# O-X-O investigation

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Introduction

O-X-O Investigation

In this investigation I will investigate the game noughts and crosses. My aim is to find out how many winning lines in a game. I will find out a rule for a regular game on a 3 by 3 grid using 3 in a row as the winning line. After I have a rule for the first grid I will then go on to find rules for several other grids and come up with a general rule. To further my investigation I will change the shape of the grid by changing the length and width and changing the number needed for a winning line. I also may investigate 3D grids and make an overall comparison between all my general rules to see if there is an overall rule for all lengths, widths, winning lines and shapes.

In this investigation on the grids the black is the outline of the grid and the red is to show how many winning lines there are. Also I will use some abbreviations in this investigation.

W=Width

L=length

Middle

48 winning lines

Size of grid WxL | N.o of squares needed for a winning line ‘T’ | N.o of winning lines ‘S’ |

3x3 | 3 | 8 |

4x4 | 3 | 24 |

5x5 | 3 | 48 |

WxL | 3 | 4(L-1) (L-2) |

To check if my rule is correct I wall use it to tell me how many winning lines would be in a 10x10 grid. I will then check this by drawing out the grid. If my rule is correct there should be 288 winning lines.

10x10

288 winning lines my rule is correct

W L | T | S |

3 x 3 | 3 | 8 |

4 x 4 | 3 | 24 |

5 x 5 | 3 | 48 |

6 x 6 | 3 | 80 |

W | 4xW² | S | What’s left? |

3 | 36 | 8 | -28 |

4 | 64 | 24 | -40 |

5 | 100 | 48 | -52 |

6 | 144 | 80 | -64 |

Overall 4W²-(12W-8)

=4W²-12W+8

=4(W²-3W+2)

=4 (W-1) (W-2)

In this case T=3 so I will write all number in my formula in relation to T in order to try and find a general formula for all squares and link my new formula to my first formula of 2W+2.

= (T+1) [W- (T-2)] [W- (T-1)]

=(T+1) (W-T+2) (W-T+1)

In investigation one I found the general formula 2W+2

My new formula is (T+1) (W-T+2) (W-T+1) which I found from investigation two

However, in investigation one T=W=L. Therefore I can relate my new formula back to investigation one by replacing T by W

= (W+1) (W-W+2) (W-W+1)

=(W+1) x 2 x 1

=2 (W+1)

=2W+1

So the formula from investigation one is a simplistic version of the formula from investigation 2

Therefore I now have a general formula for when the grid is a square and the length of the winning line is a variable

## Investigation 3

For the next part of this investigation I am going to change the shape of the grid from a square to a rectangle. I will keep the length of the rectangle 3 and only change the width of the rectangle. I will use 3 as the number of squares to get a winning line.

3x4

Conclusion

4x5

17 winning lines

4x6

24 winning lines

31 winning lines

Size of grid WxL | N.o of squares to get winning line ‘T’ | N.o of winning lines ‘S’ |

4x5 | 4 | 17 |

4x6 | 4 | 24 |

4x7 | 4 | 31 |

4xL | 4 | 7L-18 |

I will now use this rule to predict a 4x10

7x10=70-18=52

If my rule is correct in a 4x10 should contain 52 winning lines.

7x10

52 winning lines

My rules is correct

### Conclusion

From this investigation I have found out a general rule for square grids when the length of the winning line is a variable. I would of found a general rule from the rectangle grids but from the patterns I choose it made it very hard to come up with one. If I were to extend this investigation further I would look more into rectangle patterns and come up with a general, I would also maybe look at some 3D grids and find out some rules from them. If I managed to find out all these rules I would work out a big general rule for the winning lines of a grid with any width, length or height if 3D.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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