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Introduction

Number Stairs

## Aim

The intention of this investigation is to find the stair total of a stair shape when I know the stair number, I should endeavour to know this stair total in any size number grid.

Introduction

Here is an example of a stair shape;

This particular stair shape would be described as a 3-step stair, as it has three tiers; there are 3 levels hence 3-step stair.

Adding all the digits within the stair shape together can identify the stair total,

I.e. the stair total in the diagram above would be 45+35+36+25+26+27= 194

This number is the stair total

To calculate the position of the stair shape I must select a particular stair number (this is what I have named it.) I have decided to use the bottom left- hand digit, which in my diagram (above) is 25 (and circled.)

I am now going to investigate how the position of the stair shape (stair number) effects the stair total, I intend to do this by systematically moving the stair number, horizontally.

Middle

3

62

4

68

I have noticed from the results that the stair total goes up by 6 each time, making it a linear sequence. From this I can predict that if the stair number was 5 the stair total would be 74 (68+6). Although I can also use this sequence to find the rule that will enable me to find the stair total in any stair shape, provided I know the stair number. This rule, when x is the stair number, is

6x  + 44

Now that I have found the rule for any 3-step stair in a 10x10 number grid. I will deepen this investigation by attempting to find the rule for any size number grid. I will do this by systematically finding the rule in a 9x9 grid then 8x8 etc but to speed up my investigation I will find this rule by using an algebraic technique (Shown below). After this I will, as before, analyse my results for a pattern.

Here is an example of the type of algebraic technique I will be using;

Conclusion

any number step stair in any number grid.

I will continue to use my algebraic technique.

4-step stair in a 10x10 number grid

x+30+x+20+x+10+x+21+x+11+x+12+x+1+x+2+x+3+x=  10x+110

4-step stair in a 9x9 number grid

x+27+x+18+x+9+x+19+x+10+x+11+x+1+x+2+x+3+x=  10x+100

4-step stair in a 8x8 number grid

x+24+x+17+x+16+x+8+x+9+x+10+x+1+x+2+x+3+x=  10x+90

4-step stair in a 7x7 number grid

x+21+x+14+x+7+x+15+x+8+x+9+x+1+x+2+x+3+x=  10x+80

4-step stair in a 6x6 number grid

x+18+x+12+x+13+x+6+x+7+x+8+x+1+x+2+x+3+x=  10x+70

Results Table for a 4-step stair

 Number grid Rule when x is stair no. 6x6 10x+110 7x7 10x+100 8x8 10x+90 9x9 10x+80 10x10 10x+70

I can see that there is a visible linear sequence the constant in the rule increases each time by 10 so I can predict that the rule for a 5x5

BETHAN WHAT HAPPENED TO  THE

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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