Number Grids

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John Saunders (R)

Number grids

This investigation involves a number grid like this one below (10x10):

The purpose of this investigation is to use this number square, to try and find a definite rule between the four corners of a square within it.

To begin with, I will look at the four shaded squares in the 10x10 in more detail, by finding the products of its opposite numbers (in this case being; ‘12 x 23’ & ‘13 x 22’).

According to these results, the difference between the two products is 10, with the product of the top right and the bottom left hand corners being the higher result between the two. I will now use five similar grids, with other numbers, to see whether this is a rule. After all, this could be just be a coincidence:

In every grid I did, the difference between the products always turns out to be the same. The difference is always 10.

The explanation for this is fairly complicated:

The reason is that for the top right and bottom left numbers, you always multiply the top right number by a number one larger than in the top left box. Without taking the different numbers on the top row into account, this would mean that the difference between the two products would now be the number in the bottom right hand corner. However, because the top left hand square is one less than the top right hand corner, you have to take away the number inside that square away from the higher product. This in theory always gives you a difference of 10. Here is an example:

Because you are multiplying 6 x 15 and 5 x 16 you are immediately making a large difference between the two products, as you are multiplying by different numbers. When you multiply by 6, you would be making a difference of 16, IF you were multiplying the 5 by the same number, but you aren’t. You are actually making a difference of 16, but minus the amount of the number in the top right hand corner – because you are finding the product with 15 and 6, and not 16 and 6. This therefore gives a difference of ‘1 x 6’ (between ‘15 x 6’ and ‘16 x 6’), which will always be the number in the top right hand corner, in this case 6.

After all the explanation, we get down to the sum 16 – 6, which gives an answer of 10. The rule always works like this, only with different products, but nevertheless you get the same difference.

This also fits in with another pattern. The average differences between the numbers that are multiplied always equal the difference between the products. In the example above, the difference between ‘5 and 16’ is 11 (16-5), and the difference between ‘6 and 15’ is 9 (15-6). The mean of these numbers is 10 ( (9 + 11) ÷ 2 ).

This rule can be explained better using algebraic formulae for the two products:

(n = number in the top left hand corner)

Top left/bottom right product:                n(n + 11)                = n² + 11n

Top right/bottom left product:                (n + 1)(n + 10)          = n² + n + 10n + 10

                                                                         = n² + 11n + 10

                                                                        (n² + 11n + 10) – (n² + 11n)

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                                                                        = 10

The products in the formula are exactly the same, except the formula for the top right/bottom left hand corner has an extra 10 added onto it. This formula does show what happens with the grids before, so therefore I can say this is a definite rule.

I am now going to investigate further with these types of rules, to see if there is a common link between different corners and the area of the grids we use. I will then place these results in a table before trying to work out a general rule between all ...

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