# You are given a 9x9 number grid with a 2x2 square on it. Investigate when pairs of diagonal corners are multiplied and subtracted from each other.

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Introduction

Douglas Carter

Mathematics Coursework

Problem:

You are given a 9x9 number grid with a 2x2 square on it. Investigate when pairs of diagonal corners are multiplied and subtracted from each other. Come up with an algebraic formula that gives the difference for any size square/rectangle on any size grid.

Method:

Starting with the problem given, a 9x9 number grid with a 2x2 square on it, I gave each number in the square its equivalent in algebra.

If the first number is a the number grid would be as follows…

This can be used to make a formula that gives the difference (Dif) when diagonals are multiplied and subtracted from each other, for a 2x2 square on a 9x9 grid, where 9 represents the width of the grid and a

Middle

I found similarities between the differences when looking at them written in different ways. For a 9x9 grid and different sized squares I got the following table of results:

This shows that the difference is relative to the grid width but increases as the size of the square increases in similar steps but faster and with more effect.

Using the formula I found that there is a pattern in the increase in difference of rectangles, such as 2x3, 3x4, etc… I have tabulated the results.

From looking at what I have tried I saw that in order to get a universal formula for the difference of any size square/rectangle on any size grid, both the grid width, width, and

Conclusion

By inserting these variables into my basic

formula above I developed the following

universal formula:

Using this formula I tested a square and rectangle on a 9x9 grid.

2x2 square on a 9x9 grid:

Dif = l(py-y-p+1)²

= lpy-ly-lp+l

= 2x2x9-2x9-2x9p+9

= 9

2x3 rectangle on a 9x9 grid:

Dif = l(py-y-p+1)²

= lpy-ly-lp+l

= 2x3x9-2x9-3x9p+9

= 18

The formula is…

(width of grid) x (area of square/rectangle) - (width of square/rectangle) - (height of square/rectangle) + (width of grid)

This gives the difference for any size square/rectangle on any size grid.

I tested the formula on a large random sized grid with a random sized rectangle.

3x9 rectangle on a 49x49 grid:

Dif = l(py-y-p+1)²

= lpy-ly-lp+l

= 3x9x49-3x49-9x49p+49

= 784

Conclusion:

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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