6 + 7 + 8 + 16 + 17 + 26 = 80
(6 x 6) + 44 = 80
s-number = 6
s-total = 80
26 + 27 + 28 + 36 + 37 + 46 = 200
(6 x 26) + 44 = 200
s-number = 26
s-total = 200
After testing the formula to see if it works, a proof must be presented to show that it works for any s-number (with a few exceptions – see “Conclusion”). If the s-number is represented as ‘n’:
As mentioned in the aim, the bottom left corner square is the s-number (or in this example, known as ‘n’
Because the numbers in the grid increases by 1 each time, I can say that the two numbers, in the same row, increases by 1, thus the diagram; ‘n + 1’ and ‘n + 2’.
As mentioned previously, the numbers in the same column have a difference of 10. Thus, ‘n + 10’ and ‘n + 20’ as shown in the diagram. I can then show: (↓)
If I add all these values (inside the squares) together:
n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20)
= 6n + 44
*Proof 1: Note that there are 6 n’s in this equation, because there are 6 squares.
Thus, I have proven that my formula ‘6n + 44’ works for every stair shape in that grid size. I can carry this investigation further by investigating the relationship between the stair totals on other number grids.
Part 2: To further investigate the relationship between the stair totals and other step stairs on other number grids. I have decided to investigate the relationship between the stair totals on other number grids when the s-number stays constant because it will be easier to observe the relationship between the s-grid (stair number grid) and the s-total.
1 + 2 + 3 + 4 + 5 + 7 = 22
s-grid = 3
s-total = 22
1 + 2 + 3 + 5 + 6 + 9 = 26
s-grid = 4
s-total = 26
1 + 2 + 3 + 6 + 7 + 11 = 30
s-grid = 5
s-total = 30
1 + 2 + 3 + 7 + 8 + 13 = 34
s-grid = 6
s-total = 34
Table of Results:
In the table, it shows that the s-total increases by 4 each time as the s-grid increases by 1. (When g = grid size,) this is because:
← Previously, I noticed that the difference between each row in the same column is the grid size.
n + (n + 1) + (n + 2) + (n + g) + (n + g + 1) + (n + 2g)
= 6n + 4g + (4)**
Note: 6n + 4g → because there are 4g’s in the formula (as shown above).
Now that I have found out the relationship between the s-number, s-grid, and the s-total, I can carry this investigation further by investigating the relationship between the other step stairs, s-number, s-grid, and s-totals.
I will investigate the relationship between the s-number, s-total, s-grid and s-shape (stair shape). The s-shape will be investigated by enlargement; 1 square increase in length, and 1 square increase in width.
The s-number remains at the bottom left corner of the stair shape. Instead of 6 squares, there are 10. Based on my previous observations, I can predict the relationship between this s-shape, the s-number, s-grid, and the s-total as: 10n + 10g + 10.
I can check my prediction:
n+(n+1)+(n+2)+(n+3)+(n+g)+(n+g+1)+(n+g+2)+(n+2g)+(n+2g+1)+(n+3g)
= 10n + 10g + 10
I shall investigate the next stair shape, which will have a height of 5 squares, and a width of 5 squares, a total of 15 squares.
n+(n+1)+(n+2)+(n+3)+(n+4)+(n+g+)+(n+g+1)+(n+g+2)+(n+g+3)+
(n+2g)+(n+2g+a)+(n+2g+2)+(n+3g)+(n+3g+1)+(n+4g)
= 15n + 20g + 20
Observation: I have noticed that the number of squares in each stair shape has a link with triangle numbers. For example: a triangle number sequence is as follows:
When the stair shape size was 3, it has 6 squares. When the stair shape size was 4, it has 10 squares, and when the stair shape size was 5, it has 15 squares. Just like the diagram on the previous page, the pattern for the stair shape size is the same pattern as triangle numbers. This is because the stair shape is a triangle:
I can predict the number of squares they are for any s-size, following the triangle number sequence.
The formula for the triangle number sequence is:
(Where ‘s’ is the s-size.)
I should test this formula to check if it works. I shall try it on an s-size of 7.
Number of squares in this diagram: 28
Since I have successfully tested out that the relationship between the s-size and the number of squares in that s-size is the same as triangle number sequence; thus the formula to work out the s-total when the s-number, s-grid, and s-size is known, can be currently worked out as:
s-total = (Note: X is the unknown)
In this case, I have used the X’s twice, only because they are of equal value. By putting the s-size value and the formula in a table, I could see why I used the X’s twice in the formula.
The co-efficient has the same value as the constant.
I then needed to find to how to find out the co-efficient and the constant. Because there doesn’t seem to have a constant difference between the co-efficient, I shall use the method ‘changing differences’ to find the formula for the co-efficient. There is a formula to find the ‘changing difference’ known as the ‘Gregory Newton Formula’.
Firstly, I would need to find the value of the co-efficient when the s-size is 0. Obviously, if the s-size is 0, the s-total will be 0, as there are no squares to add up its value. To show how I worked out the difference in order to use the ‘Gregory Newton Formula’, I made a table of differences:
After producing the table of differences, I was able to carry out the ‘Gregory Newton Formula’ and found out how to find ‘X’:
However, I must test this formula to see if it works. I shall test it out on an s-size of 6.
n+(n+1)+(n+2)+(n+3)+(n+4)+(n+5)+(n+g)+(n+g+1)+(n+g+2)+(n+g+3)+(n+g+4)+(n+2g)+(n+2g+1)+(n+2g+2)+(n+2g+3)+(n+3g)+(n+3g+1)+(n+3g+2)+(n+4g)+(n+4g+1)+(n+5g)
= 21n + 35g + 35
It appears that this section of the formula works. Now the new current formula is:
s-total =
Important: However, this formula has its limitations. For example, the step stair may be placed on a certain area that causes the step stair to be off the grid. This means that the formula cannot calculate the s-total as there are no numbers off the grid. Perhaps another limitation can be found if it cannot calculate negative numbers.
I shall test it out on an s-size of 7, s-grid of 8, and the s-number is 20.
20 + 21 + 22 + 23 + 24 + 25 + 26 + 28 + 29 + 30 + 31 + 32 + 33 + 36 + 37 + 38 + 39 + 40 + 44 + 45 + 46 + 47 + 52 + 53 + 54 + 60 + 61 + 68
= 1064
Now to test the formula out on negative numbers:
(-12)+(-13)+(-14)+(-6)+(-7)+(0)
= -52
It appears that negative numbers do not work with this formula, thus the formula is limited to positive integers and that the stair shape must not have any part of it exceeding the grid. However, the formula works excluding the limitations. In conclusion, the relationship between the s-number, s-grid, s-size, and s-total is:
s-total =
(With certain restrictions as mentioned above)
However, this formula can be simplified because as previously mentioned, the ‘X’s were in 2 parts of the formula, thus it can be simplified. The first part of the equation where the formula for triangle number lies, can also be simplified to:
An extention of this investigation, could be to investigate the relationship between negative numbers, or transformation of the stair shape.
