Number Grids.

This grid shows the numbers 1 to 100. Two of the corners diagonally opposite each other on the grid can be multiplied together to make a number. The other to corners can be multiplied to give another number. There should be a difference between these two figures. The differences between the two numbers are what I want to look at. I can compare the difference between grids of different sizes.

I plan to start off with a small 2x2 grid.

I am apple to work out the difference using the standard multiplication method.

1x12=12

2x11=22

I then have to take away the two answers.

22-12= 10

Therefore the difference for my first example is 10. I have noticed that there is three sums that need to be figured out and as these numbers get larger the sum will become harder.

I will need a new way to get around this problem so I try to introduce algebra.

This grid (above) represents a 2x2 grid; it can be used with the correct formula to work out any difference for a 2x2 grid. I then tried the formula to fin the two numbers multiplied together.

N?(n+11), but could this be improved? I tried tuning the formula into one formula.

(n+1)(n+10) when I expanded the brackets, it reveals the difference for any 2x2 grid.

(n+1)(n+10=n?+11n+10 the formula seems to work and shows that the answer is +10. But will the formula continue to work as the sizes or the grids become larger?

I decided to try a 3x3 grid.

This time I wont start from 1, this should not be a problem as the formula is meant o work with any sized grid.

I will use the formula first

to make sure it works.

According to the formula +40 is the correct answer.
(n+2)(n+20)=n?+20n+40

So to make sure that 40 is the correct answer I will check using the standard method.

31x53=1643

33x51=1683

1683-1643= 40

It is confirmed and the formula does work.

I am now going to try a 4x4 grid using the same principles as the other two grids I have tried.

The answer for the grid using the basic method is-

7x40=280

10x37=370

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