
Ultimate Tic Tac Toe - sid6376
http://mathwithbaddrawings.com/2013/06/16/ultimate-tic-tac-toe/
======
quesera
I think the Orwin gambit can be extended to win the game every time.

\- Force opponent to fill center miniboard, as he describes.

\- Force opponent to fill (e.g.) northeast corner in the same way. Opponent
now has taken two miniboards, and you have none, but you are one turn away
from taking each of the remaining seven.

\- Pick SW corner of SW corner. You have taken SW corner miniboard. Opponent
is forced to play in same SW miniboard, already won by you.

\- Pick SW corner of S. You have taken S miniboard. Opponent is forced to play
in SW corner again, already won by you.

\- Pick SW corner of SE corner. You have taken SE corner miniboard.

\- Done. You win.

Like regular tictactoe, there is an advantage to going first. Unlike regular
tictactoe, the advantage can't be compensated for. Otoh, the second player can
use the same strategy with a little more carefulness, as long as they start
early.

So either player can force the other into a protracted certain loss, unless
there's an agreement or a rule against it. That's no fun.

EDIT: actually, you can win every time, in far fewer moves, and not using the
Orwin gambit at all. It's not necessary to force your opponent to _fill_ any
of the miniboards, not even the center.

I think this will win in ten moves and never lose driver control (excuse the
notation): C/C, C/SW, C/S, [opponent takes C], C/SE, NE/SW, NE/S, NE/SE,
[opponent takes NE], SW/SW [you take SW], SW/S [you take S], SW/SE [you take
SE, and win]. A variation can be used by either player early in the game, but
whoever starts with control would be foolish to lose it.

If this is a game played by mathematicians, either I'm wrong, or there are
additional rules. :)

EDIT2: C/C (first move above) is unnecessary. Nine moves. Perfect inning.

~~~
matchu
Here's an implementation that pretty much does exactly that. If someone can
beat this bot, lemme know: [https://www.khanacademy.org/cs/in-tic-tac-toe-
ception-perfec...](https://www.khanacademy.org/cs/in-tic-tac-toe-ception-
perfect/1681243068)

And here's James Irwin's original implementation, auto-set to a Monte-Carlo AI
and with a rules variation that (to my knowledge) prevents Perfect Bot from
winning. Play with the config variables at the top for some other AIs and
2-player games and whatnot. [https://www.khanacademy.org/cs/in-tic-tac-toe-
ception/167633...](https://www.khanacademy.org/cs/in-tic-tac-toe-
ception/1676336506)

~~~
tensaix2j
Does the AI always have to make the first move?

~~~
matchu
For Perfect Bot's strategy to work, yeah, but the environment is very open to
configuration. Set xIsComputer to false and yIsComputer to "monte" for a fun
challenge in which you go first.

------
michaelfeathers
My favorite "mathematician game" is Sprouts:
[http://nrich.maths.org/2413](http://nrich.maths.org/2413)

First heard about it in a column in Scientific American by A. K. Dewdney. The
rules are simple and it is rather fun. In one of his columns he talked about
playing a toroidal version where a line going off one edge of a rectangular
page comes back in on the opposite side.

~~~
kernel_sanders
My Dad, a physicist, taught me that game as a kid. I have great memories of
him producing a pen at any restaurant we might have been at, and he and I
passing any waiting time playing sprouts on napkins or paper scraps. I since
play it with my kids.

~~~
michaelfeathers
Did you ever learn about the variant called 'Brussels Sprouts' and the trick
to winning?

~~~
coldpie
Please explain :)

~~~
michaelfeathers
If I remember correctly, instead of only allowing three edges to emanate from
a vertex, you can have four. All other rules are the same. Apparently, in that
set up, the the first player either always wins or loses depending upon
whether the number of initial vertices is odd.

------
willvarfar
My daughters play a variant of tic-tac-toe with a physical board and just
three stones each.

I brute-forced this with a little lunchbreak program and the visualisation
output from graphviz was ... 340 MEGAPIXELS!

I blogged about it:
[http://williamedwardscoder.tumblr.com/post/35858593837/tic-t...](http://williamedwardscoder.tumblr.com/post/35858593837/tic-
tac-toe)

~~~
fduran
That's how we used to play as kids, there's an algorithm so that the first
mover always wins.

------
shardling
A weird version of tic-tac-toe my friends and I came up with years ago: you
play without a board. Any move is valid so long as you could superimpose a
standard tic-tac-toe grid over the results. (You assume the marks are at the
center of their respective squares; you allow the board to be at any scale or
angle, but you can't skew it.)

