Un = 6n + 44
n = stair number
If I’m correct I believe this formula will be able to find any other 3 – step stair vertically on the 10 x 10 grid.
Therefore to find the total for stair number 91 I will simply use the formula, which will be U91 = 6*91 + 44 and the answer is 500.
Diagonally
Un = 6n + 44
n = stair number
If I’m correct I believe this formula will be able to find any other 3 – step stair diagonally on the 10 x 10 grid.
Therefore to find the total for stair number 100 I will simply use the formula, which will be U100 = 6*100 + 44 and the answer is 644.
I have compared the formulae for the 3 – step stairs horizontally, vertically and diagonally which are all the same. From this I have found a new and easier formula to use using the 3 – step stair shape which when added makes the same formula as 6n + 44.
4 – Step stairs and 5 – Step stairs
I will now be continuing my investigation by doing 4 and 5 – step stairs horizontally on the 10 x 10 grid as the formula will be the same for vertically and diagonally as seen in the 3 – step stairs. I will then compare them to 3 – step stairs see if I can find an overall formula.
4 – Step Stairs horizontally
Un = 10n + 110
n = stair number
If I’m correct I believe this formula will be able to find any other 4 – step stair horizontally, vertically and diagonally on the 10 x 10 grid.
Therefore to find the total for stair number 10 I will simply use the formula, which will be U10 = 10*10 + 110 and the answer is 210.
5 – Step stairs horizontally
Un = 15n + 220
n = stair number
If I’m correct I believe this formula will be able to find any other 5 – step stair horizontally, vertically and diagonally on the 10 x 10 grid.
Therefore to find the total for stair number 10 I will simply use the formula, which will be U10 = 15*10 + 220 and the answer is 370.
Un = [s(s+1)/2]n + [(s-1)*s*s(+1)/6]*11
If I’m correct I believe this formula will be able to find any other step stair horizontally, vertically and diagonally on the 10 x 10 grid.
Therefore to find the total for stair number 10 for a 10 – step stair I will use the formula, which will be U10 = [10(10+1)/2]n + [(10-1)*10*(10+1)/6]*11 and the answer is 715.
Grid size for 3 – step stairs
Using g as the grid size I will see whether there is a formula that links the 3 – step stairs together with the size of the grid.
9 x 9 grid
The formula for the 9 x 9 grid for 3 – step stairs is Un = 6n + 40
n = stair number
Therefore to find the total for stair number 9 I will simply use the formula, which will be U9 = 6*9 + 40 and the answer is 94.
8 x 8 grid
The formula for the 8 x 8 grid for 3 – step stairs is Un = 6n + 36
n = stair number
Therefore to find the total for stair number 8 I will simply use the formula which will be U8 = 6*8 + 36 and the answer is 84.
I will now compare the 8 x 8 grid to the 9 x 9 and 10 x 10 to see whether there is a formula connecting them with the 3 – step stairs.
The overall formula for the grid size and the 3 step stairs horizontally, vertically and diagonally is Un = 6n + 4g + 4
n = stair number
g = grid number
Therefore to find the total for stair number 5 on a 5 x 5 grid I will simply use the formula U5 = 6*5 + 4*5 + 4 and the answer is 54.
From the comparison between the grid size and position of 3 - step stair, I have found a new and easier formula to use using the 3 – step stair shape which when added makes the same formula as 6n + 4g + 4.
Grid size for 4 – step stairs
9 x 9 grid
The formula for the 9 x 9 grid for 4 – step stairs is Un = 10n + 100
n = stair number
Therefore to find the total for stair number 9 I will simply use the formula, which will be U9 = 10*9 + 100 and the answer is 190.
8 x 8 grid
The formula for the 8 x 8 grid for 4 – step stairs is Un = 10n + 90
n = stair number
Therefore to find the total for stair number 8 I will simply use the formula, which will be U8 = 10*8 + 90 and the answer is 170.
The overall formula for the grid size and the 4 step stairs horizontally, vertically and diagonally is Un = 10n + 10g + 10
n = stair number
g = grid number
Therefore to find the total for stair number 6 on a 6 x 6 grid I will simply use the formula U6 = 10*6 + 10*6 + 10 and the answer is 100.
Grid size for 5 – step stairs
9 x 9 grid
The formula for the 9 x 9 grid for 5 – step stairs is Un = 15n + 200
n = stair number
Therefore to find the total for stair number 9 I will simply use the formula, which will be U9 = 15*9 + 200 and the answer is 335.
8 x 8 grid
The formula for the 8 x 8 grid for 5 – step stairs is Un = 15n + 180
n = stair number
Therefore to find the total for stair number 8 I will simply use the formula, which will be U8 = 15*8 + 180 and the answer is 300.
The overall formula for the grid size and the 5 - step stairs horizontally, vertically and diagonally is Un = 15n + 20g + 20
n = stair number
g = grid number
Therefore to find the total for stair number 7 on a 7 x 7 grid I will simply use the formula U7 = 15*7 + 20*7 + 20 and the answer is 265.
Un = [s(s+1)/2]n + [(s-1)*s*(s+1)/6]g + [(s-1)*s*(s+1)/6]
Pascal's Triangle
Triangular numbers – 1 , 3 , 6 , 10 , 21 , 28 ……
Un = n*(n+1)/2
Tetrahedral numbers – 1 , 4 , 10 , 15 , 35 , 56 ……
Un = n*(n+1)*(n+2)/6
Hyper tetrahedral numbers – 1 , 5 , 15 , 35 , 70 ……
Un = n*(n+1)*(n+2)*(n+3)/24
I have realised that our formula involves triangular and tetrahedral numbers along with hyper tetrahedral number, which can be found on Parcel’s triangle. Their relationship with one another is triangle number have a difference of square numbers, tetrahedral number have a difference of triangle numbers and hyper tetrahedral number have a difference of tetrahedral numbers.
