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Introduction

Maths Course Work TASK B: Diagonal Difference Aim: The aim of this investigation is to find the diagonal difference taken from small nxn grids, like 3x3. Then to experiment with larger grid sizes to find a pattern or formula to deduce further results Introduction: I will be compiling grids taken from an 8x8 grid like the one below; the grids will start at 2x2 then work up to the full 8x8. (Note it is impossible to find the diagonal difference in a 1x1 grid because their needs to be four or more numbers). 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 This is the full 8x8 grid. From this grid I will be taking smaller grids. The shaded numbers is just an illustration of the 3x3 grid. 10 11 12 18 19 20 26 27 28 This is the 3x3 grid; I will be multiplying the opposite corners then finding the difference. For example 10 x 28 = 280 12 x 26 = 312 312 - 280 = 32 Note take away the smaller number from the bigger number, this is to ensure you get a consistent result. ...read more.

Middle

30 31 34 35 36 37 38 39 40 33 34 35 36 37 38 39 42 43 44 45 46 47 48 41 42 43 44 45 46 47 50 51 52 53 54 55 56 49 50 51 52 53 54 55 58 59 60 61 62 63 64 57 58 59 60 61 62 63 16 x 58 - 10 x 64 = 288 15 x 57 - 9 x 63 = 288 The Diagonal Difference from the 7x7 grids = 288 I've decided to leave out the 8x8 grid until I find a formula, I will then use the formula to find out the diagonal difference of an 8x8 grid then check using the above process (long hand). Now I will try to find a formula to predict the diagonal difference in an 8x8 square. 8 2x2 Grid 208 16 216 24 224 32 3x3 Grid 232 40 240 48 248 56 256 64 264 72 4x4 Grid 272 80 280 88 288 7x7 Grid 96 296 104 304 112 312 120 320 128 5x5 Grid 328 136 336 144 344 152 352 160 360 168 368 176 376 184 384 192 392 ?8x 8 Grids? 200 6x6 Grid 400 I noticed when I compared the 8 times table to the diagonal differences taken from the above grid sizes, (2x2, 3x3, 4x4, 5x5, 6x6, 7x7) that they all fitted into the 8 times table, and that there might be pattern concerning how many PRIME numbers in between each square grid. ...read more.

Conclusion

The 2 times table also came into play, it was this table that helped me plot where the results were likely to be. The formula to work out the next grid (3x3 = 8 + 8 x 3) reference did work and was useful, though it required previous diagonal difference value from smaller grids and I couldn't manipulate it so that It incorporated a more universal use. Instead I came up with ((n-1)�xy) this formula did work, and It also works for other main grid sizes. Overall, my investigation was a success, and I did complete my Aim. Rough Notes These are my notes which include a graph; these notes were created based on my background Knowledge which helped me lead up to the final formula. A Graph to Show the Diagonal Difference of nxn squares My graph shows my results clearly, the x-axis (horizontal) shows the grid sizes from 2x2 - 8x8, the Y-axis (vertical) shows the diagonal difference from 0 up to 500, this also relates to the 8 times table (page 5). From graph you can see how the gap in-between the values increases as you move up to the bigger grids like 8x8. Results Table 2x2 Grids 8 3x3 Grids 32 4x4 Grids 72 5x5 Grids 128 6x6 Grids 200 7x7 Grids 288 8x8 Grids 392 These are the same results gathered from all my methods, they all proved consistent. ?? ?? ?? ?? Maths Course Work - 02/05/2007 - By Andrew Barnes. ...read more.

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