The first two moves have no constraint, so you can just start with an X and O
already on the board. And after a few moves there might be only one implied
board possible, after which it reduces to the normal game. But I always
thought it was an interesting twist.

~~~
lotharbot
Somebody posted another tic tac toe variant on HN some time ago, where the
squares were numbered 1-9 and the goal was to get any three squares that
summed to 15. It turns out this is equivalent to playing tic tac toe with
sides that wrap around (like a torus).

~~~
Someone
No, that's 100% equivalent to tic tac toe ([http://en.wikipedia.org/wiki/Tic-
tac-toe#Variations](http://en.wikipedia.org/wiki/Tic-tac-toe#Variations))

(To see why no torus version exists: on a '3x3' torus, all squares are
identical, so you have must four ways to get three in a row from each starting
number. Also, the row/column/diagonal sum must be 15. However, starting with
9, there are only two ways to get 15 with: 1 and 5 and 2 and 4)

The variant with nine words, where selecting three words that share a letter
wins the game, IMO is even better.

~~~
lotharbot
I may have misremembered the details of the game, or conflated it with another
game.

What I remember clearly is that the game was equivalent of tic tac toe on a
torus -- one could win with a "diagonal" of the form

.|x|.

x|.|.

.|.|x

as well as any of the traditional tic tac toe win conditions.

~~~
Someone
Tic tac toe on toruses is fairly popular as an educational toy, and there also
is serious research on it, for example
[http://www.ux1.eiu.edu/~kwolcott/TTT.pdf](http://www.ux1.eiu.edu/~kwolcott/TTT.pdf).
You might have mixed the 'on a torus' part with the 'using a magic square'
part. No problem, but the two do not mix that way (maybe, on larger boards?
That 4x4 Dürer square
([http://mathworld.wolfram.com/DuerersMagicSquare.html](http://mathworld.wolfram.com/DuerersMagicSquare.html))
has zillions of 'add up to 34' sets of cells. There may be even better ones of
larger size)

------
dreen
There is another great variant of Tic Tac Toe you can play on a Torus (a donut
shape):
[http://upload.wikimedia.org/wikipedia/commons/1/17/Torus.png](http://upload.wikimedia.org/wikipedia/commons/1/17/Torus.png)

The way to play is to "warp" each side of the board to the side opposite of
it. For instance, playing on a 6x6 board with a win condition of length 5:

    
    
        .|X|.|.|.|.
        X|.|.|.|.|.
        .|O|.|.|.|.
        .|O|.|.|X|.
        .|.|O|X|.|.
        .|.|X|O|.|.
    

is a win for X, because the upper and lower ends of the board are synonymous.

I can't find any material about this on the net, I just played it in school on
boring lessons (but more commonly we played for 5 in a row on an infinite
board, I personally prefer the Torus)

------
arh68
> Whichever square he picks, that’s the board you must play in next.

And what about the dual of this game, where whichever board you pick
determines the square he plays next?

------
nawitus
Does anyone else play the 'infinite board' tic-tac-toe which requires you to
get five in a row? It's what I used to play as a student, and it's pretty
well-known at least where I live. Nobody bothers with the 3x3 version, but the
five in a row version is pretty exciting and requires plenty of strategy and
thinking ahead.

~~~
bierik
I think this is called "Gomoku" (just on a quasi-infinite board):
[http://en.wikipedia.org/wiki/Gomoku](http://en.wikipedia.org/wiki/Gomoku) I
used to play this a lot when I was still at the university

~~~
seri
Yup, Gomoku is a much more difficult game than it looks. Indeed, the game is
challenging enough that there is a world championship every odd year. You can
find the games from the last world championship here:
[http://renju.net/media/games.php?gameid=45016](http://renju.net/media/games.php?gameid=45016)

I am amazed by the fact that Gomoku can be so hard to master with rules so
simple you can explain to a five year old. And unlike Chess where there are a
hundred year of theories to learn from before you can get going, Gomoku is
still new. After a few weeks studying the standard surewin openings, you can
expect to see things in a very different light and the game will get much more
interesting.

Perhaps Gomoku is best known for programmers as a problem to solve. But it is
nowhere near being solved. In fact, the best software are weaker than many top
players.

I find Gomoku hits the sweet spot when it comes to my desire to play board
games. It doesn't consume much of my time. I am always excited to find those
long and obscure wins. I think the game needs more love from programmers like
me.

