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Step Stairs - number stairs.

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Introduction

NUMBER STAIRS This is a 10 by 10 Number Grid: 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 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 Fig. 1 Marked on the grid is a stair shape; this is called a 3-Step Stair. The total of the numbers inside the stair shape is: 25 + 26 + 27 + 35 + 36 + 45 = 194 The Stair Total for this 3-step stair is 194. Part 1 For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. Part 2 Investigate further the relationship between the stair totals and other step stairs on other number grids. 4w+1 4w+2 4w+3 4w+4 5w 3w+1 3w+2 3w+3 3w+4 4w 2w+1 2w+2 2w+3 2w+4 3w 1w+1 1w+2 1w+3 1w+4 2w 0w+1 0w+2 0w+3 0w+4 1w w = width Fig. 1 On every size of number grid the formula in each block above will be the right formula for the corresponding number on any grid. ...read more.

Middle

25+26+27+35+36+45 = 194 Also from these formulae, when x is substituted with 7 and y is substituted with 7 (7 and 7 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 446 is calculated. 67+68+69+77+78+87 = 446 This confirms the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a 10 by 10 grid. 57 58 59 60 61 62 63 50 51 52 53 54 55 56 43 44 45 46 47 48 49 36 37 38 39 40 41 42 29 30 31 32 33 34 35 22 23 24 25 26 27 28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7 Fig. 1 From the formula (6wy-2w+6x+4), when x is substituted with 2 and y is substituted with 2 (2 and 2 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 86 is calculated. 9+10+11+16+17+23 = 86 From the formula (6wy-2w+6x+4), when x is substituted with 4 and y is substituted with 6 (5 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 266 is calculated. 39+40+41+46+47+53 = 266 This confirms that the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a number grid of any size. ...read more.

Conclusion

+ [(wy-w+x)+2w+2] + [(wy-w+x)+3w] + [(wy-w+x)+3w+1] + [(wy-w+x)+4w] This can be simplified to: 15wy+5w+15x+20 From these formulae, when x is substituted with 3 and y is substituted with 4 (3 and 4 being the coordinates of the keystone of the step stair on the grid above), an answer of 585 is calculated. 27+28+29+30+31+35+36+37+38+43+44+45+51+52+59 = 585 This confirms the formula 15wy+5w+15x+20 can be used to calculate the stair total of a 5-step stair on a number grid of any size. 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 Fig. 1 The formula for the 2-step stair is: (wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+w] This can be simplified to: 3wy-2w+3x+1 From these formulae, when x is substituted with 2 and y is substituted with 3 (2 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 63 is calculated.: 18+19+26 = 63 This confirms the formula 3wy-2w+3x+1 can be used to calculate the stair total of a 2-step stair on a number grid of any size. Elia Bols 10WD Maths (Mrs Carr) ...read more.

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