Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite
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Introduction
Introduction:
The mathematical investigations that are about to be undertaken are all under one puzzle called Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite Corners.
There are a few basic procedures to follow to achieve a basic understanding of the whole puzzle.
A box consisting of numbers from 1 to 100, a 10 by 10 grid (arranged in a regular pattern) will aid in initiating an understanding for this piece.
Procedure:
 Place borders of four lines in order to enclose numbers arranged in a given grid. The enclosed numbers should form a perfect square.
 Multiply the numbers that are found diagonally opposite and placed in the four corners of the box.
 From the products obtained after multiplying, find the difference between them.
An example is demonstrated on the next page.
Below is a 10 by 10 grid. Here the numbers are arranged in 10 columns.

Middle
 A 6 by 6 grid
 A 7 by 7 grid
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 ruley (n−1)² is going to be tested on the above findings.
In a 3×3 square (from a 6 by 6 grid): y (n−1) ² = difference 6 (3 – 1)² = difference 6 × 2² = difference Therefore difference = 24  In a 5×5 square (from a 7 by 7 grid): y (n−1) ² = difference 7 (5 – 1)² = difference 7 × 4² = difference Therefore difference = 112 
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.
Conclusion
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In an 8×8 square (below) obtained from a 15 by 15 grid, the difference is:
124  125  126  127  128  129  130  131 
139  140  141  142  143  144  145  146 
154  155  156  157  158  159  160  161 
169  170  171  172  173  174  175  176 
184  185  186  187  188  189  190  191 
199  200  201  202  203  204  205  206 
214  215  216  217  218  219  220  221 
229  230  231  232  233  234  235  236 
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
SolutionCheck:
y (n−1) ² = difference
10 (19 – 1)² = difference
10 × 18² = difference
Therefore difference = 3240
Below is a 13 by 13 grid.
1  2  3  4  5  6  7  8  9  10  11  12  13 
14  15  16  17  18  19  20  21  22  23  24  25  26 
27  28  29  30  31  32  33  34  35  36  37  38  39 
40  41  42  43  44  45  46  47  48  49  50  51  52 
53  54  55  56  57  58  59  60  61  62  63  64  65 
66  67  68  69  70  71  72  73  74  75  76  77  78 
79  80  81  82  83  84  85  86  87  88  89  90  91 
92  93  94  95  96  97  98  99  100  101  102  103  104 
105  106  107  108  109  110  111  112  113  114  115  116  117 
118  119  120  121  122  123  124  125  126  127  128  129  130 
131  132  133  134  135  136  137  138  139  140  141  142  143 
144  145  146  147  148  149  150  151  152  153  154  155  156 
157  158  159  160  161  162  163  164  165  166  167  168  169 
In a 10×10 square (below) obtained from a 13 by 13 grid, the difference is:
1  2  3  4  5  6  7  8  9  10 
14  15  16  17  18  19  20  21  22  23 
27  28  29  30  31  32  33  34  35  36 
40  41  42  43  44  45  46  47  48  49 
53  54  55  56  57  58  59  60  61  62 
66  67  68  69  70  71  72  73  74  75 
79  80  81  82  83  84  85  86  87  88 
92  93  94  95  96  97  98  99  100  101 
105  106  107  108  109  110  111  112  113  114 
118  119  120  121  122  123  124  125  126  127 
1× 127 = 127
10 × 118 = 1180
1180 – 127 = difference
Therefore difference = 1053
SolutionCheck:
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 = differenceis the link between size of square, the grid size and the difference.
This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.
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Here's what a teacher thought of this essay
This is a well thought out and demonstrated algebraic investigation. To further develop this a general form for other rectangles within the grid (not just squares) should be investigated.
Marked by teacher Cornelia Bruce 18/04/2013