Facebook page:
[https://www.facebook.com/GomokuWorld](https://www.facebook.com/GomokuWorld)
Play here:
[http://www.playok.com/en/gomoku/](http://www.playok.com/en/gomoku/) Or here:
[http://fumind.com](http://fumind.com)

~~~
fc2
Gomoku has already been solved actually, maybe you were thinking of Go?
There's a revised version called Connect6 though, in case you are interested.

[https://en.wikipedia.org/wiki/Connect6](https://en.wikipedia.org/wiki/Connect6)

~~~
seri
Under free opening rules, Gomoku has been proved to be sure-win for black, aka
the first player. This work done by Victor Allis could be part of the reason
why Gomoku is undeservedly treated as a toy board game.

We have to remember, however, that this proof is computer search based. There
was never an algorithm to find the best move given a random position. In other
words, it only proves that these certain openings will guarantee black a win.

Those who loved the game came up with new opening rules. The world
championships in 1989 and 1991 used the pro rule. After some evolutions, swap2
now became the standard. Under Gomoku swap2, the first player puts two black
stones and one white stone on the board, and the second player can either pick
one color, or puts two more stones and gives back the power to choose color to
the first player.

I know about Connect6, and also a branch called Renju. But I find Gomoku more
attractive. You should totally try it sometimes.

------
Delfino
Neat variant. I made something similar for the last ludum dare:
[http://madelfino.github.io/LD26/postcompo.html](http://madelfino.github.io/LD26/postcompo.html)

~~~
M4v3R
Nice! Managed to beat it, but it wasn't easy:
[http://dl.dropbox.com/u/1127246/Screenshots/5ths.png](http://dl.dropbox.com/u/1127246/Screenshots/5ths.png)

------
speeder
When my dad was at university he made just out of curiosity a AI to play Tic
Tac Toe that started knowing nothing about the game, and tried to learn from
playing against the player, until it became impossible to beat it.

Later someone complained with university administrators that my dad was
"playing games" in the computer lab, and he got banned from it :/

But I guess this version might be even more interesting to make a AI test or
something!

~~~
garethadams
The only winning move is not to play

------
mrspeaker
I don't understand the clarifying rule #2: "What if my opponent sends me to a
board that’s full?" Isn't that impossible, as there is a maximum of 9 ways to
be sent to each local area?

~~~
cousin_it
I think you can get sent to a full board only if the game started on that
board.

~~~
jbri
Of course, there's a cascading effect where it's now possible to also overflow
the board you chose with your free move.

------
joeyrobert
Another quick implementation of this game:
[http://joeyrobert.org/projects/ultimatetictactoe/](http://joeyrobert.org/projects/ultimatetictactoe/)

------
alphaoverlord
Perhaps the game is less deterministic if you can choose any board as soon as
the board is won - not when it is filled. Then, you can only force 3 moves
before you have to give up control. Then approaches like the Orwin gambit
would not work (it is too costly to lose a board if it only lets you choose
three spots.)

------
Kurtz79
[http://xkcd.com/832/](http://xkcd.com/832/)

Randall Munroe could do a "small" update to this based on these rules...
although it would take some time.

~~~
lucb1e
A "small" update? I think this is more like the difference between checkers
and chess. But yes, relevant xkcd

~~~
tripzilch
> I think this is more like the difference between checkers and chess

I'm not sure what you mean by this?

BTW are you aware that despite it has only one type of stone and simpler
rules, checkers is in fact the more complex game, to write an AI for?

~~~
pong_
I guess you refer to International checkers (being from the Netherlands).
Still a challenge for computers, given that the brute-force solution for
American checkers doesn't work well for this larger board.

Much harder to program an AI for than chess, certainly. The computer never
managed to beat the world champion in a match last year.

~~~
tripzilch
Cool, I didn't know there were multiple versions of checkers. I'll have to
check (hah) out that solution for the American version though, always
interesting (as well as what the differences in rules are). Off to Wikipedia,
I guess ;-)

------
yashg
This is brilliant! Has anybody created an online version of this?

~~~
gambogi
I can bet you there will be now

~~~
ronaldx
It doesn't observe a win and for now you have to play against yourself, but:

[http://xoxo.gl/ultimate](http://xoxo.gl/ultimate)

------
deletes
Important question, can this game be solved, like ordinary tic-tac-toe. Is
there a solution that always leads to draw of defeat.

~~~
k2enemy
Zermelo's theorem[1] says "yes," although I don't know which it is (between
draw, p1 win, p1 lose).

[1]
[https://en.wikipedia.org/wiki/Zermelo%27s_theorem_%28game_th...](https://en.wikipedia.org/wiki/Zermelo%27s_theorem_%28game_theory%29)

~~~
CJefferson
In fact, it can't be a p1 lose, because the extra piece p1 gets can never be a
disadvantage (you can prove this more carefully). The question is then is it a
p1 win, or draw?

