# Maths Opposite corners

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Introduction

Maths Coursework

Opposite Corners

My task is to investigate a number grid. If you take a 2x3 rectangle and place it on a 10x10 number grid the diagonal difference of the numbers inside is 20. I want to first investigate whether the diagonal difference is always 20, no matter where the rectangle is, and prove this. I will then investigate further by changing the size of the number grid and of the rectangle.

First I investigated two rectangles to see if the diagonal difference of both was 20.

Both of these diagonal differences were 20.

I decided to make the rectangle larger by one square wider and one deeper to see what happens.

Again I investigated two rectangles to see if the diagonal difference is the same for both.

Both came out as 60 so the formula is:

x²+23x-(x²+60+23x)=60

So far the diagonal differences are both factors of twenty, meaning the first digit is an even number (When there are two digits) and the number ends in zero.

Middle

I decided to make a formula for the diagonal difference of my first box by calling the number in the top left corner ‘x’:

If I call the number in the top left hand corner ‘x’ then I can work out the rest of the numbers. The top right hand number has to be ‘x+2’. This is because in a 2X3 box this number will always be 2 more than the one on the top left. The bottom left will be ‘x+10’ because every step down in a 10X10 grid adds 10 to the number. The bottom right must be ‘x+12’ because it is two higher than ‘x+10’. I multiply ‘x’ with ‘x+12’ just like before and do the same with ‘x+2’ and ‘x+10’. The products of these numbers are:

x(x+12)=x²+12x

(x+2)(x+10)=x²+20+12x

The difference between these numbers is:

x²+12x-(x²+20+12x)=20

This shows that the answer is always 20.

Conclusion

(x+m-1)(x+pn-p)-x(x+pn+m-p-1)= x²+xpn-xp+mpn+mx-mp-x-pn+p-x²-xpn-xm+xp+x

= pnm-pn-pm+p

My final formula for the diagonal difference of a rectangle/square of any size on a grid of any size is pnm-pn-pm+p. This proves my prediction that ‘p’ will replace the number ten in my earlier formula. I’ll prove this by using the formula on my first rectangle:

m=3, n=2, p=10. 60-20-30+10=20 The diagonal difference is 20.

I will now try my formula with a 4X5 rectangle on a 20X20 grid to prove that it works with different sized grids and boxes.

Diagonal difference= 1X84=84

= 4X81=324

=324-84

= 240

Diagonal difference= pnm-pn-pm+p

= 20X5X4-20X5-20X4+20

= 240

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