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.
Reblogged this on {"code":"java"} and commented:
A new dimension to traditional Tic-Tac-Toe..
Sunuvagun! I’d *just* invented that not five days ago, at some chain restaurant, trying to entertain a gaggle of tweens! Brilliant!
In our games, though, there were no restrictions on which large-square you had to play — you could play any unoccupied small-square..
If you won a large square, you immediately claimed it *and went again*.
If it ended up as a tie, that ended your turn. The next player could claim it as a win (or not). If that player didn’t claim it, you could on your subsequent turn.
Enjoy the blog.
I coded it with an alpha-beta based algorithm and it seems much more stronger than the Monte Carlo one used by Khan Academy, mostly ending with a win, sometimes with a tie, for a shorter thinking time.
C code can be found there : http://cschlum2.free.fr/ticTacToeUlt.c
Reblogged this on DisOrganizer v0.1 and commented:
Wenn man aus Tic-Tac-Toe mit der Hinzugabe einer Dimension was machen kann, wie bekäme das dann wohl anderen Spielen?
Brilliant. My son who is an avid chess player will love this as training for his chess.
Reblogged this on Mercurial Twister and commented:
Adding Spices to the Traditional Tic-Tac-Toe.
I’ve gave it a shoot too
http://www.kongregate.com/games/nazywam/ultimate-tic-tac-toe
We used to play a version of this that was Connect Four. You could play with as many people as you like, four or more was best. Everyone had their own board, and every turn you had to put one of your marks in your home board, and one in any other board you chose. You won by getting a normal connect four.
It got intense because you could see that someone would win on their next turn, but if it was still far away, you could pass the buck and assume somebody else would step up and block them. I wish there was a flash or Facebook version of the game, I think it could be pretty fun.
What if you have 3 tied spaces in a row, and you’re playing the version where the tied spaces count as both X and O?
This would be extra challenging if you extended it to three dimensions.
Here’s an Android app: https://www.facebook.com/pages/Inception-Tic-Tac-Toe/145431495644200?ref=stream
Wow! This game is really cool! I bought the paid version and it gets really hard. I really recomend this app to everybody.
Years ago I had a game for the original Nintendo called 3-D Tic Tac Toe. You had four boards that were each 4×4 and there were some nefarious rules that made it a real challenge to win. It was also lots of fun. And it’s not difficult to replicate on paper, either.
This is brilliant. Have been testing out a couple of variations of the game, including different winning conditions.
Who wins the most 3 in a rows across all nine boards, and who wins the most overall grids – certainly makes for some interesting strategy changes.
Thanks for the mention! I’ve actually added online multiplayer this week, and the new rules are coming in the next days 🙂 ~Niek Haarman
Awesome game, thanks for sharing this with us!
I highly recommend this alternative rule for “What happens if one of the small boards is a tie?”
Once a small board is completely filled, resulting in a draw, that board is then CLEARED, and the player who was sent there gets to make the first move on the newly-empty small board.
The strategy around this would be quite interesting – you’d need to be careful about sending your opponent to an almost-full board they can’t capture. Instead of sending them there for a “useless” move (and a guaranteed next square location), they’d instead be able to make the first move on a newly-cleared board, sending you to any square they choose.
This would also ensure that games don’t get bogged down with a bunch of “dead” squares, because the larger squares would never end up being “tied” – they’d either be in play, or definitively belong to one player or the other.
For the larger board, I also recommend a secondary way of counting victory. Obviously, if any player gets three large squares in a row (vertical, horizontal, diagonal), they win. However, if the larger board ends up in a draw, then the player who claims the most large squares wins.
This allows for more games to remain competitive up to the end, and provides some interesting endgame dilemmas. For instance, one player is close to getting three in a row, while the other player no longer has a chance for three in a row but can potentially capture more squares – meaning that both players still have something to play for, and defend against.
Plus, fewer games would end up in a draw this way. In fact, I’d need to modify my recommendation above for the small squares – if each player has claimed four large squares, leaving only one unclaimed square left on the board, and that last small square ends up in a draw, then the game ends in a draw right there. (No sense in clearing & restarting that one last square over and over, as then you’re just playing regular tic-tac-toe!)
In our implementation, the cats game is actually pretty rare if your strategy is solid. Our AI will strategically “trade” a win in one square for a win in a more strategic square (from its viewpoint) when playing the three in a row winning condition. We randomized the ‘best move’ algorithm to make the game difficult but winnable after you get enough practice. Try your luck and see how you fare … remember if at first you don’t succeed try try again 🙂
I hate to break too all of you but tic tac toe is a no win game. Neither player can win. If either player wins that would show a lack of intellect on the part of the loser.
NON SENS
What a fresh new game for my repertoire. Thanks.
I’m not saying that we were the only inventors, but this sounds exactly like the game my high school computer science class came up with around 2000. We also considered what you have dubbed the Orlin Gambit, but usually the person trying it chickened out before seeing it all the way through (peer pressure and jeering at its finest). Anyway, to counter it we considered another rule that never really caught on – each player has 2 or 3 “free moves” that they can use at any point in time. It is enough to break this gambit (choose a random square and send the perpetrator to the middle), but not enough to cause a breakdown in the strategizing.
nice game !! super tic tac toe 🙂
You mentioned “no through the internet play yet” so I made this. http://tictac9x.kissr.com/. This was fun to play with my usually unimpressed twelve-year-old, especially when he kept beating me. I really like your explanation style. Thanks for sharing!
Having read this I thought it was rather enlightening.
I appreciate you finding the time and energy to put this content
together. I once again find myself spending a significant amount of time both reading and commenting.
But so what, it was still worth it!
oh man i’m definately gonna give this a go! thanks for sharin’ it!
I haven’t played you version very much, but your gambit really is “too strong”. The version I’ve played is like the one on Khan Academy. If a player is sent to an already won box, they get to go wherever they want which would stop your gambit from working. This rule changes the strategy of the game a lot though. I haven’t played my version enough to find out if there are ways to force wins, so I don’t know if this variation is any better.
As a mathematician and game designer I’m absolutely enamored with this.
Trivia time … I’ll ship a Tic Tac Toe Ten board and t-shirt out to the first person who can tell me how many UNIQUE variations of Tic Tac Toe Ten can be played in the mode where 1st player to win one square is the winner of the game … State your assumptions 🙂
Reblogged this on Phasers and Spells and commented:
An awesome take on a simple game.
Nice to see that.
Back in the 80s we had a board game that used two dice to determine the range of squares you could fill, 5 sq. counted as a win, etc.
Suggestion: in the event of a draw, the square goes to whoever has the most marks in that square.