I know these are the formulas for each grid size using a 3 step stair.
So the overall formula for a 3 step stair on any sized grid =
g x g = 6n + 4g + 4
I am now going to investigate different sized step stairs on different sized grids.
I will now try to find a formula for any 2 step stair on any sized grid.
2 step stair on a 10x10 grid
There is a difference of 3 so the formula must start 3n. I also know this as there are 3 steps in the stair. 1 x 3 = 3. 14 – 3 = 11.
2 x 3 = 6. 17 – 6 =11
The formula for a 2 step stair on a 10x10 grid = 3n + 11.
2 step stair on a 9x9 grid
Formula for a 2 step stair on a 9x9 grid = 3n + 10
2 step stair on a 8x8 grid
Stair number x 3 = 3. 12 – 3 = 9 so the formula for a 2 step stair on a 8x8 grid = 3n + 9
I can now see a relationship between the formulas. I can see that the number you add on to the three n decrease one when the grid size gets smaller. I can put my findings into a table.
The number you add on is 1 less than the grid size. This allows me to use the grid size in the overall formula as you need to add the grid size then add another 1.
The formula for 2 step stair on any grid size = g x g = 3n + g + 1
I will now try to find a formula for any 4 step stair on any sized grid.
4 step stair on a 10x10 grid
There is a difference of 10 between the terms so the first part of the formula must be 10n. Once again the number of steps in the stair is equal to the difference between the terms.
Using the step stair total 120 as my example I will work out the rest of the formula. 1(stair number) x 10 = 10. 120 – 10 = 110.
The formula = 10n + 110.
The grid size = 10 so 10x10 = 100, this means the formula for a 4 step stair on a 10x10 grid = 10n + 10g + 10
4 step stair on a 9x9
4 step stair on a 8x8 grid
4 step stair on a 7x7 grid
To make sure the formula 10n + 10g + 10 works I will try to find out the total for a 4 step stair on a 8x8 grid with 1 as the stair number.
1 x 10 = 10. 10 + 10 x 8 = 90. 90 + 10 = 100. Which is the step stair total for a 4 step stair on a 8x8 grid with 1 as the stair number.
Therefore the formula for any 4 step stair on any sized grid = g x g = 10n + 10g + 10
I will now try to find a formula for any 5 step stair on any sized grid.
5 step stair on a 10x10 grid
There’s a difference of 15 and there are 15 steps in the stair so the first part of the formula must be 15n.
1 x 15 = 15. 235 – 15 = 220 so the formula is 15n + 220
Grid size = 10 so 10 x 10 = 100, 220 – 100 = 120 so the formula =
15n + 10g + 120
5 step stair on a 9x9 grid
To find out if the formula 15n + 10g + 120 works:
Stair no. = 1 so 1 x 15 = 15 + 10 x 9 = 105 + 120 = 225 which is the stair total.
5 step stair on a 8x8 grid
To find out if the formula 15n + 10g + 120 works:
Stair no. = 1 so 1 x 15 = 15 + 10 x 8 = 95 + 120 = 215 which is the stair total.
5 step stair on a 7x7 grid
To find out if the formula 15n + 10g + 120 works:
Stair no. = 1 so 1 x 15 = 15 + 10 x 7 = 85 + 120 = 205 which is the stair total.
The formula for a 5 step stair on any sized grid =
15n + 10g + 120
From the formulas I got I realised they were all triangular numbers. This means for the overall formula I’m trying to find the triangular number formula must be in it. ½ s (s + 1). I will now factorise the formulas to get the triangular number on its own.
10x10 grid
9x9 grid
8x8 grid
7x7 grid
I am now going to make it so that all the numbers after n + can be divided by 3.
10x10 grid
Form here I can see a relationship between the numbers after n. They go up 11 whenever the step size gets bigger. This means the second part of the formula has to have 11s and divided by 3 in it. We already know the triangular formula is ½ s ( s + 1 ) whilst n is always in the other formulas so ½ s (s + 1) (n + 11s and another part make up the formula. The difference between the formulas is 11. This means when I times 11 by s I always need to take away 11 as that is the pattern. So 11 x 2(step size) -11 = 11. I will then divide 11 by 3 and add n.
Formula =
s ½ s (s+1) (n + 11s – 11)
3
I will now test the formula. 3 = step size, n = 1.
0.5 x 3 = 1.5 x 4 =6
11 x 3 = 33 – 11 = 22 divided by 3 = 7.3 + 1 = 8.3 x 6 = 50.
The formula works
The formula is the same for all other step sizes apart from the number you times by s is always one bigger than the grid size which changes.
9x9 grid
s ½ s (s+1) (n + 10s – 10)
3
8x8 grid
s ½ s (s+1) (n + 9s – 9)
3
7x7 grid
s ½ s (s+1) (n + 8s – 8)
3
I will now put all the formulas together in a table which shows the formulas for each grid size.
gxg ½ s (s + 1) (n + ( g + 1 ) x s – ( g + 1 )
3
In the formula now I have the grid size, step size and the step stair number. I represented the grid size in this formula as I knew the totals for each grid size were always one more than the grid size. I represented this as g + 1.
Overall Formula = ½ s (s + 1) x (g + 1) x s – (g + 1) + n
3
Conclusion:
From my results I have found a formula that will find the step stair total for any sized step on any sized grid with any step stair number.
Here are two examples to show that my formula works:
If I wanted to find the step stair total for a 3 step stair on a 10x10 grid I can use the overall formula.
Firstly times the step size by 0.5, which = 1.5. Then add the step size to 1 this gives you 4. Times the 4 by 1.5 which gives you 6. Then do the second part of the equation. g + 1 = 11. Times the 11 by s – (g + 1). This gives you 11 x 3 – 11. This equals 22. Divide the 22 by 3 which leaves you 7.3 requiring. As the stair number is 1 add this onto 7.3 leaving you with 8.3. Then finally times the 8.3 requiring by 6 which gives you 50 which is the step stair total for a 3 step stair on a 10x10 grid.
I will now find the step stair total of a 2 step stair on a 8x8 grid using the formula.
Overall Formula = ½ s (s + 1) x (g + 1) x s – (g + 1) + n
3
0.5 times 2 = 1
2 + 1 = 3
3 x 1 = 3
9 x 2 – 9 = 9 3 = 3
3 + 1 = 4
4 x 3 = 12. The step stair total for a 2 step stair on a 8x8 grid.
I obtained my results by working through different stages. Firstly I found the formula for a 3 step stair on a 10x10 grid; I then found the formula for a 3 step stair on different sized grids. I kept doing this changing the step stair size and the grid size as I went on. Then I had all the formulas and introduced the grid size into them along with step size and step stair number. I could do this as I found the formulas all started with the triangular formula. All I had to do then was factorise the formulas so they all related i.e. I divided them all by 3. This enabled me to combine the triangular formula with the formula I had found from my findings to make an overall formula.
