# Tic-Tac-Toe squared

Want to play a game of Tic-Tac-Toe that's genuinely challenging and hard? Try "Ultimate Tic-Tac-Toe," in which each square is made up of another, smaller Tic-Tac-Toe board, and to win the square you have to win its mini-game. Ben Orlin says he discovered the game on a mathematicians' picnic, and he explains a wrinkle on 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. (And whichever square you pick will determine which board he plays on next.)...

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.

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1. austinhamman says:

doesn’t mater if you nest them, whoever takes the top right corner wins or ties. so take the top right corner of the top right corner, and the top right corner of any sector you want.

1. Jake Wasserman says:

Choosing the top right corner in each little square just guarantees that your opponent will easily capture the top right corner on the big square.

The rules quoted here allow a first mover win by forcing the opponent into the same sub-board over and over.  The site lists a few modifications that further complicate that.

1. dragonfrog says:

I’m not sure that’s true.  Let’s say I’m trying to keep you pinned in the top right corner.

I start by playing the top right square of the center board.

You take the center square of the top right board.

I’m back in the center square, and now I can’t force you into the top right corner anymore – I have to send you somewhere else.  And anytime I send you somewhere where the center square is free, you can send me back to the center board, and have to send you somewhere else…

1. Jake Wasserman says:

I’m not 100% sure it’s true, but only because I can’t do the math myself.  I’m willing to take the author’s word for it, though, especially when I can try (and repeatedly fail) to beat that strategy against the program using it here:

Here’s the algorithm, induced from the program’s play:
1. Sacrifice middle board to claim 7 other center squares.
2. Sacrifice the eighth edge board to claim that square in all other edge boards, beginning with the board opposite the eighth edge board.
3. When 2nd player chooses the center square in the eighth edge board, use the free move to claim the opposite board.
4. Continue returning 2nd player to the eighth edge board and its opposite.  Eventually 2nd player will run out of choices to avoid a loss.

And having done that induction, I’m now 100% sure.  1st player always wins, by the originally stated rules.

1. Jonathan Roberts says:

Variation 5D would help to keep the game elegant but winnable:

If a player is sent to a mini-square that has already been won, they must play in that mini-square (if not empty), but then they get to go again.  If a player is sent to a totally filled square then they may go anywhere.

2. Martijn says:

If you want that strategy, you start by taking the top-right square of the top-right board. Your opponent can’t send you to the top-right board, so wherever he sends you, there you pick the top-right square again.

This will allow the first player to easily dominate and win the game, and that’s why you need a rule to prevent it: when a board has already been won, you can’t be forced to play there (wasting your move). Instead, you get a free pick of which board you play.

2. Martijn says:

I played this a while ago with my brother. The most important limitation is that if a board has already been won, or it’s full and it’s a draw, then you can’t be forced to play there anymore. If you should be forced to play there, instead you can pick which board to play.

This rule prevents all the easy win strategies, and adds a new layer of strategy onto the game.

Quite often I chose not to win a board because doing so would allow my brother a free choice which he’d use to win a more important board.

I think it still ended in a draw, though. I suspect you either get an easy win strategy for the first player, or if you manage to prevent that with the right rules, you get a draw.

2. Dan Hibiki says:

that would work on any corner?
it’s all relative.

2. bnschlz says:

Why stop at two layers? Go fractal!

1. knappa says:

That came to my mind too, but the problem is that the natural progression gives you a Cantor set of positions and there won’t be any 3 positions in a row in which to put your X’s or O’s. Arbitrary finite depth would work though.

1. rastronomicals says:

I must not understand what you’re saying.  No reason you can’t use the board that’s shown in the picture and make it the top left, say, square of a larger board.

And then when playing you start at the bottom and move up from the smallest to largest, smallest to largest.

What obstacle is it I’m not seeing?

2. Antinous / Moderator says:

Tic-Tac-Tesseract?

1. Kenmrph says:

Frac-Tac-Tal?

1. Petzl says:

Fric-Fract-Toe?

1. Why I love boing boing #38: Read this article and immediately raced to comments to make some fractal based tic tac toe wordplay only to find at least four people have already beaten me to it.

3. Very cool idea I am excited to try with my son (or officemates).  I am so happy to discover new brain games we can play with pen and paper (or whiteboard).

4. Kenmrph says:

Psh. I’ve been doing this for years, but with chess.

1. retepslluerb says:

You mean with 64 boards where each board represents a square in a superboard?

How do you move the superboards without toppling the pieces?

1. Martijn says:

It’d be hell with go; keeping track of 362 boards.

2. Gilbert Wham says:

We tried to figure out chess where you could move pieces through time once, but the implementation proved beyond us (though we were high).

1. PhasmaFelis says:

There’s a real-time strategy game called Achron where time travel is an essential game mechanic. Like, you can send your tanks back in time to hit the enemy base while their defenses are still weak, but then they can send their tanks back to reenforce themselves in the past, etc etc. It sounds insanely awesome and way too complicated for me.

3. Kenmrph says:

Oh I don’t use actual physical boards… I just keep track of it by memory.

2. I’ve been doing it with Monopoly. I am so bored.

5. ryuthrowsstuff says:

Used to do this in high school after a friend read about it in a popular facing book on physics and math. Its a hell of a lot of fun, mostly because as you figure it out you end up coming up with additional complications. Until you can’t remember the rules of the game you’re playing. In the end the game becomes less about the mechanics of tic tac toe and more about coming up with new variations to confuse your friends.

6. trentboyett says:

So wait…with the stated rules, could I end up needing to play a mini board that was already decided as won or lost?

1. dragonfrog says:

Sounds like it.

On one hand, that could be an advantage to your opponent because your move is ‘wasted’.  On the other hand, it could free you up to move entirely on the basis of where you send your opponent, since there’s nothing to lose on that board.

1. AlexG55 says:

Though if a board’s been won, a lot of the squares on it will be full so you’re actually quite limited on where you can send your opponent.

1. dragonfrog says:

In principle, as few as three of the squares could be taken – it may be that only one player has ever been sent to that board

2. Martijn says:

Our conclusion was that you need a rule preventing that. Once a board has been decided, being forced to play there gives you a free move.

7. Petzl says:

Clarifying rules (from website):

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.

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!)

1. Martijn says:

I think case 1 should also lead to a free choice.

1. surreality says:

He says so in an edit further down the page:
“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.”

2.  Given that there are only nine possible ways that anyone could be sent to a single board (each representative position in the nine boards) …
in what circumstance would it be expected to be possible to be sent to a full board? That would be the tenth visit.
I suspect that that point can’t be reached.

1.  Player 1 plays upper left of upper left. Player 2 plays anywhere on upper left board.  Player 1 plays upper left of board he is sent to.  Player 2 plays anywhere on upper left board.  Continue until last move, in which case we have:

122  1xx  1xx
222  xxx  xxx
222  xxx  xxx

1xx  1xx  1xx
xxx  xxx  xxx
xxx  xxx  xxx

1xx  1xx  1xx
xxx  xxx  xxx
xxx  xxx  xxx

Player 2’s turn, and she was just sent to a full upper left board

8. This week I thought of a Tic Tac Toe game where the center square has to be won with a game of dots and boxes.

It was Tic-Tac-TARDIS, because it’s bigger on the inside.

9. leeta says:

What happens if the one of the inner boards is a draw?