Opposite Corners Investigation
Length: 3505 words (10 doublespaced pages) Rating: Red (FREE)                                  
Opposite Corners Investigation
This investigation is about finding the difference between the products of the opposite corner numbers in a number square. There are three variables which I can change whilst doing my investigation, they are the size of the grid, the shape of the grid and the numbers inside the grid. I am going to start by looking only at number squares with consecutive numbers Consecutive Numbers =================== 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 x 13 = 52 1 x 16 = 16 Difference = 36 I am now only going to type the corner numbers, as the other numbers are irrelevant 0 3 12 15 0 x 15 = 0 3 x 12 = 36 Difference = 36  2 1 10 13 1 x 10 = 10 2 x 13 = 26 Difference = 36 The difference seems to be the same, for these 3 the answer is 36 but this isn't proof. Let X stand for the start number which can be any real number. X X + 3 X + 12 X + 15 (X + 3) (X + 12) = X2 + 3X + 12X + 36 = X2 + 15X + 36 X (X + 15) = X2 + 15X Difference = 36  So, the difference between the products of the opposite corner numbers in a 4x4 number square is 36. What about a 3x3 number square?  X  X + 2  X + 6  X + 8  (X + 2) (X + 6) = X2 + 2X + 6X +12 = X2 + 8X +12 X (X + 8) = X2 + 8X Difference = 12 So, the difference between the products of the opposite corner numbers in a 3x3 number square is 10. What about Other squares? X This investigation does not work with a square size of 1x1, as the square does not have four corners. X X + 1 X + 2 X + 3 (X + 1) (X + 2) = X2 + X + 2X +2 = X2 + 3X +2 X (X + 3) = X2 + 3X 2 X X + 4 X + 20 X + 24 (X + 4) (X + 20) = X2 + 4X + 20X + 80 = X2 + 24X + 80 X (X + 24) = X2 + 24X = 80 X X + 5 X + 30 X + 35 (X + 5) (X + 30) = X2 + 5X + 30X + 150 = X2 + 35X + 150 X (X + 35) = X2 + 35X 150 Square size Difference Factors 2x2 2 2x1 3x3 12 3x4 4x4 36 4x9 5x5 80 5x16 6x6 150 6x25 10x10 ? ? Text Box: N is the length of the square NxN N(N  1)2 Predict + check Looking at the patterns of numbers from my tables of results it appears for a grid size of NxN the difference is N(N  1)2. I predict that for a 10x10 grid the difference will be 10 x 92 = 10 x 81 = 810. I will check by drawing. X X+9 X+90 X+99 (X + 9) (X + 90) = X2 + 9X + 90X + 810 = X2 + 99X + 810 X(X + 99) = X2 + 99X 810 The check shows that the predicted formula is correct. But this is not proof. These are the equations that I have figured out for each corner. [IMAGE]X X+(N1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X+N(N1) X+N(N1)+(N1) (X + (N1)) (X + N(N1)) = X2 + X(N1) + X(N(N1)) + N(N1)(N1) X (X + N(N1) + N1)) = X2 + X(N1) + X(N(N1)) N(N1)(N1) =N(N1)2 This formula is the same as before so I have proved my prediction. Grid within a grid ================== The formula that I have figured out works for any sized square with a consecutive number grid but what about a grid within a grid? 1 2 3 4 5 6 7 8 9 10 P is the length of the outer grid. R is the length of the inner grid. X is the start number. 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 I'm now going to see whether the corners have any algebraic relation to each other. [IMAGE] [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'll write this out in equation form 35 X + (R  1) [IMAGE]36 [IMAGE] X + P (R1) [IMAGE]45 X + (R1) + (P(R1)) [IMAGE]46 [IMAGE] 45 is the starting number + 10 so I need to find a way to add 10 X X + (R  1) 68 is the starting number + 2 so I need to find a way to add 2. [IMAGE][IMAGE][IMAGE]66 68 [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'll write this out in equation form [IMAGE][IMAGE]86 88 X + (R1) + (P(R1)) [IMAGE] X + P (R1) [IMAGE] [IMAGE] X + (R  1) X [IMAGE]17 20 This number is the starting number + the top right corner + the bottom left corner so I'll write this out in equation form 47 is the starting number + 30 so I need to find a way to add 30 [IMAGE][IMAGE] X + (R1) + (P(R1)) [IMAGE]47 50 [IMAGE] X + P (R1) The algebraic terms for the corners seems to be the same for any outer square so I'll now put these terms into the square and find the difference in algebraic terms. [IMAGE] X X+(R1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE] [IMAGE]X+P(R1) X+P(R1)+(R1) (X + (R1)) (X + P(R1)) = X2 + XP (R1) + X(R1) + P(R1)(R1) = X2 + XP(R  1) + X(R  1) + P(R1)2 X (X + P(R1) + (R 1 ) = X2 + XP(R  1) + X(R  1) = P(R1)2 Predict + check Looking at the results I believe that inside a square PxP the difference of the products of opposite numbers in a inner square sized RxR = P (R1)2. I predict that, for a 6x6 square inside a 10x10 square that the difference will be 10 x (61)2 = 10 x 25 = 250. I will check by drawing. 