Notakto

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

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.

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.

Style:

Next move: Player 1 Player 2

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 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?

Although a seemingly obvious variation of Noughts and Crosses, the game of Notakto appears to have arisen only very recently. The earliest reference I could find was from 2010 — by Bob Koca, a Mathematics Professor and Backgammon champion. It is also clearly related to the game Connect 4, and the 1×n version is called Treblecross.

An interesting question to consider is how does the computer choose which tile to flip? The algorithm employed is called the Minimax algorithm, and is an important algorithm to learn for anyone interested in artificial intelligence and computer game play (e.g. for Checkers, Chess, Go).

Game Theory is the branch of mathematics that deals with the analysis of games, or more generally, any decision-making situation involving cooperating or competing parties. Therefore the principles of game theory apply not only to games such as cards and chess, but also to real-world problems in economics, politics, and warfare.

Mathematicians and Computer Scientists consider the class of games similar to Notakto as m,n,k-games since they involve lines of length k on boards of size m×n. These games are important tools in the study of Game Theory.