A similar situation arises in the game 'hex', except in that game there is no
draw, so it has to be a p1 win!

~~~
Someone
_" because the extra piece p1 gets can never be a disadvantage (you can prove
this more carefully)"_

Please do so, as I do not see that this is trivially true. I see two tricky
cases:

\- your opponent must play on the board where p1 is placed, and that board
would have room iff p1 weren't present => Addition of p1 gives him a 'move
anywhere' move.

\- you must play on the board where p1 is placed, and that board would have
room iff p1 weren't present. The normal argument 'move anywhere and assume
that that 'anywhere' is where your first move went, and you just played p1'
does not help here, as changing the first move may change where your
opponent's first move could have gone.

Except from an exhaustive search, I do not see how to prove that you can
prevent either case.

~~~
CJefferson
Yes, you are right.

Actually, the fact you force your opponent isn't the problem (I don't think)
but the fact that the presence of that piece might later give p2 a free move,
where previously they wouldn't means it doesn't work.

Sorry, that's what I get for thinking I didn't need to figure out all the
details carefully!

------
jjcm
The one I play with my friends is usually a 4-in-a-row, 4 level board. You can
win on any individual level, or by creating a straight line of 4 that
intersects plays on each level (so playing in the top right of each level
would net you a win, for instance). It forces the players to keep a three
dimensional model in their head and opens up the board for more
counters/strategy.

Anyone care for a game?

    
    
        level 1    level 2    level 3    level 4
        .|.|.|.    .|.|.|.    .|.|.|.    .|.|.|.
        .|.|.|.    .|.|.|.    .|.|.|.    .|.|.|.
        .|.|.|.    .|.|.|.    .|.|.|.    .|.|.|.
        .|.|.|.    .|.|.|.    .|.|.|.    .|.|.|.

------
ronaldx
I made a quick HTML demo for playing through the game at
[http://xoxo.gl/ultimate](http://xoxo.gl/ultimate)

~~~
joelthelion
Now add an AI :)

~~~
ronaldx
You think so? I'm not very keen to work on an AI if there's proof of a quick
win.

------
irunbackwards
We made this into an Android application about a year ago:
[http://superttt.com](http://superttt.com)

------
lkozma
I recall seeing this before, called "crazy" tic tac toe:

[http://tictactoe-cssi.appspot.com/](http://tictactoe-cssi.appspot.com/) via:
[http://neil.fraser.name/news/2011/07/16/](http://neil.fraser.name/news/2011/07/16/)

~~~
deanclatworthy
The first of these doesn't follow the same rules as OP. It seems that you can
retake a board if you get a new row in a box that has already been won.

------
h4pless
I haven't tested all the cases but this method seems to win every time I've
tested it against myself:

Start on an edge-center container [N,E,S,W], marking it with it's
corresponding sub-board space. Employ the Orwin gambit to fill the initial
edge container. When your opponent selects the last space in the first
container, wherever you are sent: pick it's corresponding sub-board space and
then employ the Orwin gambit again and then repeat the routine until the game
is finished. Your opponent gets to pick your next moves but because they
eventually must send you to a filled square on the second and third rounds,
you have the ability to control the game's end.

Starting in the center gives you a tactical disadvantage because it only
leaves you 4 paths to victory compared to your opponents 8 with him/her in a
position affecting 4 lines, whereas by starting on a side piece, you have 7
paths to victory and your opponents position only benefits him on 2 lines.

------
tictactoeten
My team and I have implemented what you call Ultimate Tic Tac Toe as what we
call Tic Tac Toe Ten. We put it out for iOS, Android, Windows Phone and have
made board game sets to go into school districts with starting in 2008. If any
of you in this community are interested in helping us to build out the backend
analytics infrastructure to help us make the academic breakdown of this game
useful for teachers and students please visit our community at
[https://www.facebook.com/tictactoeten](https://www.facebook.com/tictactoeten)
and drop me a message or DM us on Twitter (@tictactoeten) ... we are also
always looking for volunteers at our school tournaments where we pit kids
against each other March Madness style with brackets and give out prizes to
the winners ... we are here in the Bay Area, hope to meet some of you guys
soon :)

cheers, @bware218

------
baritonehands
I created a version of this game back in 2008, for the 3 major mobile
platforms and PC:

PC: [http://www.tictactoeten.com/play](http://www.tictactoeten.com/play)

iOS: [https://itunes.apple.com/us/app/tic-tac-toe-
ten/id317168510?...](https://itunes.apple.com/us/app/tic-tac-toe-
ten/id317168510?mt=8)

Android:
[https://play.google.com/store/apps/details?id=com.livelovele...](https://play.google.com/store/apps/details?id=com.livelovelearn.tictactoeten&hl=en)

Windows Phone: [http://www.windowsphone.com/en-
us/store/app/tictactoe10/d84e...](http://www.windowsphone.com/en-
us/store/app/tictactoe10/d84eac98-7a6f-e011-81d2-78e7d1fa76f8)

------
AUmrysh
What if you played it where the square selected is the direction you move to
go to the next board. If the player selects the center square on a board, that
board must be played again. If the player selects the left square, the board
to the left must be played next. If a corner square is picked, you play the
board to that diagonal direction.

If the corner is not attached to a board, you roll the board around as though
the ends are connected (or as though it's an infinitely repeating tiling of
the same game). For example, the bottom right square on the bottom right board
makes the next board the top left.

The center-right square on the center-right board would result in the next
board being the center-left board, and so on.

Would there be a gambit in this ruleset?

edit: So, it looks like the same gambit applies, instead of selecting the
center piece you just pick the one pointing toward the center every time.

------
yarekt
[http://graffitiwall.co.uk/ultimatetictactoe](http://graffitiwall.co.uk/ultimatetictactoe)

My quick implementation of this game

Edit: For anyone who arrived 10 seconds after i posted that link, The board
resembled the ultimate tic tac toe, and then quickly degraded into a paint
fight

