There is an obvious relationship between the answers I have got again for different size grids therefore I should now attempt to write a formula for a 5 by 5 step in any size grid.
10 by 10→ 15N + 220
9 by 9 → 15N + 200
8 by 8 → 15N + 180
7 by 7 → 15N + 160
6 by 6 → 15N +?
? Should equal 140. To prove my formula I will check it works in a 6 by 6 grid.
I think that with these four different grid sizes I should be able to construct a suitable formula for any 6 by 6 step stair in any grid.
10 by 10→ 21N + 385
9 by 9 → 21N + 350
8 by 8 → 21N + 315
7 by 7 → 21N + 280
6 by 6 → 21N +?
? Should equal 140. To prove my formula I will check it works in a 6 by 6 grid.
I now have a reasonable amount of results. I can now calculate any of the flowing steps in any grid: 3 step
4 step
5 step
6 step
What I need to think about is how to continue my investigation even further. In order to calculate a final formula for any step stair in any grid I will need to calculate a one step stair in any grid and a two so than I can begin looking for a relationship between the final formulas.
The formula to calculate triangle number is n (n+1)/2. e.g. 3 (3+1)/2 = 6 which is the third triangular number. I need this formula to then be multiplied by N which is the corner number. So I have decided to call the corner number X instead of N this way I can use the simplified formula for triangular numbers. The formula I have constructed is TnX which is a simplified version of N (N+1)/2 x X
So far I have the first par of my formula I now need to think about what is added on to it. I know that my final formula will definitely have (y+1) on the end because this is consistent in all the formulas therefore I need to begin looking at the number before the (y+1).
1 step stair = 1N + 0(y+1)
2 step stair = 3N + 1(y+1)
3 step stair = 6N + 4(y+1)
4 step stair = 10N + 10 (y+1)
5 step stair = 15N + 20(y+1)
6 step stair = 21N + 35(y+1)
So far my final formula looks like this:
TnX +? (y + 1)
The numbers are increasing by the sum of the previous triangular numbers. ? Basically is the sum of all the triangular number less than its one. For example if the size of the step is 5 then it’s the 1st triangle number + 2nd the 3rd and the 4th (1+3+6+10) which is 20.
This number is equal to the ? under five.
I could write this as a formula:
(∑Tn < X) (y+1)
This is the final formula for the second part of the investigation I already have the first so if I were to write out the complete formula it would be.
TnX+ (∑Tn < TnX)(y+1) N = the of the shape and X = the corner number Y= size of the grid (But there is another way)
By using the cubic formula I will be able to calculate the relationship between these numbers in red.
1 step stair = 1N + 0(y+1)
2 step stair = 3N + 1(y+1)
3 step stair = 6N + 4(y+1)
4 step stair = 10N + 10 (y+1)
5 step stair = 15N + 20(y+1)
6 step stair = 21N + 35(y+1)
an3 +bn2 +cn+ d
0 1 4 10
1 3 6
2 3
1
a+b+c+d 8a+4b+2c+d 27a+9b+3c+d 64a+16b+4c+d
0 1 4 10
7a+3b+c 19a+5b+c 37a+7b+c
1 3 6
12a+2b 18a+2b
- 3
6a
1
a= 1/6
b= 0
c= -1/6
d= 0
TnX + (y+1)
I now have a final formula which hopefully works all I now need to do is test it. To begin with I will imagine I have a 3 by 3 step stair in a 10 by 10 grid. I know the answer for the formula should be 6x +44.
T3X= 6x
24
= ----- = 4
6
6x+ 4 (10+1) = 6x + 44
If the corner number is one then the answer would be 50 which is the correct answer.
1+2+3+11+12+21=50
