Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted.

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Jonathan Ransom

Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted

Introduction

The purpose of this investigation is to explore the answer when the products of opposite corners on number grids are subtracted and to discover a formula, which will give the answer in all cases. I hope to learn some aspects of mathematics that I previously did not know.

The product is when two numbers are multiplied together.

There is one main rule: the product of the top left number and the bottom right number must be subtracted from the product of the top right and bottom left numbers. It cannot be done the other way around.

I plan to achieve my aim by attempting a series of different methods and comparing the answers to see if they are the same. The first method I will do is to draw diagrams, using excel. These will give me a clearer picture of what is involved in this investigation. I will write all the calculations of how I worked out the answer beside each diagram.

There are many different types of number grid, but I will start with square grids. I will also assume that the starting number is one and that the numbers in the grid are consecutive. The numbers in the grid will go horizontally from left to right.

Examples

2 x 2 Grid          

3 x 3 Grid

4 x 4 Grid

No patterns are visible yet so I have done two more diagrams.

5 x 5 Grid

6 x 6 Grid

Using the same method:

(6 x 31) – (1 x 36)

186 – 36 = 150

Answer   = 150

I do not need to draw any more diagrams because it is clear what is being done to get the answer and I now have enough data to make a table of results. This is another method where I can look for any obvious patterns in the data.

Table of Results

I chose to put the second, third and fourth column in so that I could easily see if there were any relationships between the numbers.

There are not many obvious patterns visible from this table and not many clues to a formula. If I draw a graph I may then be able to work out other number grid answers and a pattern might present itself from the graph.

From the graph, I have realised that you cannot find out other answers accurately because the line is not linear. It is a curve so the next answer could be anything. Drawing a line of best fit would not really be accurate enough. Another method is the Difference method. I had not come across this until this investigation. This method is new to me:

I made the first difference column to see if there was a pattern in the increase of the numbers. I then carried on doing difference columns until the number increase was constant.

As the numbers in difference column 3 are all the same, the formula will include a term N in difference column 3. As the numbers are all 6 the formula may include + or – 2. It is two because the difference is 6 and it is in difference column 3. So 6/3 = 2. Also there may be a term in N6.

There are no direct patterns from this table.

I predict that the answer to a 7 x 7 grid will be 252.  

If I take a number grid, 3 x 3 for example, and replace the numbers in the grid with algebraic expressions, then I may be able to work out a formula. I will assume the starting number is always one, for this case.

                                                                       

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The calculation for grid one is:

(3 x 7) – (1 x 9)

21 – 9 = 12

In grid two it will follow the same rule. I shall call the answer A.

A = N (N²- N + 1) – (1 x N²)

I can now use this formula to work out the

Answer with a 7 x 7 grid.

7 (7² - 7 + 1 – (1 x 7²)

7 (49 – 6) – 49

(7 x 43) – 49    

301 – 49 = 252

If I ...

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