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Introduction

Philip Spicer Maths coursework: Number Stairs 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 49 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 The problem: I have to find a theory that links the relationship between the stair total and the position of the stair shape on the grid. I plan to do this is by comparing the grid width against the stair number (for this stair the bottom left hand square) to find an equation that relates it to the stair total ( the sum of all the numbers in the stair). e.g. stair number 55: 55+56+57+65+66+75 = 374 (stair total) ...read more.

Middle

I plan on rotating the standard stair through 360� and make recordings on a 10x10 grid, tabulating the results: Rotation (�) Stair No. (x) Stair total Equation 0 25 104 6x+4g+4 90 25 114 6x-4g+4 180 25 106 6x-4g-4 270 25 186 6x+4g-4 Now I will move on and find equations which represent different sizes of stairs and whether there is a link between which may enable me to establish a solution. To calculate an accurate equation I will have to use the same stair number (the number in the bottom-left of the grid in order to find a link. If I did not do this and used different stair numbers the results would not be correct and a correct solution may not be found: For this I have used a base of 33: 2 step-stair X+g X X+1 3 step stair: X+2g X+g X+g+1 X X+1 X+2 4 step stair: X+3g X+2g X+2g+1 X+g X+g+1 X+g+2 X X+1 X+2 X+3 5 step-stair: X+4g X+3g X+3g+1 X+2g X+2g+1 X+2g+2 X+g X+g+1 X+g+2 X+g+3 X X+1 X+2 X+3 X+4 At this point the only similarity I can note is that there is an equal number of 'g's to the end numbers and that the 'x's are all triangular numbers: Maybe this is a major point?? ...read more.

Conclusion

Using the sequence from the page above I will look and find the differences between the numbers which should allow me to calculate the formula: So If I compare the two sequences I can calculate the values of the letters: From these results I should be able to calculate my final formula. This is just the basic formula but I need to factorise it before I can reach the answer. / n� + n� - / n Factorisation: I will need to then factorise and prove this formula before I can say I have completed this project. To prove it I will have to put it into different stair sizes and find out whether the equation works: / n� + n� - / n n( / n� + n - / ) n( n� + n - 1) 6 n( n + 1)( n - 1) 6 21 11 12 1 2 3 So this a 3 step stair. N = 3 From previous work I can see that the equation for a three step stair is 6x+4g+4. So if this is to work, the answer to this equation should be 4: ( 2 ) 3 ( 4 ) 24 = 4 6 6 This proves that this equation must be correct. ...read more.

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