THE MATHS
A solved draw
With best play from both sides neither player can force a win, so the game is a theoretical draw. Mathematicians call it a solved game.
See the numbers
THE MATHS OF A PLAYGROUND GAME
Nine squares,
no winner.
Play noughts and crosses perfectly and it always ends in a draw.
The game has been fully worked out by mathematicians. There are 255,168 ways a game can unfold, a short list of rules that never loses, and a Cambridge computer that played it back in 1952.
DRAW with perfect play
Pick your next move- Win if you can
- Block if you must
- Make two threats at once
- A solved game, in plain English
THE SHORT ANSWER
Small board,
big tree.
Nine squares and eight winning lines.
Every possible game has been counted.
STRATEGY
A rule list that never loses
A 1972 priority list by Newell and Simon tells you what to do each turn: win, block, fork, block a fork, then centre and corners.
Try the rules
HISTORY
An early computer game
In 1952 a Cambridge PhD candidate wrote OXO, noughts and crosses for the EDSAC computer, with moves picked on a telephone dial.
Meet OXO
FOUR HABITS TO TRY
Never lose
again.
You cannot force a win against someone who plays perfectly, but you can make sure you never lose, and pounce when they slip.
Scan for your own win first
Before anything else, check every line for two of your marks and an empty square.
Then look for their threat
If they have two in a line, block it straight away. A missed block is the usual way games are lost.
Set up a fork
Aim for a move that makes two lines of two at once. Your opponent can only stop one of them.
Know any opening is safe
X can win or force a draw from a corner, edge or centre start. A corner leaves the opponent the fewest safe replies.
TWO NUMBERS WORTH KNOWING
Lots of games,
very few endings.
One figure counts every possible game,
one counts the different ways it can end.
255,168
Possible games of noughts and crosses
Counting every order of moves, without merging mirror images, there are 255,168 games. Of these, 131,184 are won by the first player, 77,904 by the second and 46,080 are draws.
EVERY GAME · GAME TREE
138 endings
Different finished boards
Treat rotations and reflections as the same and there are 138 finished boards: 91 won by X, 44 won by O and only 3 drawn.
ENDINGS · UP TO SYMMETRY
The first player wins far more of the random-play games, yet with perfect play from both sides the result is always a draw. A second player cannot have a winning strategy, because an extra mark on the board can never hurt the first player.
WHAT SHOULD I DO NOW?
Pick the situation,
get the move.
Perfect play follows a strict order of priorities. Choose what you see on the board to find the move, and how high it sits on the list.
Based on the Newell and Simon (1972) strategy as summarised in the Wikipedia article on tic-tac-toe.
OXO, 1952
The computer
that played back.
Alexander Douglas was a PhD candidate at the University of Cambridge when he wrote OXO for the EDSAC computer. It showed the board on a cathode-ray tube driven as a 35 by 16 dot grid, and players chose a square with a mechanical telephone dial.
The player decided who went first by dialling 0 or 1. It was made to support his dissertation on human-computer interaction, and few people outside Cambridge ever played it.
Questions
Questions people ask
Can you always win at noughts and crosses?
No. If both players play perfectly, neither can force a win and the game ends in a draw. You can only win if your opponent makes a mistake.
How many different games are there?
Counting every order of moves, there are 255,168 possible games. If you treat rotations and reflections as the same, there are 26,830 games and 765 essentially different positions.
Is it better to start in the centre or a corner?
X can win or force a draw from any opening. A corner leaves the opponent the fewest replies that avoid losing; one study cited on Wikipedia suggests the centre may be best against imperfect players.
What about bigger versions?
They behave differently. On a 3×3×3 cube the first player can force a win, in as few as four moves, starting in the central cube. The 4×4×4 game Qubic was proved a first-player win in 1980 using over 1,500 hours of computer time.
How old is the game?
Three-in-a-row boards have been traced to roofing tiles from around 1300 BC in ancient Egypt, and a Roman version called terni lapilli was played around the first century BC.
Sources
- Norman Do, How to win at tic-tac-toe, Gazette of the Australian Mathematical Society (2005)
- Jesper Juul: 255,168 ways of playing tic tac toe
- Wikipedia: Tic-tac-toe
- Computer History Museum: Timeline of computer history, 1952
- PONG-Story: Noughts and Crosses, the oldest graphical computer game
Updated