# investigate how many winning lines there are in a 7x9 grid.

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Introduction

Connect 4

I am going to investigate how many winning lines there are in a 7x9 grid.

The following diagrams show how many winning lines there are in the grid, with each coloured line representing 1 winning line.

There are 24 winning horizontal winning lines as shown in the diagram opposite.

There are 21 vertical winning lines in a 7x6 grid as shown opposite.

There are 24 diagonal winning lines in a 7x6 grid. As diagonal lines go both ways I multiplied the number of lines shown by 2 in order to achieve the correct result. I felt this was the easiest method to use as to draw 2 sets of lines on one grid would be very confusing, and to draw 2 grids would have been very time consuming.

I am now going to investigate if there are patterns of winning lines within grids. In order to work systematically I am going to begin my investigation with square grids as this will involve using only one variable, and I will gradually complicate matters when moving onto rectangles and the use of 2 variables.

Square Grids

I am now going to investigate winning lines in square grids.

Middle

Diagonal Rule

If we refer back to our winning lines table, we once again find that the first difference is not a constant. I also know that whatever rule I find will be multiplied by 2 to account for both diagonal directions.

Rule: 2(n-3)²

I predict that in grid size 7x7 the number of diagonal winning lines will be 32.

In order to check my prediction I will have to draw out the 7x7 grid.

The number of winning lines shown opposite is 16, however when multiplied by 2 (thus accounting for the fact that diagonal lines can go both ways) we reach the predicted answer of 32. So, rule is correct.

Justifying the Rule

2(n-3)²

No winning lines can be put in these shaded areas for the simple fact that it is 3 squares wide and a line lasting 4 squares cannot fit in there. So, if you multiply n-3 by n-3 you get the number of winning line starting squares, therefore the number of winning lines.

Rectangular Grids

I am now going to extend my investigation by exploring the possibilities of rectangles.

I have put my results in a table to enable me to spot patterns.

n = horizontal grid size m = vertical grid size

Winning Lines | ||||

m | n | Horizontal | Vertical | Diagonal |

6 | 5 | 12 | 15 | 12 |

7 | 5 | 14 | 20 | 16 |

8 | 5 | 16 | 25 | 20 |

Conclusion

Connect Any

I am now going to investigate what these rules would be if they were connect any, not just connect 4.

I predict that in the place of the ‘-3’, in all of the rules, there will be c(connect number) -1. This is because each winning line is composed of one starting square followed on by the number of squares remaining in the connect number.

So, the rules for connect any in rectangles would be:

Horizontal winning lines : m(n-(c-1))

Vertical winning lines : n(m-(c-1))

Diagonal winning lines : 2(m-(c-1))(n-(c-1))

The rule for total winning lines, connect any in a rectangle would be:

(m-(c-1))n + (n-(c-1))m + 2(n-(c-1))(m-(c-1))

I will now check the total rule.

I predict that for connect 3, in a 4x3 grid, the number of winning lines would be 14 altogether.

I will now draw out the grids to check my prediction.

There are 14 winning lines altogether, therefore my rules are correct.

Justifying ‘c-1’

As already explained each winning line has one starting square, so you subtract this to get the number of squares which need to be taken away in the rules. To get the total rule I added all of the other rules together.

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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