= 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 the differences using an algebraic formula.
Here is the investigation for a 3 x 3 grid…
The rule explaining this is the same as the rule for the two by two grid. This is because as in the two by two grid, it is the same equation, but for the top right and bottom left hand corners, you multiply the smaller number in the 70’s by an extra 2 in the 50’s. This leaves a gap of 2 x 73, which equals 142. This is, of course, with the top right hand corner number taken away from this total, but because the difference between 71 and 73 is 2 (and not 1), the number in the top hand corner that is taken away is doubled, to be ‘2 x 53’ (106). This total eventually leaves 146 – 106, which following the rule does equal 40.
This explained with a formula is as followed:
(n = number in top left hand corner)
Top left/bottom right hand product: n (n + 22) = n² + 22n
Top right/bottom left hand corner: (n + 2) (n + 20) = n² + 2n + 20n + 40
= n² + 22n + 40
The rule has an explanation and the formula shows this rule is true, because the formula for the two products is the same except for the 40 added on to the product of the top right and bottom left hand corner. As they did for the two by two grid, the examples and the formula have turned out all to be the same, giving the difference of 40.
Here is the investigation for a 4 x 4 grid…
The rule for this again is the same, only with minor alterations. As with the three by three, it is the same equation, but for the top right and bottom left hand corners, you multiply the same number in the bottom row by an extra 3 in the top row. This leaves a gap of ‘3 x 40’, which equals 120. This is, of course, without taking the difference of the numbers in the top row into account, but because the difference between 37 and 40 is 3 and not 2, the number in the top hand corner that is taken away is trebled to be ‘3 x 10’ (30). This total eventually leaves 120 – 30, which following the rule does equal 90.
This explained with a formula is as followed:
(n = number in top left hand corner)
Top left/bottom right hand product: n (n + 33) = n² + 33n
Top right/bottom right hand corner: (n + 3) (n + 30) = n² + 3n + 30n + 90
= n² + 33n + 90
This again clearly shows that because the formula shows the pattern with my examples, that this is a definite rule.
Now, I have noticed a pattern between the 2x2, 3x3 and 4x4 patterns. The pattern concerned between the differences is 10, 40 and 90. If you look at the first digits of these numbers, they read 1, 4 and 9, which are all square numbers. This would make perfect sense, because the areas we are using of number grids are square, and the difference between the products of the 2x2 is 10, and not 1 (meaning all the square numbers would be 10 times larger than they would be).
So, my rule that I am predicting is that the differences go up in square numbers, multiplied by 10. This would mean a 5x5 grid would have differences of 160. Lets see if it does this:
Here is the investigation for a 5x5 grid…
The rule for this again is the same, only with minor alterations. This is because as in the four by four, it is the same equation, but for the top right and bottom left hand corners, you multiply the same number in the bottom row by an extra 4 (in the top row). This leaves a gap of ‘4 x 45’, which equals 180. This is, without taking the difference of the numbers in the top row into account, but because the difference between 41 and 45 is 4 and not 3, the number in the top hand corner that is taken away is quadrupled to be 4 x 5 (20). This total eventually leaves 180 – 20, which following the rule does equal 160.
This explained with a formula is as followed:
(n = number in top left hand corner)
Top left/bottom right hand product: n (n + 44) = n² + 44
Top right/bottom right hand corner: (n + 4) (n + 40) = n² + 4n + 40n + 160
= n² + 44n + 160
As I thought, the difference between the products of a 5x5 is 160. I will now draw up a table of all my results and so I can work out a formula for all the differences:
TABLE OF RESULTS:
Here are the results for the differences, but with a formula for a square, which could have any number of rows and columns. This would prove my theory (the square number theory in italics) correct if I can come up with a formula for the results.
The formula for t can be used for any number, and the difference will always come out correct no matter what number you replace it with. The formula will also tell us that the square number theory is correct, as it fits all the differences between 2 x 2 and 5 x 5 (in the table).
This next table represents the two formulae I found for each grid. From these different formulae, I will try and determine a pattern between the numbers to create an overall formula (as I did in my first table) to be used for any number of rows and columns of a grid.
These rather complicated formulae in the t x t row represent what any number of rows/columns in a grid would produce in the same way as in 2 x 2 etc. These will not necessarily help in determining patterns, but it does help if we want to determine the rule again using higher number grids quicker.
