I have now got the complete formula:
T = x + (x + 10) + (x + 20) + (x + 1) + (x + 2) + (x + 11)
This formula is very long and can be simplified to:
T = 6x + 44
I am now going to look at 3 step stairs on different sized grids.
I will start by looking at some 3 step stairs on a 9x9 grid to see if there is any relationship between the stair total and the position on the grid. Here are some examples.
The general pattern seen when looking at my examples is the higher up the grid you go the higher the stair total. I can also see that every time the stairs are moved along the grid 1 you add 6 to the stairs total. This is shown in the results table I have drawn below.
I am now going to look at the relationships of the numbers inside the stairs.
From my stairs we can see that the number above x is 9 more than x. So I can start to construct an algebraic formula:
Total (T) = x + (x + 9)
We add by 9 because it is a 9x9 grid so each row up is 9 more than the square below and each row down is 9 less than the square above.
I can also see by looking at the number 2 squares above x it is going to be x + 18 because it is a 9x9 grid so each time you go up a row it is 9 more.
The number which is one square right of x is going to be x + 1. This is because on a 9x9 grid there are 9 squares in each row so each square to the right you add 1. This is the same for any sized grid. So I have now got the formula:
T = x + (x + 9) + (x + 18) + (x + 1)
The number 2 squares right of x is going to be x + 2 the reason for this is as you go right along the grid you add 1 so if you go 2 squares right you add 2.
I have now got the formula:
T = x + (x + 9) + (x + 18) + (x + 1) + (x + 2)
By looking at the number diagonal of x I can see it is going to be x + 10 the reason for this is that as you go along a row on a 9x9 grid you add 1 and as you go up a row on a 9x9 grid you add 9 so it is x + 10.
I have now got the complete formula:
T = x + (x + 9) + (x + 18) + (x + 1) + (x + 2) + (x + 10)
This formula is very long and can be simplified to:
T = 6x + 40
I am now going to look at some 3 step stairs on a 8x8 grid to see if there is any relationship between the stair total and the position on the grid. Here are some examples.
The general pattern seen when looking at my examples is the higher up the grid you go the higher the stair total. I can also see that every time the stairs are moved along the grid 1 you add 6 to the stairs total. This is shown in the results table I have drawn below.
I am now going to look at the relationships of the numbers inside the stairs.
From my stairs we can see that the number above x is 8 more than x. So I can start to construct an algebraic formula:
Total (T) = x + (x + 8)
We add by 8 because it is a 8x8 grid so each row up is 8 more than the square below and each row down is 8 less than the square above.
I can also see by looking at the number 2 squares above x it is going to be x + 16 because it is a 8x8 grid so each time you go up a row it is 8 more.
The number which is one square right of x is going to be x + 1. This is because on a 8x8 grid there are 9 squares in each row so each square to the right you add 1. This is the same for any sized grid. So I have now got the formula:
T = x + (x + 8) + (x + 16) + (x + 1)
The number 2 squares right of x is going to be x + 2 the reason for this is as you go right along the grid you add 1 so if you go 2 squares right you add 2.
I have now got the formula:
T = x + (x + 8) + (x + 16) + (x + 1) + (x + 2)
By looking at the number diagonal of x I can see it is going to be x + 9 the reason for this is that as you go along a row on a 8x8 grid you add 1 and as you go up a row on a 8x8 grid you add 8 so it is x + 9.
I have now got the complete formula:
T = x + (x + 8) + (x + 16) + (x + 1) + (x + 2) + (x + 9)
This formula is very long and can be simplified to:
T = 6x + 36
By investigating 3 step stairs on a 10x10 grid a 9x9 grid and a 8x8 grid I am going to find a formula which can find the total of any 3 step stair on any sized grid.
I found that instead of adding say 10 on a 10x10 grid I called it G, which stands for grid size, which is 10 for a 10x10 grid.
So I can start construct an algebraic formula:
T = x + (x + G)
The number 2 rows above x on a 10x10 grid is 20 which is 2 times G so I can call it 2G
I have now got the formula:
T = x + (x + G) + (x +2G)
For the number diagonal of x it is 1 row up so, X + G, and one row across so, X + G +1.
I have now got the formula:
T = x + (x + G) + (x +2G) + (x + G + 1)
The number which is one square right of x is going to be x + 1. This is the same for any sized grid.
I have now got the formula:
T = x + (x + G) + (x +2G) + (x + G + 1) + (x + 1)
The number 2 squares right of x is going to be x + 2 the reason for this is as you go right along the grid you add 1 so if you go 2 squares right you add 2.
I have now got the complete formula for any 3 step stair on any sized grid:
T = x + (x + G) + (x +2G) + (x + G + 1) + (x + 1) + (x + 2)
This formula is very long and I can simplify it to:
T = 6x + 4G + 4
