• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month   # 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

Extracts from this document...

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

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

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

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.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month ## 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

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Number Stairs, Grids and Sequences essays

1.  ## opposite corners

5 star(s)

So I will now use algebra to see if I am correct. n n+2 n+20 n+22 (n+2)(n+20) - n(n+22) (n2+20n+2n+40) - (n2+22n) 40 This proves that the answer is always 40 when a 3 x 3 square is placed on a 10 x 10 grid.

2.  ## Opposite Corners Maths investigation.

2 x 4 1. 77 x 84 = 6468 74 x 87 = 6438 30 2 x 5 1. 78 x 84 = 6552 74 x 88 = 6512 40 It appears that every time I increase the width by one, the difference increases by ten.

1. ## Number Stairs

grid it is on, and that it is only the end part of the formula (the + something bit) that changes on each size grid. The co-efficient of x is always a triangular number as the stair shapes make a triangle shape (Fig.

2. ## Number stairs

+ (x+10) + (x+11) + (x+1) + (x+2) In conclusion the algebra formula to find the total inside the 3-step stairs for a 10 by 10 Number Grid is: T= 6x + 44 I am going to test my formula for this portion of a 3-step stair: T=6x + 44 T= (6 x 35)

1. ## Number Stairs

| 42 11 | 2 | 48 From these results I can determine the following general stair shape: n+2g n+g n+g+1 n n+1 n+2 From which, in turn, I can determine the following formula: S= 6n +4g +4 This is because there are 6 ns in my shape, 4 gs in my shape, and a 4 in my shape.

2. ## Number Stairs.

6 and every time you move the stair shape one square to the left you decrease the stair total by 6. Now we are going to see what happens when we move the stair shape up one square on the grid and if there is a pattern.

1. ## Number Grids Investigation Coursework

15 23 24 25 33 34 35 (top right x bottom left) - (top left x bottom right) = 15 x 33 - 13 x 35 = 495 - 455 = 40 So in this square, the difference between the products of the opposite corners equals 40.

2. ## For other 3-step stairs, investigate the relationship between the stair total and the position ...

and adding the values we get -350 12 x 12 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 • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 