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  • Level: GCSE
  • Subject: Maths
  • Word count: 5603

Number Grid

Extracts from this document...

Introduction

Higher Tier Task - Number 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 Use the following rule: find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and the top right numbers in the square. Calculate the difference between these numbers. INVESTIGATE! I will begin with 2x2 windows on a 10x10 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 34 x 25 = 850 49 x 58 = 2842 24 x 35 = 840- 48 x 59 = 2832- 10 10 88 x 97 = 8536 63 x 72 = 4536 87 x 98 = 8526- 73 x 62 = 4526- 10 10 3 x 12 = 36 6 x 15 = 90 2 x 13 = 26- 5 x 16 = 80- 10 10 For all of these windows you can see ...read more.

Middle

When multiplied outside of the brackets: n�+ 8n + n + 8 - n�- 9n This can be simplified to: n�+ 9n + 8 - n� - 9n = 8 The n� and 9n cancel each other out therefore you are left with 8. I will now substitute a random 2x2 window size into this equation. n n+1 n+8 n+9 44 45 52 53 44�+ (9 x 44) + 8 - 44�-(44 x 9) = 8 1936 + 396 + 8 - 1936 - 396 = 8 2340 - 2332 = 8 n n+1 n+8 n+9 I predict that when doing 3x3 window on an 8x8 grid the difference will 32 because the sum of the opposite corners subtracted from each other in 2x2 window on the 8x8 grid was 4/5 of the difference for a 2x2 window on a 10x10 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 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 26 x 12 = 312 57 x 43 = 2451 10 x 28 = 280- 41 x 59 = 2419- 32 32 39 x 53 = 2067 21 x 7 = 147 37 x 55 = 2035- 23 x 5 = 115- 32 32 I was correct, it was 32, therefore the difference for a 4x4 window size should be 72. ...read more.

Conclusion

Therefore if you put 5x5 windows anywhere on a 10x10 grid with an increment of 3 the difference will be equal to 1440. We can again show the algebraic method of working out the difference. n n+3 n+6 n+9 n+12 n+30 n+33 n+36 n+39 n+42 n+60 n+63 n+66 n+69 n+72 n+90 n+93 n+96 n+99 n+102 n+120 n+123 n+126 n+129 n+132 We can change this into an equation: (n+12)(n+120)-n(n+132) When multiplied out of the brackets: n�+ 120n + 12n + 1440 - n�- 132n= 1440 This can be simplified to: n�+ 132n + 1440 - n�- 132n = 1440 Again the n� and 132n cancel each other out therefore you are left with 1440. This is the pattern I have seen. Window size Difference First Difference Second Difference 2x2 90 270 3x3 360 180 450 4x4 810 180 630 5x5 1440 180 810 6x6 2250 180 990 7x7 3240 180 1170 8x8 4410 180 1350 9x9 5760 This is a formula I have recognised for grids with an increment: The increment� multiplied by the difference for the same size window on a 10x10 grid with an increment of 1, for example: On a 10x10 grid with an increment of 1 anywhere a 2x2 window was placed the difference was 10. On a 10x10 grid with an increment of 2 anywhere a 2x2 window was placed the difference was 40. 2�= 4 4 x 10 = 40 And, on a 10x10 grid with an increment of 3 anywhere a 2x2 window was placed the difference was 90. 3�= 9 9 x 10 = 90 Therefore, a 2x2 window placed anywhere on a 10x10 grid with an increment of 4 the difference would equal 160. 4�= 16 16 x 10 = 160 Conclusion This is the formula to find the difference for any window size on any size grid with any increment. (n - 1)(n - 1) x w x i� = difference 'w' means width of the grid i� means the increment squared This could be simplified to: (n - 1)�x w x i�= difference Robert Bland Maths Coursework January 2004 ...read more.

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