INVESTIGATING USING DIFFERENT SHAPES:
I will now investigate these number patterns in the same way, only using different shapes other than squares:
I will use the different shaded shapes to work out rules between the numbers.
As you can see, the pattern between the numbers is that the middle (white) number squared always comes to one larger than the two numbers either side of it, and four larger than the numbers on the outskirts of the grid.
This is because, using this example, if you were to multiply the 5 by 4, and not 3, then the difference between the numbers, and the middle number squared would be 4. But, because you’re multiplying by 3, you have to subtract 3 from this number, to account for the changes in the sum. This leaves the sum 4 – 3, which gives us 1, which is what I found the difference to be.
This works in the same way with the numbers on the outskirts. If you were to multiply 6 by 4, and not 2, then you would get a difference of 8.
Using a formula, I can explain the rule too:
(n = middle (white) number)
Middle no. squared: n² = n²
Numbers on outside of white number: (n – 1)(n + 1) = n² + n – n –1
= n² - 1
Numbers on outskirts of grid: (n – 2)(n + 2) = n² +2n – 2n – 4
= n² - 4
Another grid we could use is a vertical column grid to try and determine a pattern between the horizontal and vertical singled rowed/columned grid, by again using the method of multiplying the numbers, which are shaded in the same colours.
This is because, using this example, if you were to multiply 37 and 27 together, rather than 37 and 17 together, you would find a difference of 370, the number that remains the same in both of these multiplications, multiplied by 10. But because you multiply 37 by 17, you must take away the amount in the middle square in the grid, multiplied by 10, in this case giving 270 (27 x 10). This leaves the calculation of 370 - 270, which equals 100. It works in exactly the same way for the two numbers at the top and bottom as well. This is why the rule works.
This shows that the gaps end up to be exactly the same as with the horizontal grids, except the vertical differences are 100 times larger than the corresponding horizontal value, so, for example, instead of there being a gap of 4, there is a gap of 400, etc.
Using the ‘n’ formula, I can explain the rule too…
(n = middle/white number)
Middle no. squared: n² = n²
Numbers on outside of white number: (n – 10)(n + 10) = n² + n – n –100
= n² - 100
Numbers on outside of grid: (n – 20)(n + 20) = n² +2n – 2n – 4
= n² - 400
TABLE OF RESULTS FOR SINGULAR ROW/COLUMN GRIDS:
n = middle number/white number
t = place in grid when middle number equals 0, numbers on outside of that equals 1 etc.
This shows that the patterns are very similar, with the only difference being what is taken away from n². In the case of the horizontal, I presume that t² is taken away, because with the original investigation, I found a square number pattern in the grids differences. For the vertical, t² is multiplied by 100, which accounts for the larger differences between each number in the grid.
INVESTIGATION OF A GRID IN A DIFFERENT SIZED NUMBER SQUARE:
You have already seen what happens when we use a 10x10 grid, but what would happen if we used a different sized grid? Above is a 9x9 grid (numbered from 1 to 81), with the same four-number grid highlighted as the first example I gave on page one.
This shows that the two products have a difference of 9. However, we need to use more examples to be sure this is always what happens, even though I know it will, as all the results were the same with the 10x10 grid:
This shows that when you multiply a 2x2 grid within a 9x9 number square, the difference between the products always comes to 9. Because of this, I am predicting that the differences between the products of a grid within an 8x8 square will be 8.
This is because in the 10x10, the difference of the products is 10, in a 9x9, the difference is 9, and so the pattern should continue, so that an 8x8 should make a difference of 8, and so on…
This shows that the two products have a difference of 8. However, we need to use more examples to be sure this is always what happens, even though I know it will, as all the results were the same with the 10x10 grid AND with the 9x9 grid:
This did follow the pattern I suggested it would do earlier on. This means that I can now find the difference for the y x y area square (not ‘n x n’ or ‘t x t’ because n corresponds to the value in the top right corner of the grid, and t corresponds to where a value is).
This also follows the other pattern that I found on page two – that the average difference between the numbers that are being multiplied is the same as the overall difference. Here is a table for this:
Average for y: ( (y + 1) + (y – 1) ) ÷ 2 = 2y ÷ 2 = y
This table is for the product differences:
y = length of any side of the square grid:
This rule is very simple. The value of y is the number of squares on any one side of the grid that is being used. For example, on a 10x10, the length of all sides is 10 squares, so y will equal 10.
