to go for a centre square when it’s a 1st move.
And there is a super good looking app for that…
https://play.google.com/store/apps/details?id=com.MoongLabs.UltimateTicTacToeHD
Good looking but using the wrong rules – despite point #5 of the description, your variant of the game forces players to keep putting marks in already-won minigames if they’re not full.
There’s another one on Google Play that has the correct rules: https://play.google.com/store/apps/details?id=org.droid_in_the_sky.ult_tic_tac_toe
Hi Don,
Thanks for the note. I just checked it seems the developer of the app has fixed it.
Happy to check out the other app also. Good to see maths based puzzles being available to common man.
http://projects.minecraftcolton.info/uttt/
Includes the updated rule set (cannot force an opponent to a completed game). Play against a friend over the internet. Does not support AI opponents or two people at the same computer.
I belive my coworker has solved the game; first player wins: https://www.joachim-breitner.de/blog/archives/604-Ultimate-Tic-Tac-Toe-is-always-won-by-X.html
Nevermind, now I see the other posts about it being solved. Got confused by the different variants.
Reblogged this on RicardosLive and commented:
Recreational Mathematics at its best!
Hi, I coded a multi user multi machine game last night.
You can have a play here http://bit.ly/tp10t
You need to register because I need to pass information across users playing in same session.
I am also working on Chrome webstore and FB app for the same.
I love this version. Simple, yet elegantly evil!
I made a version for Windows Phone, its still in development but it has the main rules in them.
http://www.windowsphone.com/s?appid=43a33644-6f99-4315-9f3e-52cb3afe4f1d
We launched an Android App for this with some modification. Below is the link.
https://play.google.com/store/apps/details?id=example.pro
Dear Mr. Ben Orling,
I am writing to you on behalf of the Argentinian Ministry of Education. We have read your note “Ultimate Tic-Tac-Toe” and we found it really interesting.
We think the game is a good exercise and a potent mathematics skills developer, that is why we have the intention to publish it in our web site http://www.educ.ar/sitios/educar/estudiantes/, so teachers and students of our country can profit it. Although the idea is to translate the text we would like to use the pictures you posted on your blog. It is important for us to say that we will mention the sources in the note.
Looking forward to your answer.
Yours faithfully,
Lucila Quiroga
Editor
Educ.ar
Portal de Educación y tecnología del Ministerio de Educación de la Nación Argentina
Hi Ms. Quiroga,
That sounds great! I’m happy to let you use my drawings. (Or, if you want to fly me down to Buenos Aires to do some new drawings in person, even better!)
Best,
Ben
Wow that was odd. I just wrote an very long comment but after I clicked
submit my comment didn’t show up. Grrrr… well I’m not writing all that
over again. Anyways, just wanted to say wonderful blog!
Sorry WordPress swallowed your comment! It can be finicky like that.
Anyway, thanks for reading!
A variation that we tried for the full square issue.
In the course of the game if a player (say A) is sent into a 3×3 square already won, the choice of the board in which the move is to be made rests with the person who won that 3×3 square.
This adds an element of further strategy in which players take advantage of the 3x3s they have already won. It also rewards aggressive strategies early on and speeds up the game.
Try it and share feedback.
That’s a cool variation. I’ll have to try it.
It sounds like it rewards early board wins, and, by the same token, punishes sacrificing a board early on.
That’s how we implemented our mobile app offerings (Tic Tac Toe Ten). Out of all the options we investigated we found that this one was the most fun and challenging. It definitely adds what we feel is the maximum level of critical thought to these games of T10.
T10 Team
Mobile app offerings? I can only find one link on your website that seems to be for a mobile app and it’s broken.
Never mind, I’ve found it. I couldn’t see it directly because it’s listed as incompatible with my Motorola Droid 3.
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i suggest him/her to pay a visit this weblog, Keep up the pleasant work.
YEA, we played this in highschool ~2006
Cant remember who 1st told us but it was a blast! We just called it 3D tic-tac-toe : D
Forgot to add wasn’t the same rules we played of cause.
Interesting game. This Android version has a great UI
https://play.google.com/store/apps/details?id=com.sqr.tictactoe
My HTML5 Version for two players, works on Desktop and Mobile and games can be paused and resumed via bookmark:
http://bejofo.net/ttt
Try this with constraints tied to multiplication facts 🙂 http://artofmathstudio.wordpress.com/2013/08/30/multiplication-tic-tac-toe