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. I will find the stair total of each stair shape and looking at the results try to notice a pattern.
I will firstly do this with a 3-step stair.
This is my first stair shape I used this particular stair shape, as its stair number is one, which seems a sensible place to start.
Stair no. – 1
Stair total – 1+2+3+11+12+21= 50
I have moved along the stair number by one and I will continue to do this so the pattern will be simpler to detect.
Stair no. – 2
Stair total – 2+3+4+12+13+22= 56
Stair no. – 3
Stair total – 3+4+5+13+14+23= 62
Stair no. – 4
Stair total – 4+5+6+14+15+24= 68
Results Table
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;
As I can see from the numbers if I convert the stair number to a x then the other numbers will be x+1 going across and then x+10 going vertically. Obviously each time you get further from the stair number the numbers you need to add to it will increase.
E.g.
This method speeds up my investigation as it takes me immediately to the rule without the drawn-out working out, here is an example of how adding together the X’s and numbers leave me with the final rule very quickly,
x+1+x+2+x+10+x+11+x+20+x= 6x+44
Here is the rule
I have continued to show the stair number as x
9x9 Grid
x+x+18+x+9+x+10+x+1+x+2= 6x+40
8x8 Grid
x+x+16+x+8+x+9+x+1+x+2= 6x+36
7x7 Grid
x+x+14+x+7+x+8+x+1+x+2= 6x+32
6x6 Grid
x+x+12+x+6+x+7+x+1+x+2= 6x+28
Results Table
Once again there is a clearly defined pattern; the constant in the rule increases each time by 4. As this is a linear sequence then a rule for any number grid can be easily calculated. The overall for a 3-step stair in any number grid is, when g is grid number,
6x+(4g+4)
Now that I have found the rule for any 3-step stair in any number grid, I can assume the next step will be to find the rule for any number step stair in any number grid.
I intend to do this by firstly finding the overall rule for a 2-step stair in any number grid, then a 4-step stair grid etc until I have enough overall rules to put in a table and analyse. Therefore I will, hopefully, meet my aim the rule for 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
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
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