# Number Grid

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Introduction

Ma1: Number Grid Introduction My task is to find a formula for any given number grid, I will then go on to find the formula for the squares and then change the squares to rectangles! I have been given a 10 x 10 number grid to start with. I will draw a square around four numbers. I will identify the different squares by using the top left number as the square number. Once I have found my square I will find the product of the top left and bottom right numbers, and then find the different of this from the bottom left and top right's product. This would be Square Number 1. 1 2 11 12 1x12 = 12 2x11 = 22 22-12 = 10 Grid One 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 Results Square Number 12 = 12x23 = 276 13x22 = 286 Difference = 286-276 = 10 Square number 18 = 18x29 = 522 19x28 = 532 Difference = 532-522 = 10 Square number 45 = 45x56 = 2520 55x46 = 2530 Difference = 2530-2520 = 10 Square number 72 = 72x83 = 5976 82x73 = 5986 Difference = 5986-5976 = 10 Square number 78 = 78x89 = 6942 79x88 = 6952 Difference = 6952-6942 = 10 Table of Results Square Number Difference 12 10 18 10 45 10 72 10 78 10 My table of results show that the differences of each square on the 10x10 grid are all 10. ...read more.

Middle

5x5 Squares 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 Results Square 1 = 1x45 = 45 41x5 = 205 Difference = 205-45 = 160 Square 6 = 6x50 = 300 10x46 = 460 Difference = 460-300 = 160 Square 25 = 25x69 = 1725 29x65 = 1885 Difference = 1885-1725 = 160 Square 32 = 32x76 = 2432 36x72 = 2592 Difference = 2592-2432 = 160 Square 51 = 51x95 = 4845 55x95 = 5005 Difference = 5005-4845 = 160 Table of results Square Number Difference 1 160 6 160 25 160 32 160 51 160 My table of results shows that each difference of my products came to 160! Prediction For square number 56 I predict that the difference of the two products will be 90 56x100 = 5600 60x96 = 5760 Difference = 5760-5600 = 160 Square Number Difference 56 160 My prediction was right the difference was 160! Formula I now think I have enough results to find the formula for the size of the square used in the grid. Square Size Difference 2 10 3 40 4 90 5 160 From this table, I can see that the differences are all square numbers multiplied by 10. Here is what I have found: 2x2 Square = 1x1x10 3x3 Square = 2x2x10 4x4 Square = 3x3x10 5x5 Square = 4x4x10 There for the formula for the square would be: 2X2 square = (2-1)�x10 3x3 Square = (3-1)�x10 4x4 ...read more.

Conclusion

Square 1 = 1x32 = 32 2x31 = 62 Difference = 62-32 = 30 Square 9 = 9x40 = 360 10x39 = 390 Difference = 390 - 360 = 30 Square 43 = 43x74 = 3182 44x73 = 3212 Difference = 3212-3182 = 30 Square 47 = 47x78 = 3666 48x77 = 3696 Difference = 3696-3666 = 30 Grid two - 8x8 Grid, 3x5 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 Results Square 1 = 1x35 = 35 3x33 = 99 Difference = 99-35 = 64 Square 5 = 5x39 = 195 7x37 = 159 Difference = 195-159 = 64 Square 10 = 10x44 = 440 12x42 = 504 Difference = 504-440 = 64 Square 14 = 14x48 = 672 16x46 = 736 Difference = 736-672 = 64 From my results, I can confirm that my predicted formula was correct. Grid Reference Predicted Difference Difference Grid one - 10x10 Grid, 3x4 Rectangles 30 30 Grid two - 8x8 Grid, 3x5 rectangles 64 64 I have now proven that for a grid any size with a rectangle any size the formula will be: D=(S-1)x(L-1)xW Formulas Any Number Grid: If I let D equal difference and G equal grid width, then my formula to know the difference of products on any size grid will be: G=D Square Any Size, Grid Any Size: If I had a square SxS on a grid W wide, and let D equal difference, then my formula would be: D=(S-1)�xW Rectangle Any Size, Grid Any Size: If I had a rectangle LxW on a grid G wide, and let the difference equal D, then my formula would be: D=(L-1)x(W-1)xG Leander Wilson Page 1 08/05/2007 ...read more.

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