In the 4 × 4 square:
34 × 67 = 2278
37 × 64 = 2368
The difference between the products is 90.
In a 5×5 square:
25 × 69 = 1725
29 × 65 = 1885
The difference between the products is 160.
In a 6×6 square:
2 × 57 = 114
7 × 52 = 364
The difference between the products is 250
In a 7×7 square:
14 × 80 = 1120
20 × 74 = 1480
The difference between the products is 360.
In an 8×8 square:
1 × 78 = 78
8 × 71 = 568
The difference between the products is 490.
In a 9×9 square:
2 × 100 = 200
10 × 92 = 920
The difference between the products is 720.
Unit 1- Investigation and Generalisation:
x represents the first number in the box
y represents the number of numbers in a column
n represents size of square (n × n)
Below is a y by y grid. Here the numbers are arranged in y columns. (An example of a square used in opposite corners):
x [x+(n−1)y+(n−1)] is subtracted from [ x+(n−1) ] [x+(n−1)y ]
Prove that:
Difference = y(n−1) ²
Where:
y = numbers arranged in y Columns and n = size of square
Since:
x represents the first number in the box
x + (n −1) represents the last number of the first row
x + (n −1)y represents the first number of the last row
x+(n−1)y+(n−1) represents the last number of the last row
And:
The formula for finding the common differences:
[ (last number of first row) × (first number of last row) ] − [ (first number of the square) × (last number of last row)]
So when the values above are inputted into the formula, a general formula for the common difference in terms of y, x and n can be found.
Common difference
=( [x + (n −1)] × [x + (n −1)y] ) − ( [x] × [x+(n−1)y+(n−1)] )
= (x² + x(n −1)y + x(n −1) + (n−1)² y) − (x² + x(n −1)y + x(n−1))
= x² + x(n −1)y + x(n −1) + (n−1)² y − x² − x(n −1)y − x(n−1)
= x² − x² + x(n −1)y − x(n −1)y + x(n −1) − x(n −1) + (n−1)²y
= (n−1)²y
The general formula for finding the common difference is y (n−1) ²
Where:
y = numbers arranged in y Columns or in a y grid and n = size of square.
Unit 2- Testing the rule, y (n−1) ², on larger squares:
Note:
The tests above are from the 10 by 10 grid found on page 2.
Solutions check can be found on Pages 6 and 7.
Unit 3- Proving the rule:
- Above is a 3×3 square with 24 and 46; and 26 and 44, diagonally opposite and also found in the corners of the box hence the name opposite corners.
24 × 46 = 1104
26 × 44 = 1144
-
The difference between the products above is 40.
y (n−1) ² = difference
10 (3 – 1)² = difference
10 × 2² = difference
-
Therefore difference in a 3×3 square = 40
and y (n−1) ² is truly the rule that links the size of square as well as the difference.
Unit 4- Trying the rule, y (n−1)² with different column arrangements and sizes of squares:
For example, in the 3×3 square below obtained from a 6 by 6 grid, the difference is:
8×22=176
10×20=200
200 – 176=24
In another example of a 5×5 square (below) obtained from a 7 by 7 grid, the difference is:
9×41=369
13×37=481
481 – 369=112
There is an obvious difference in the differences of products of the multiplied values, in the opposite corners from the above grids as compared to the 10 by 10 grid. Now the rule y (n−1)² is going to be tested on the above findings.
Unit 5 - Extra Tasks:
In this unit the difference for the following squares are going to be determined. For example:
(i) 3×3 from the 20 by 20 grid.
(ii) 8×8 from the 15 by 15 grid.
The difference obtained from square size n×n is 3240, from a 10 by 10 grid; the value of n is to be found.
Also for a grid arranged in 13 columns, the difference for a square size, 10×10, is to be considered.
Solutions:
Below is a 20 by 20 grid.
In a 3×3 square (below) obtained from a 20 by 20 grid, the difference is:
230 × 272 = 62560
232 × 270 = 62640
62640 – 62560 = difference
Therefore difference = 80
Solution check:
y (n−1) ² = difference
20 (3 – 1)² = difference
20 × 2² = difference
Therefore difference = 80
(ii) Below is a 15 by 15 grid
In an 8×8 square (below) obtained from a 15 by 15 grid, the difference is:
From the previous page,
124 × 236 = 29264
131 × 229 = 29999
29999 - 29264 = difference
Therefore difference = 735
Solution Check:
y (n−1) ² = difference
15 (8 – 1)² = difference
15 × 7² = difference
Therefore difference = 735
Solution:
3240 = 10 (n – 1) ²
3240÷10 = (n – 1) ²
19 = n
Solution Check:
y (n−1) ² = difference
10 (19 – 1)² = difference
10 × 18² = difference
Therefore difference = 3240
Below is a 13 by 13 grid.
In a 10×10 square (below) obtained from a 13 by 13 grid, the difference is:
1× 127 = 127
10 × 118 = 1180
1180 – 127 = difference
Therefore difference = 1053
Solution Check:
y (n−1) ² = difference
13 (10 – 1)² = difference
13 × 9² = difference
Therefore difference = 1053
Conclusion
Therefore the main aim of this coursework has been dealt with. The formula y (n−1) ² = difference is the link between size of square, the grid size and the difference.
