Higher-Dimensional Tic-Tac-Toe | Infinite Series

Regular tic-tac-toe can get a bit boring -- if both players are playing optimally, it always ends in a draw. But what happens if you increase the width of the board? Or increase the dimension of the board? Or increase both?

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The standard game of tic-tac-toe is too easy. How can we, as mathematicians, play with the combinatorics of tic-tac-toe? There are (at least) three easy ways to modify the game of tic-tac-toe: increase the width of the board - like *this* 5x5 board - increase the dimension of the board - like *this* 3x3x3 board - or increase both, like this 4x4x4 board.

Challenge Winner of the How the Axiom of Choice Gives Sizeless Sets:
For Your Math

Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow

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Special Thanks: Benjamin Houston-Edwards and Nathan Kaplan

Mathologer Video:

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And thanks to Mauricio Pacheco and Nicholas Rose who are supporting us on Patreon at the Lemma level!

[Bodyknock]

A really interesting variant I remember learning as a kid from Games Magazine is a variant called Order and Chaos. It's played typically on a 6x6 board with both players able to play either X or O however they like. One player, Order, is trying to form a line of either five Xs or five Os (Order wins in either case). The Chaos player wins if Order doesn't manage to make any lines of five in a row before the board is filled.

This variant is significantly trickier than standard tic tac toe games because, unlike a traditional tic tac toe game where every mark you place is beneficial, in Order and Chaos extra Xs or Os may or may not assist you depending on how they're arranged and whether or not you are Order or Chaos.

And unlike normal tic tac toe the standard method of proof that the first player can always win or force a draw doesn't work with this game. This is in part because of the game's asymmetry - the first player's win condition isn't the same as the second player's win condition, so if the first player simply "copies the second player's strategy" (eg First player is Order and is using a winning strategy for Chaos) it won't work.

A really fascinating tic tac toe variant!

[Reddles]

I can think of some other variations that might also be interesting:
1. Increase the number of players.
2. Allow the length of a winning line to vary separately from the grid size, eg. you could have a 4x4 board where you only need 3 in a row to win.
3. Increase the dimension of the winning 'line', for example a 3d grid where you need to completely fill a 3x3 plane to win.

[santiagovelamorales]

"If you´d never played 4by4by4by4 tic tac toe, pause here and give it a try"

[ImaginaryMdA]

Play a few games? What? You think I have friends?

[BALAGE]

The best game rule is having an infinite board (a notebook page irl) and aiming for five in a row.

[rngwrldngnr]

The other obvious change that would keep it an abstract strategy game would be to increase the number of players.

[MrHatoi]

*1966* "I bet in the future we'll have 3D chess!"
*2017* "We made 3D tic-tac-toe!"

[LNRDR]

My favorite variant of Tic-Tac-Toe is Ultimate Tic-Tac-Toe, where you have a large Tic-Tac-Toe board with nine smaller boards, one in each big square. The rule is that the second player has to play in the same big square as the corresponding small square of the previous player (so if P1 plays in the upper-right square of any small board, P2 must play in the upper-right board for their next turn). Whoever takes a small board wins that area, and "sending" an opponent to that board afterward allows them to play wherever they want. Whoever gets three small boards in a row wins!

I know there's a winning strategy, but it's so complicated that I don't know what it is and most people I meet don't either, so it's a really fun game every time I play!

Putting boards inside each other is another way to "expand" the Tic-Tac-Toe board, on top of increasing the side length and the number of dimensions.

[ObjectsInMotion]

In high school we would play 4 x 4 x 4 x 4 often, but the win was given to the person with the *most* 4s in a row, not the first.

[hallfiry]

Haven't watched the video yet, but I must say this: My desk-neighbour and I used to play mental 4D-tic-tac-toe in 12th and 13th class during boring classes. The dimensions were:
left, middle, right
top, middle, bottom
back, middle front
yesterday, today, tomorrow

[AngryArmadillo]

7:30

An exhaustive proof (even one done by a computer) is a rigorous proof. It is just as valid as any deduction.

[EdSmiley]

That brings me back to my childhood. I and a friend in the boy scouts would sketch out four dimensional tic tac toe games in the dirt and play. (You use 9 regular tic tac toe boards.) Other friends thought we were weird.

[AxcelleratorT]

My wife and I built a 4x4x4 tic tac toe board. We quickly learned that "x" always won. So, to make the game more interesting we built random chance into the start of the game.

We did this by making 64 tiles, each with the coordinates of one unique space on the board. At the beginning of game play, each player would draw three tiles at random. So for instance, "x" might draw tiles with coordinates (3, 1, 4), (4, 2, 2), and (3, 1, 3) and "o" might draw tiles with coordinates (1, 1, 2), (4, 3, 1), and (2, 2, 2). Each player would then place one of their markers in each of the spaces specified on the tiles.

After this random start, game play would return to normal with each player taking turns placing a game piece in the location of their choosing.

[PlayTheMind]

*Tadashi Tokieda*
VS *Matt Parker*
_epic-tac-toe_

[Anthony-cnll]

A 1x1 board will also guarantee that the first player always wins.

[irwainnornossa]

For the slight chance, that someone will actually read this:
This topic is very close to my hearth. I’ve spend some quality time to getting know 4D space more intimately. And one of products of this, was my version of 4D 3T (Tic-Tac-Toe for short). The rules were different though. I don’t know, how elsewhere, but in my county 3T are played on 2D board, usually 19×19, and winning pattern is line of length 5. Now, the number of surrounding fields of one particular field is 3^n – 1, where n is the dimension. So usually it’s 8, which leaves us with 4 directions. In higher number of dimensions this “combination” number rises exponentially, obviously. This lead to 4D game with the rule of winning 6-long line. It’s usually played in tesseract of edge-length 9 or better, 11. How that look like? Field of fields. You have 11×11 grid of 11×11 grids. It’s surprisingly easy to work with, because you don’t need to visualize something, what our brain cannot visualize: 4D space. You work with what you know, you only need to think about the relations between 11×11 fields. Of course, length 11 is kind-of arbitrary, it might be different. Why odd number? So you have the central point.
In similar fashion, you can put together 5D or even 6D field, but it gets insanely large really fast.
I say it all the time. Higher dimensions are fun. I don’t like our 3D space, I want more dimensions. 4, that would be awesome. Because if you have tesseract of edge-length 1, the full-space diagonal line is 2 units long.

[jaimeduncan]

This series continues to be extraordinary, an the Phd is very good at presenting the information

[macronencer]

Love this video! When I was a kid (about 11?) I wrote out all possible games for the standard 3x3 (ignoring symmetries), to verify that the strategy worked. I encoded the squares as digits from 1 to 9. It's possible I still have that little red notebook somewhere, with pages and pages of digit sequences... I hope so.

[GroovingPict]

you can also create a variant where you have a big 3x3 board, and inside each square of that is another 3x3 board. And a player can go in any square of any of the smaller boards when it's their turn. Once a small board has been won by either X or O, that large square is turned into an X or O respectively and of course the object is to win the big board.

The common rules that we played by in highschool (when sitting in boring physics lessons):

- 5 in line wins
- "board" is unlimited