A game of "Noughts and Crosses", but without the noughts!

The rules of the game

Just like Noughts and Crosses, this game is played between two players, adding a symbol to the board in turns. However, unlike Noughts and Crosses, the players use the same symbol (a cross), and can both contribute to the same line. Interestingly, although this is a simplification of Noughts and Crosses, it actually improves the game by making a draw impossible instead of practically guaranteed.

Notakto is usually played in misére mode where the goal is to avoid making a line of a specific length, but can also be played in normal mode where the goal is to make such a line.

Note that in the game below, each player's symbol is drawn in a different colour, but this has no significance for the game—a winning line can be of any combination of colours.

Now let's play

Choose a size for the board, a target length for a winning line, and a style of play (i.e. misère or normal). Click a blank tile to place an X, or hit the Computer button to let the computer make the next move.


Next move:  Player 1 Player 2      


How can mathematics help us understand this game?

Consider a 3×3 board, with target length also 3.

  • How many lines are possible?
  • Which square is best for the initial move if playing in normal mode? Why?
  • Which square is best for the initial move if playing in misère mode?
Now extend your analysis to consider an n×n board, with target length n.
  • Now how many lines are possible?
  • How might you describe mathematically the difference between placing an X in a corner, along an edge, or internally?
  • Does this mathematical description help you devise a strategy for this game?
You might also like to consider the situation where the winning line is of length n−1, or when the board is of size m×n. As the analysis is extended, the difficulty increases rapidly. This is common for combinatorial problems of this type.