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• Level: GCSE
• Subject: Maths
• Word count: 2865

# 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:

1. Place borders of four lines in order to enclose numbers arranged in a given grid. The enclosed numbers should form a perfect square.
1.   Multiply the numbers that are found diagonally opposite and placed in the four corners of the box.
1. 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.

 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

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 = difference6 (3 – 1)² = difference6 × 2² = difference Therefore difference = 24 In a 5×5 square (from a 7 by 7 grid):y (n−1) ² = difference7 (5 – 1)² = difference7 × 4² = differenceTherefore 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

1. ### 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

4 star(s)

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

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