# Number Grids.

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Introduction

Number Grids The diagram shows a 10*10 grid, a rectangle has been shaded on the 10*10 grid. I will find the diagonal difference between the products of the numbers in the opposite corners of the rectangle. 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 Opposite numbers in the rectangle are:- 54 and 66 56 and 64 56*64=3584 54*66=3564 .�. The Diagonal Difference = 3584 - 3564 = 20 Study I have studied some more 3*2 rectangles and I have found this:- 12 13 14 22 23 24 12*24=288 14*22=308 Diagonal difference =308 - 288=20 74*86=6364 76*84=6384 Diagonal difference =6384 - 6364=20 74 75 76 84 85 86 27*39=1053 29*37=1073 Diagonal difference =1073 - 1053= 20 27 28 29 37 38 39 So from this I conclude that all 3*2 rectangles have a diagonal difference of 20. ...read more.

Middle

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 I will begin by doing some 3*2 rectangles 21*32= 672 23*30=690 Diagonal difference=690 - 672 =18 21 22 23 30 31 32 57*68= 3876 59*66=3894 Diagonal difference=3894 - 3876 =18 57 58 59 66 67 68 So I conclude that all 3*2 squares on a 9*9 grid have a diagonal difference of 18. I have tested the other way by using a 2*3 rectangle. 1*20=20 2*19=38 Diagonal difference = 38 - 20 = 18 1 2 10 11 19 20 40*59=2360 41*58=2378 Diagonal difference = 2378 - 2360 = 18 40 41 49 50 58 59 So all 3*2 and 2*3 squares have a diagonal difference of 18. I will study more rectangles on a 9*9grid and then draw up a table. ...read more.

Conclusion

3*2 rectangle on an 8*8 grid I use my formula to predict that: - (3-1)*(2-1)*8 =2*1*8 =16 2*12=24 4*10=40 Diagonal difference = 40 - 24=16 2 3 4 10 11 12 My prediction is right. So using my formula I predict that a 3*3 square on an 8*8 grid will be (X-1)*(X-1)*8 (3-1)*(3-1)*8 =2*2*8 =32 46*64=2944 48*62=2976 Diagonal difference = 2976 - 2944 =32 46 47 48 54 55 56 62 63 64 So my prediction and formula are also right. A pattern A pattern has shown up in the formulas so I made this table: - Grid rectangle Formula 10*10 (R-1)*(C-1)*10 9*9 (R-1)*(C-1)*9 8*8 (R-1)*(C-1)*8 7*7 (R-1)*(C-1)*7 6*6 (R-1)*(C-1)*6 5*5 (R-1)*(C-1)*5 Grid Square Formula 10*10 (X-1)*(X-1)*10 9*9 (X-1)*(X-1)*9 8*8 (X-1)*(X-1)*8 7*7 (X-1)*(X-1)*7 6*6 (X-1)*(X-1)*6 So on a 6*6 grid I would expect a 3*2 rectangle to be: - (3-1)*(2-1)*6 =2*1*6 =12 So 8*16=128 10*14=140 Diagonal difference = 140 - 128 =12 8 9 10 14 15 16 Also a 3*3 square on a 6*6 grid would be: - (X-1)*(X-1)*6 => (3-1)*(3-1)*6 =2*2*6 =24 3*17=51 5*15=75 Diagonal difference = 75 - 51 =24 3 4 5 9 10 11 15 16 17 Both predictions and formulas are correct so I conclude that the overall formula is: - R= rows C= columns X= square N� G= grid Rectangle: - (R-1)*(C-1)*G ...AND... Square: - (X-1)*(X-1)*G ...read more.

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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