N+20
N+10 N+11
N N+1 N+2
I can now work out a general by adding together all the N’s:
N+N+N+N+N+N= 6N
And then adding that to all the other numbers added together:
10+20+11+1+2=44
There for the algebraic rule is 6N+44
I can prove this by using the S-number 1. If you put 1 in the formula it is 6x1+44, which gives us 50. If you add 1+2+3+11+12+21 it also gives you 50, so the rule works.
I will prove it again with another example. If you put the S-number 2 into the rule it gives you 56. If you add 2+3+4+12+13+22 it also gives you 56. So the rule works.
I am now going to find general rules for different sized step stares. I will then try and find a link between step stair size and the S-total.
2x2 step stair
In this example the S-total is 14, as 1+2+11=14. I am now going to find how the S-number links with the rest of the numbers in the step stair (we will call the S-number N).
N+10
N N+1
I am going to find the general rule as I did previously:
This will give me 3N+11.
I can prove this by using the example S-number 1. Using the rule the S-total comes to 14, by adding all of the numbers within the step stair it also comes to 14. Therefore the rule works.
I can prove this again by using the S-number 2. Using the rule this gives me 17, by adding up all the numbers within the step stair this also gives me 17. Therefore the rule works.
4x4 step stair
In this example the S-total is 120 as 1+2+3+4+11+12+13+21+22+31=120. I can now try and find a link between the S-number and the rest of the numbers within the step stair.
N+30
N+20 N+21
N+10 N+11 N+12
N N+1 N+2 N+3
Using the previous method I can determine that the rule is
10N+110
I can prove this by using the first example. Using the rule the S-total is 120, adding all the numbers within the step stair also gives you 120. So the rule works.
I am now going to find a relationship between grid size and the S-total. I am going to use a standard 3x3 grid.
11x11 Grid
I am going to find the rule like I have done in the previous exercises for several grid sizes. I am now going to draw a table of my results:
