Computational complexity[ edit ] The computational complexity of a game describes the asymptotic difficulty of a game as it grows arbitrarily large, expressed in big O notation or as membership in a complexity class.
History[ edit ] Games played on three-in-a-row boards can be traced back to ancient Egypt where such game boards have been found on roofing tiles dating from Noughts and crosses analysis BCE.
It was called terni lapilli three pebbles at a time and instead of having any number of pieces, each player only had three, thus they had to move them around to empty spaces to keep playing.
Another closely related ancient game is Three Men's Morris which is also played on a simple grid and requires three pieces in a row to finish,  and Picariaa game of the Puebloans.
The different names of the game are more recent. The first print reference to "noughts and crosses" nought being an alternative word for zerothe British name, appeared inin an issue of Notes and Queries.
The US renaming of "noughts and crosses" as "tic-tac-toe" occurred in the 20th century. The Tinkertoy computer, made out of almost only Tinkertoys, is able to play tic-tac-toe perfectly. Combinatorics[ edit ] When considering only the state of the board, and after taking into account board symmetries i.
A combinatorics study of the game shows that when "X" makes the first move every time, the game is won as follows: In each grid, the shaded red X denotes the optimal move, and the location of O's next move gives the next subgrid to examine.
Note that only two sequences of moves by O both starting with center, top-right, left-mid lead to a draw, with the remaining sequences leading to wins from X.
Optimal strategy for player O. Player O can only force a win or draw by playing in the centre first. A player can play a perfect game of tic-tac-toe to win or at least, draw if each time it is his turn to play, he chooses the first available move from the following list, as used in Newell and Simon's tic-tac-toe program.
If the player has two in a row, they can place a third to get three in a row. If the opponent has two in a row, the player must play the third themselves to block the opponent.
Create an opportunity where the player has two threats to win two non-blocked lines of 2. Blocking an opponent's fork: If there is only one possible fork for the opponent, the player should block it.
Otherwise, the player should block any forks in any way that simultaneously allows them to create two in a row.
Otherwise, the player should create a two in a row to force the opponent into defending, as long as it doesn't result in them creating a fork. For example, if "X" has two opposite corners and "O" has the center, "O" must not play a corner in order to win.
Playing a corner in this scenario creates a fork for "X" to win. A player marks the center. If it is the first move of the game, playing on a corner gives the second player more opportunities to make a mistake and may therefore be the better choice; however, it makes no difference between perfect players.
If the opponent is in the corner, the player plays the opposite corner. The player plays in a corner square. The player plays in a middle square on any of the 4 sides. The first player, who shall be designated "X", has 3 possible positions to mark during the first turn.
Superficially, it might seem that there are 9 possible positions, corresponding to the 9 squares in the grid. However, by rotating the board, we will find that in the first turn, every corner mark is strategically equivalent to every other corner mark. The same is true of every edge side middle mark.
For strategy purposes, there are therefore only three possible first marks: Player X can win or force a draw from any of these starting marks; however, playing the corner gives the opponent the smallest choice of squares which must be played to avoid losing.
The second player, who shall be designated "O", must respond to X's opening mark in such a way as to avoid the forced win. Player O must always respond to a corner opening with a center mark, and to a center opening with a corner mark.
An edge opening must be answered either with a center mark, a corner mark next to the X, or an edge mark opposite the X. Any other responses will allow X to force the win. Once the opening is completed, O's task is to follow the above list of priorities in order to force the draw, or else to gain a win if X makes a weak play.
More detailedly, to guarantee a draw, O should adopt the following strategies: If X plays corner opening move, O should take center, and then an edge, forcing X to block in the next move.
This will stop any forks from happening. When both X and O are perfect players and X chooses to start by marking a corner, O takes the center, and X takes the corner opposite the original.‘Noughts and Crosses” by Malorie Blackman is a novel which follows the lives and experiences of two characters, Callum and Sephy.
Throughout the book Blackman deals with a number of issues including relationships, alcohol abuse, power abuse, depression and violence. Perhaps the most famous novel written by Russian author Fyodor grupobittia.comally in Russian under the title Prestuplenie i nakazanie (Преступлéние и наказáние).
A moody university student named Raskolnikov murders an old moneylender who has been exploiting her clients, but accidentally also kills someone else.
He struggles with the ramifications of his actions through. How does the structure of Noughts and Crosses help to reinforce the key themes in the novel? In this essay I will be analysing and evaluating the structural techniques used in Malorie Blackman s language throughout the novel, and identifying different layers of meaning of her writing using textual evidence.5/5(1).
Final instalment of Noughts and Crosses - play version - SoW (lessons, resources and a knowledge organiser). The unit includes non fiction lessons (including reading strategies such as stop and jot and read, cover, recap), GCSE skills looking at language and .
A booklet I gave to my Year9s once we’d read part of Noughts and Crosses by Malorie Blackman (we’d read up to just after the trial of Ryan). It includes a List of 11 tasks from which they can choose at least 2 to complete to show their understanding of.
1 Noughts & Crosses by Malorie Blackman: a textual analysis The essay below explores a short extract from Noughts and Crosses by Malorie Blackman.