Steps: This is how many steps there are in the stair shape.
Finding a formula the Stair Total
A formula can be used to work out the numbers in a stair total. I am now going to find out how to get the formula.
The above stair shape can be expressed further in terms of the stair number X.
For example see below:
St(X) = X+X+1+X+2+X+10+X+11+X+20
= 6X+44
St(X) is the replacement of St(25) so St(X) is the total of the values in the stair total
A check can be used to see if the above expression is correct for all of the numbers in a stair shape.
Checking the Formula
So I will check by using X=14.
I will use this representation of X and use it in the stair shape below and see what stair total it generates.
St(X)= 6X+44
X=14
St(14)= 6(14)+44
St(14)= 128
A check can be done by doing it manually ie by adding up all of the numbers in the stair shape, the answer I should get should match with the stair total I got using the formula method.
St(14)=14+15+16+24+25+34
St(14)=128
This does match the answer from the formula but to make sure I will use a different stair number and see if both answers will match again.
So I will check by using X=15.
I will use this representation of X and use it in the stair shape below and see what stair total it generates.
St(X)= 6X+44
X=15
St(15)= 6(15)+44
St(15)= 134
A check can be done by doing it manually i.e. by adding up all of the numbers in the stair shape, the answer I should get should match with the stair total I got using the formula method.
St(15)= 15+16+17+25+26+35
St(15)=134
In the two examples above I have used the stair number formula to check if the numbers in the stair number agrees with the stair total.
Changing the grid width
When changing the grid width then that means that the numbers you are going to pick will also change. This is because you are compressing the grid to make it smaller or larger. If changing the grid width then that will change the position of the numbers it generates in the grid.
The formula will change if changing the grid width because when expanding or compressing the grid of numbers. You are choosing different numbers each time. So this in turn will give a different formula.
St(X) = X+X+1+X+2+X+10+X+11+X+20
= 6X+44
Finding a Stair Total Formula for any grid width
I am now going to show what will happen if I change the grid width.
I arrived at the below stair shape by noticing that every time you go up a row an extra W is added and every time going across a column an extra 1 is added.
St(X,W,3) = X+X+1+X+2+X+W+X+W+1+X+2W
= 6X+6+4W
St(X,W,3) = 6X+4+4W
Changing the number of steps, p
Now I am going to see what will happen if I change the number of steps of the stair shape. By this I mean you can add on extra blocks and I want to see how much this effects the formula for the stair total.
St(X,X,1)=X
=X+X+1+X+W
=3X+1+W
St(X,W,2)= 3X+1+W
=X+X+1+X+2+X+W+X+W+1+X+2W
=6X+6+4W
St(X,W,3)= 6X+4+4W
=X+X+1+X+2+X+3+X+W+X+W+1+X+W+3+X+3W+X+2W+1 +X+3W
=10X+16+10W
St(X,W,4)=10X+10+10W
=X+X+1+X+2+X+3+X+4+X+W+X+W+1+X+2+W+X+W+3+X+2W+X+2W+1+X+2W +2+X+3W+X+2W+1+X+4W
=15X+20+20W
St(X,W,5)
Now I am going to put all of the above information into a table, by doing this I can see what patterns develop by adding on blocks to the stair shape.
Triangular Numbers
From the above stair total I have noticed that there is triangular numbers when increasing the stair total. I am now going to symbolize the triangular numbers.
=
=
=
=
=
To try and workout the next set of triangular numbers I am going to try and find a formula to do this. The following formula below is the formula to work out the next set of triangular numbers.
T(p)= PxP+1
2
T(p)= P(p+1)
2
I derived the formula for T(p) by using the table below
Redraw the above table using the notation
Translation
The translation is .
St(X,W,3,2,0)=6X+4W+16
There is an extra 12
The translation is.
St(X,W,3,0,3)=6X+22W+4
There is an extra 18W
The translation is.
St(X,W,3,2,3)=6X+22W+16
There is an extra 18W+12
The translation is.
St(x,w,3,a,0)=6X+4W+6a+4
There is an extra 6a
When translating by (a,0) each block moves across by ‘a’ units
therefore you have to add an ‘a’ to the stair shape.
As the formula is different by 6a this has appeared as you are moving each block by ‘a’ then there will be 6 blocks you are moving.
The translation is.
St(X,W,3,b,0)=6X+4bW+4W+4
There is an extra 6bw
Every time the block moves up by 1 unit you add 1W. As you are moving up by ‘b’ units then you add bW. If you go up by ‘b’ then you are going up by bW.
The translation is.
St(X,W,3,a,b)=6X+6bW+4W+6a+4
There is a difference of 6bW and 6a
The reason for the difference is that as you move up by 1 unit upwards then you have an extra bW, as you are going across by 1 unit 6 times then there is an extra 6a’s