14 19 64 69 19 x 64 = 1216 14 x 69 = 966 = 250 My equation is right. I Have noticed that the height of the outer square is irrelevant in the formula so this formula will also work for squares inside rectangles. Rectangles I have worked out the formula in number squares, but what about number rectangles? 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 10 x 31 = 310 N is the width M is the height X is the starting number 1 x 40 = 40 = 270 I'm now going to see whether the corners have any algebraic relation to each other. [IMAGE] [IMAGE] 1 10 [IMAGE][IMAGE]21 30 X + (N1) + (M1)N 21 is the starting number + 20 so I need to find a way to add twenty. [IMAGE][IMAGE] X + (M1)N [IMAGE] [IMAGE] 1 10 [IMAGE][IMAGE]41 50 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] [IMAGE] [IMAGE] 41 is the starting number + 40 so I need to find a way to add forty. X + (M1)N X X+(N1) These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X+(M1)N X+(N1) + (M1)N These are the equations that I have figured out for each corner. (X + (N1)) ((X + (M1)N) =X2 + XN(M1) + X(N1) + N(N1)(M1) X(X + (N  1) + (M1)N)= X2+ XN(M1) + X(N1) Difference = N(N1)(M1) Check Using my equation I predict that for a rectangles sized 7x5 the difference will be N(N1)(M1) = 7(71)(51) = 7 X 6 X 4 = 168. 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 7 X 29 = 203 1 X 35 = 35 Difference = 168 My equation is correct Rectangles inside rectangles 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 A is the length of the outer grid B is the height of the outer grid C is the length of the inner grid D is the height of the outer grid X is the starting number [IMAGE] 50 is the starting number + 3 so I need to find a way to add three. X + (C1) [IMAGE][IMAGE]47 50 [IMAGE][IMAGE]62 65 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically X + (C1) + A(D1) [IMAGE] [IMAGE] [IMAGE] [IMAGE] [IMAGE]67 72 X + (C1) + A(D1) [IMAGE][IMAGE]97 102 [IMAGE] [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically 97 is the starting number + 30 so I need to find a way to add thirty. 95 is the starting number + 4 so I need to find a way to add four. [IMAGE] X + (C1) X [IMAGE][IMAGE]91 95 [IMAGE][IMAGE]106 110 [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically 106 is the starting number + 15 so I need to find a way to add fifteen. [IMAGE] X + (C1) + A(D1) X [IMAGE] [IMAGE] 28 is the starting number + 2 so I need to find a way to add two. X + (C1) [IMAGE]26 28 [IMAGE][IMAGE]41 43 [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] X + (C1) + A(D1) X + A(D1) These are the equations that I have figured out for each corner. [IMAGE]X X+(C1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X+A(D1) X+(C1)+A(D1) (X + (C1)) (X + A(D1)) = X2 + XA(D1) + X(C1) + A(D1)(C1) (X) (X + (C1) + A(D1)) = X2 + XA(D1) + X(C1) Difference = A(D1)(C1) Check Using my equation, I predict that inside a 6x5 rectangle a 3x2 inner rectangle will have a difference of 6(21)(31) = 6 x 1 x 2 = 12. 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 8 10 14 16 10 x 14 = 140 8 x 16 = 128 Difference = 12 My equation is correct. Looking at these results I have realised that squares are actually rectangles where the length = width so I am now not going to treat squares and rectangles differently. Text Box: X is the starting number. N is the length. M is the height. S is the arithmetic progression. Patterns What happens when I change the pattern inside a rectangle? Arithmetic Progressions 8 is the starting number + 6 so I need to find a way to add six. [IMAGE] [IMAGE] [IMAGE] 2 8 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE][IMAGE][IMAGE]18 24 X + (SN(M1) + S(N1)) [IMAGE] [IMAGE] 6 is the starting number + 4 so I need to find a way to add four. [IMAGE] X + S(N1) [IMAGE]2 6 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] X + (SN(M1) + S(N1)) [IMAGE][IMAGE]14 18 [IMAGE] X + SN(M1) 14 is the starting number + 12 so I need to find a way to add twelve. 