# 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:

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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. However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five.

Middle

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

Fig. 1

The formula for a 3-step stair is:

(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+2w]

This can be simplified to:

6wy-2w+6x+4

From these formulae, when x is substituted with 5 and y is substituted with 3 (5 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 194 is calculated.

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.

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 |

Conclusion

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 5-step stair is:

(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+3] + [(wy-w+x)+4] +

[(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+w+2] + [(wy-w+x)+w+3] +

[(wy-w+x)+2w] + [(wy-w+x)+2w+1] + [(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.

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 |

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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