~~~
speeder
Interesting that hackers instead of working or reading HN are very interested
in fighting over the board (ie: gray color people try to erase it, pink color
try to draw a board, and blue color trying to play the game...)

~~~
danielweber
"Someone keeps stealing my letters"
[http://www.lunchtimers.com/](http://www.lunchtimers.com/)

------
frogpelt
There's a way to prevent the Orlin gambit. Allow the player with the last
three in row to take another turn.

This creates a means of alternating who goes first.

I'm not sure whether this gives too much advantage to the other player or not.

~~~
akincisor
It does. He can now force _you_ to now place pointless moves in the center
square.

------
GotAnyMegadeth
My friends used to play something similar with connect 4. You don't expect to
meet people who are good at connect 4, but these two guys beat me 100% of the
time...

------
gwern
Reminds me a bit of
[http://en.wikipedia.org/wiki/Fire_and_Ice_%28board_game%29](http://en.wikipedia.org/wiki/Fire_and_Ice_%28board_game%29)
/ [http://www.boardgamegeek.com/boardgame/4643/fire-and-
ice](http://www.boardgamegeek.com/boardgame/4643/fire-and-ice) \- look at the
board.

------
sudhirj
Can anyone else hear the quiet roar of a thousand not-so-silent fans as they
struggle to dissipate heat caused by furious Xcode and RubyMotion
compilations?

------
gamegoblin
Another interesting way to play tic-tac-toe that makes it a lot harder:

There is a bag of numbers 1-9. Players take turns moving numbers from the
central bag to their own private bags, removing that number from play.
Whenever a player has a subset of exactly 3 numbers that sums to 15, they win.

This is isomorphic to tic-tac-toe since the magic square for a 3x3 board has
rows/columns/diagonals that sum to 15.

------
solox3
It happened a while back, when Bobby Kehres told my colleague and I about the
game. [https://github.com/1337/Tic-Tac-Toe-
Extreme](https://github.com/1337/Tic-Tac-Toe-Extreme)

------
rfisher1968
I made this a while back. It a HTML5 game made with construct2.
[http://jenpop.com/game.php?gurl=games/uttt](http://jenpop.com/game.php?gurl=games/uttt)

~~~
roryokane
Your game is just Tic-Tac-Toe on a 6x6 board. It is not the same game as the
Ultimate Tic-Tac-Toe described in the article.

------
crimzonphox
My favorite Mathematics game is:
[http://en.wikipedia.org/wiki/Connect_Six](http://en.wikipedia.org/wiki/Connect_Six)

Mathematically fair

------
randren
Kid-tested. The 7-year-old loves it. Thank you for saving me from plain-old
Tic Tac Toe games with my kid.

------
minikomi
What happens to the game if you switch order when you take a board?

------
iguana
Someone needs to write a JavaScript implementation of this!