12 is the starting number + 9 so I need to find a way to add nine. [IMAGE] X + S(N1) [IMAGE] X [IMAGE]3 12 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] X + (SN(M1) + S(N1)) [IMAGE][IMAGE]39 48 X + SN(M1) [IMAGE] To get the answer to the equation, you have to multiply the rectangle going up in consecutive numbers by the arithmetic number you are using this is because when you use consecutive numbers, you are actually going up in arithmetic progression of 1. [IMAGE] X is the starting number. N is the length of the rectangle. M is the height of the rectangle. S is the arithmetic progression you are using X X+S(N1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X+SN(N1) X+(SN(N1)+S(N1)) (X + S(N1)) (X + SN(M1) = X2 + XSN(M1) + XS(N1) + SN(M1) S(N1) X (X + (SN(M1) + S(N1) = X2 + XSN(M1) + XS(N1) Difference = SN(M1) S(N1) Check For a table 6x6 with an arithmetic progression of nine I predict that the difference will be (9X6(61)) X (9(61) = (54 X 5) X (9 X 5) = 270 X 45 = 12150 9 54 279 324 54 X 279 = 15066 9 X 324 = 2916 Difference = 12150 My equation was right. Geometric Progressions There is a pattern for arithmetic progressions but what about geometric progressions? X is the starting number. N is the length of the rectangle. M is the height of the rectangle. 16 is the starting number + 14 so I need to find a way to add fourteen. [IMAGE] XN [IMAGE][IMAGE]2 16 This number is the top right corner X the bottom left corner divided by the starting number. X [IMAGE] (XN) (XN(M1)) [IMAGE][IMAGE]8192 65536 X(XN(M1)) [IMAGE] [IMAGE] [IMAGE] 2 8 [IMAGE] This number is the top right corner X the bottom left corner divided by the starting number. [IMAGE][IMAGE]16 64 (XN) (XN(M1)) [IMAGE] [IMAGE] XN X [IMAGE][IMAGE]3 27 This number is the top right corner X the bottom left corner divided by the starting number. [IMAGE][IMAGE][IMAGE]2187 19683 (XN) (XN(M1)) [IMAGE] [IMAGE] These are the equations that I have figured out for each corner. X XN These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X(XN(M1)) XN (XN(M1)) X is the starting number. N is the length of the rectangle M is height of the rectangle. (XN) (X(XN(M1))) = X(XN) + XN (XN(N1)) (X) (XN (XN(M1))) = X(XN) + XN (XN(N1)) Difference = 0 Check For a 3x3 table with a geometric progression of 7 I predict that the difference is 0. 7 343 823543 40353607 343 X 823543 = 282475249 7 X 40353607 = 282475249 Difference = 0 My equation is right. Arithmetic Progressions in grids within grids I have looked at grids inside a grid and I have also looked at arithmetic progressions but what happens when I put the together? 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 A is the length of the outer grid. B is the height of the outer grid. C is the length of the inner grid. D is the height of the inner grid. X is the starting number. S is the arithmetic progression. 138 is the starting number +2 so I need to find a way to add two. X [IMAGE] [IMAGE][IMAGE]136 138 X + S(C1) + SA(D1) [IMAGE][IMAGE]156 158 [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] 156 is the starting number +20 so I need to find a way to add twenty. [IMAGE] X X + A(C1) [IMAGE][IMAGE]24 28 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE][IMAGE]44 48 [IMAGE] [IMAGE] X + S(C1) + SA(D1) X + SA(D1) [IMAGE] [IMAGE] X + S(C1) [IMAGE]88 92 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE] X + S(C1) + SA(D1) [IMAGE][IMAGE]128 132 [IMAGE] 128 is the starting number +40 so I need to find a way to add fourty. X + SA(D1) [IMAGE] [IMAGE] [IMAGE]12 20 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE][IMAGE][IMAGE]52 60 X + SA(D1) [IMAGE] X + S(C1) + SA(D1) These are the equations that I have figured out for each corner. [IMAGE] A is the length of the outer grid. B is the height of the outer grid. C is the length of the inner grid. D is the height of the inner grid. S is the arithmetic progression. X is the starting number. X X + S(C1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]X + SA(D1) X + S(C1) + SA(D1) (X + S(C1)) (X + SA(D1)) = X2 + SAX(D1) + SX(C1) + SA(D1)S(C1) (X) (X + S(C1) + SA(D1)) = X2 + SAX(D1) + SX(C1) Difference = SA(D1)S(C1) Check Using my formula I predict that for a 6x4 outer grid with an arithmetic progression of 5 the difference of the product of the opposite corners inside an inner grid of 3x2 will equal 5x6(21)5(31) = 30(1)5(2) = 30 x 10 = 300. