maths coursework
...33 34 35 36 37 38 39 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 N Equation total Total 1 6N+44 = 50 50 75 6N+44= 494 494 28 6N+44= 168 168 The table above shows my equation was correct, as all randomly chosen numbers equalled the same as the equation. I then decided to try to find the same equation for a 9 by 9 grid, but still using a 3-step stair. 100 91 92 93 94 95 96 97 98 99 82 83 84 85 86 87 88 89 90 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 So for this size grid my results were: N Total Difference 1 46 6 2 52 6 3 58 6 4 64 6 I noticed the difference between the totals when N was increased by one, which were once again 6. So I subtracted 6 from the 1st term (46), which left me with 40. Therefore the equation for this size grid was 6N+40. I tested it out on a few different numbers: the table is shown above. N Equation Total Total 1 6N+40= 46 46 5 6N+40= 70 70 77 6N+40= 502 502 I was pleased to see this equation also worked. So I moved on to an 8 by 8 grid. 97 98 99 100 89 90 91 92 93 94 95 96 81 82 83 84 85 86 87 88 73 74 75 76 77 78 79 80 65 66 67 68 69 70 71 72 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 Using the same method as before I went about trying to work out the equation. N Total Difference 1 42 6 2 48 6 3 54 6 Once again the difference was 6, so I subtracted this from the 1st term (42), which left me with 36. Therefore the equation would be 6N+36. N Equation total Total 1 6N+36= 42 42 14 6N+36= 120 120 83 6N+36= 454 454 Having completed the equation for these grid sizes I began to notice a pattern: Grid Size (6N+X) where X is… Difference between Xs 10 by 10 44 4 9 by 9 40 4 8 by 8 36 4 I decided it might be a good idea to see if this pattern was relevant or just coincidence. So I worked out for a 5 by 5 grid the equation should be, Grid size X 10 44 (-4) 9 40 (-4) 8 36 (-4) 7 32 (-4) 6 28 (-4) 5 24 Therefore the equation should be 6N+24. 96 97 98 99 100 91 92 93 94 95 86 87 88 89 90 81 82 83 84 85 76 77 78 79 80 71 72 73 74 75 66 67 68 69 70 61 62 63 64 65 56 57 58 59 60 51 52 53 54 55 46 47 48 49 50 41 42 43 44 45 36 37 38 39 40 31 32 33 34 35 26 27 28 29 30 21 22 23 24 25 16 17 18 19 20 11 12 13 14 15 6 7 8 9 10 1 2 3 4 5 I chose these 3 step stairs to check if my working was correct. N Equation total Total 1 6N+24= 30 30 38 6N+24= 252 252 67 6N+24= 426 426 I was very glad to see my prediction was correct. From this I was able to create the following chart: Grid Size Equation 3 6N+16 4 6N+20 5 6N+24 6 6N+28 7 6N+32 8 6N+36 9 6N+40 10 6N+44 11 6N+48 12 6N+52 Now I had done this I realised I must figure out an equation that could get the total from a 3-step stair on any size grid. I couldn’t see any pattern or trend from the table above that would help me in this specific equation. Knowing I had labelled the bottom left hand number N I drew out this diagram: N+2X N+X N+X+1 N N+1 N+2 So by looking at the above table I added up all of the Ns and there are 6, there are 4 Xs and then there’s some left over numbers, which add up to 4. So from this I can make the equation: 6N+4X+4 If this was the right equation N would still stand for the bottom left-hand cell, and X would represent t...