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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

#### 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.

Middle

30

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

10wy-0w+10x+10 can be used to calculate the stair total of a 4-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 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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