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 15 25 45 55 25 x 45 = 1125 15 x 55 = 825 Difference = 300 My equation is correct. Geometric progressions in grids within grids 21 22 23 24 25 26 27 28 29 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 2100 After doing my original study on geometric progressions I have realised that it is easier to keep the numbers in powers form. [IMAGE] [IMAGE] 212 213 [IMAGE] This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically NX + (C1) + A(D1) [IMAGE][IMAGE]222 223 [IMAGE] [IMAGE] [IMAGE] 244 245 246 247 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically [IMAGE][IMAGE]254 255 256 257 254 is the starting number X 210 so I need to find a way to multiply the start number by 210. [IMAGE][IMAGE] NX + A(D1) NX + (C1) + A(D1) 276 is the starting number X 24 so I need to find a way to multiply the start number by 24. [IMAGE] [IMAGE] NX + (C1) [IMAGE]272 273 274 275 276 This number is the starting number + the top right corner + the bottom left corner so I'm going to write this algebraically NX + A(D1) 282 283 284 285 286 [IMAGE][IMAGE] NX + (C1) + A(D1) [IMAGE][IMAGE]292 293 294 295 296 292 is the starting number X 220 so I need to find a way to multiply the start number by 220. These are the equations that I have figured out for each corner. [IMAGE]NX NX+(C1) These are the equations that I have figured out for each corner. These are the equations that I have figured out for each corner. [IMAGE][IMAGE]NX+A(D1) NX+(C1)+A(D1) NX + (C1) X NX + A(D1) = N2X + (C1) + A(D1) NX X NX + (C1) + A(D1) = N2X + (C1) + A(D1) Difference = 0 Check For a 7x3 outer grid a 4x2 inner grid with a geometric progression of 7 will be 0. 71 72 73 74 75 76 77 78 79 710 711 712 713 714 715 716 717 718 719 720 721 73 74 75 76 710 711 712 713 76 + 710 = 716 73 + 713 = 716 Difference = 0 My equation is correct. Spirals X is the starting number. N is the length of the rectangle. M is the height of the rectangle. What would happen if I spiralled into the centre? [IMAGE] [IMAGE] 6 is the starting number + 5 so I need to find a way to add five. 1 2 3 4 5 6 20 21 22 23 24 7 11 is the starting number +10 so I need to find a way to add ten. 16 is the starting number +15 so I need to find a way to add fifteen. 19 32 33 34 25 8 [IMAGE][IMAGE]18 31 36 35 26 9 X+(N1)+(M1) X+2(N1)+(M1) [IMAGE][IMAGE]17 30 29 28 27 10 16 15 14 13 12 11 [IMAGE] [IMAGE] 1 9 [IMAGE] 11 is the starting number +10 so I need to find a way to add ten. X+(N1)+(M1) [IMAGE][IMAGE]19 11 X+2(N1)+(M1) [IMAGE] [IMAGE] X [IMAGE] 10 is the starting number +2 so I need to find a way to add two. [IMAGE]8 10 [IMAGE] 14 is the starting number +6 so I need to find a way to add six. [IMAGE] 12 is the starting number +4 so I need to find a way to add four. X+(N1)+(M1) X+2(N1)+(M1) [IMAGE][IMAGE]14 12 [IMAGE] These are the equations that I have figured out for each corner. X X+(N1) [IMAGE] These are the equations that I have figured out for each corner. [IMAGE] These are the equations that I have figured out for each corner. X+2(N1)+(M1) X+(N1)+(M1) (X+(N1)) (X+2(N1)+(M1) = X2+X(N1)+ X(M1)+2X(N1) +2(N1)2+(M1)(N1) (X) (X+(N1)+(M1) = X2+X(N1)+X(M1) Difference = 2X(N1)+2(N1)2+(M1)(N1) Check For a 7x5 rectangle with a starting number of 3 I predict that the difference will be 2x3(71)+2(71)2+(51)(71) = 6(6)+2(6)2+(4)(6) = 36+2(36)+24 = 36+72+24 = 132. 3 9 19 13 9 x 19 = 171 3 x 13 = 39 Difference = 132 My equation is correct. How to Cite this Page
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"Opposite Corners Investigation." 123HelpMe.com. 19 Apr 2014 <http://www.123HelpMe.com/view.asp?id=120558>. 
