A formula for a 2x2 square in a 10x10 large number grid would be:
a+1 = (a x a+11) + 10
I have a prediction that the difference between the products of the top left and bottom right and the products of the top right and bottom left is 10, because it is a 10x10 grid. I will see if this is true by changing the 2x2 number square, to a 3x3 number square to see if the difference is still 10 in a 10x10 large number grid. I will work in the same way, using all the corner numbers in the same way as before.
42 x 64 = 2688
44 x 62 = 2728
67 x 89 = 5963
69 x 87 = 6003
4 x 26 = 104
6 x 24 = 144
I have noticed that the difference between the two products is not 10 therefore my prediction was incorrect. The difference between the two products was 40, an increase of 30. Also one of the results was not an even number. Now that I know that the difference in a 2 x 2 number square in a 10 x 10 large number grid is not 10 because it was it was in a 10 x 10 large number square, I will try and investigate further into the 10 x 10 large number grid, and try to find another pattern. I will now investigate 4 x 4 number squares in a 10 x 10 large number grid.
32 x 65 = 2080
35 x 62 = 2170
I have noticed that the difference between these two values is 90 and that they are multiples of 10.
6 x 39 = 234
9 x 36 = 324
I have noticed that the difference between these two values is again 90.
66 x 99 = 6534
69 x 96 = 6624
Again, the difference is 90. At this point in my investigation I have noticed that the differences are going up on order of the odd numbers, multiplied by 10. I will show this in a table.
I will now investigate 5 x 5 number squares in a large 10 x 10 number grid, and try to see if there are any more patterns. I predict that the difference will be 160, as this will be 70 (7 x 10) more than the last difference
I will investigate the 5 x 5 grid like I have all the other grids, finding the products of the top left and bottom right numbers, and the top right and bottom left numbers.
1 x 45 = 45
5 x 41 = 205
The difference between these two values is 160, 70 more than last time.
51 x 95 = 4845
55 x 91 = 5005
The difference between these two values is again 160.
The difference is again 160. At this point in my investigation I have noticed another pattern, which I shall now write down. Also my previous prediction was correct.
I have noticed that the differences are going up in order of 10 multiplied by square numbers. For example, in the 2 x 2 number square, the difference is 10, which is 10 multiplied by 1 squared. In the 3 x 3 number square, the difference was 40, 10 multiplied by 2 squared. I will now show these findings in a table to show them more clearly. I will then try and put my findings into a formula, then test it in 6 x 6 number squares to see if it would work for all sizes of number squares in a 10 x 10 number grid.
I have also noticed that the number squared and multiplied by 10 is always 1 less than the size of the grid. For example, 1 is 1 less than 2, 2 is 1 less than 3 and so on.
I will now investigate 6 x 6 number squares, to see if this pattern carries on. I predict that the difference will be 250, 10 multiplied by 5 squared. If this is true, so will be my prediction about the difference between the differences going up in order of odd numbers multiplied by 10.
1 x 56 = 56
6 x 51 = 306
The difference is, as I predicted, 250, which is 10 multiplied by 5 squared, I will see if this is the same for 2 more sets of 6 x 6 number squares.
5 x 60 = 300
10 x 55 = 550
So far my prediction has been correct. I will do one mores et just to be 100% certain.
42 x 97 = 4074
47 x 92 = 4324
I am now certain that my prediction was correct, and that all number squares within a 10 x 10 large number grid will follow this format. I will now try and put this into a formula.
The rule is that the grid size (n), take away one, squared, multiplied by 10 equals the difference (d) between the products of top left and bottom right, and top right and bottom left numbers. So –
I am happy with this formula and confident that it will work for all number squares in a 10 x 10 large number grid. Also, I think that it is 10 multiplied by (n – 1) squared, because it is in a 10 x 10 large number grid. I predict that in a 9 x 9 number grid, the formula will be –
I will now investigate this by finding the products of the top left and bottom right numbers, and top right and bottom left numbers in, firstly, a 2 x 2 number square in a 9 x 9 large number grid.
12 x 22 = 264 The difference between these two
13 x 21 = 273 products is 9. In the 10 x 10 grid the first difference was 10, now in the 9 x
9 grid the first difference is 9.
8 x 18 = 144 The difference is again 9.
9 x 17 = 153
59 x 69 = 4071 Again, the difference is 9.
60 x 68 = 4080
So far my prediction is correct. 9 = 9 multiplied by 1 squared. For my prediction to stay correct, the next difference in the 3 x 3 number squares should be 36, 9 multiplied by 2 squared.
I will now investigate 3 x 3 number squares to see if my prediction is correct and see if there are any more patterns emerging.
58 x 78 = 4524
60 x 76 = 4560
The difference between these two products is, as I predicted, 36. I will see if this is true for two more sets of 3 x 3 number squares.
6 x 26 = 156
8 x 24 = 192
The difference is again 36.
46 x 66 = 3036
48 x 64 = 3072
The difference is again 36, my prediction was correct.
Now that my predictions have so far been correct, I will now only do two examples for each set of number squares, to save a lot of time. I will now investigate 4 x 4 number squares, remembering that if my prediction is to be correct, the next difference will have to be 81, 9 multiplied by 3 squared.
32 x 62 = 1984
35 x 59 = 2065
The difference between these two products is 81, as I predicted. I am already beginning to for a formula in my head for 9 x 9 large number grids.
10 x 40 = 400
13 x 37 = 481
The difference between the two products is again 81. At this point in my investigation, I have made another finding.
The differences in a 9 x 9 large number square are going up according to the numbers in the 3 times table squared. For example, in a 2 x 2 number square the difference was 9, which is 3 squared. In the 3 x 3 number square the difference was 36, which is 6 squared. When I have done the 5 x 5 number square, I will put all of my results into a table to show them more clearly, and to make it easier to find a formula. For my prediction to continue to be correct, the next difference will have to be 144, 9 multiplied by 4 squared. 144 is also 12 squared which means that my other theory is also correct.
39 x 79 = 3081
43 x 75 = 3225
The difference between these two products is 144. I will make sure that this is true by doing one more example of the 5 x 5 number square.
3 x 43 = 129
7 x 39 = 273
The difference is again 144, I am now certain that my original prediction is correct. I will now put all of my findings for the 9 x 9 large number grid into a table.
The formula for a 9 x 9 large number grid, where n = size of number square (e.g. 2 = 2 x 2) is –
From this I think that I can safely say that I have come up with a general formula, for any size large number grids, where n = size of number squares, and k = size of large number grid (e.g. 10 = 10 x 10)
I do not have time to complete any other examples for any other size large number grids, so I will make an assumption that this formula will definitely work for any other size large number grids. E.g. in a 20 x 20 large number grid, the formula for the differences in the different number squares would be
Conclusion
In this investigation I have managed to a) find a formula for the differences in a 10 x 10 large number grid, b) find a formula for differences in a 9 x 9 large number grid, c) found a general formula for any size large number grid and d) noticed other patterns in the numbers in the large number grids that I investigated. Had I not run out of time, I could have worked with different shape large number grids or number squares, and seen what patterns there were there. I conclude that the formula for finding differences between products of top left + bottom right, and top right and bottom left numbers in any size large number grids, when n = size of number square and k = size of large number grid, is -
Kieran Paul 10E
Mrs Johnson
