Investigating Number Stairs

So we get 6S+ 47 for 11x11 and 6S+49 for 12x12

Since we have the formulae for 10x10, 11x11 and 12x12, we can make a general formula for finding the stair total for a 3-step stair for any grid size. We can do this since it increases in a linear fashion (+2 every time).

We get the formulae: 6S (S as the stair number) + 4G (G as the grid number) + 4

Test: (6 x 25) + (4 x 10) + 4 = 194 (this works with 10x10 and hence works since the total is 194)

Test 2: (6x25) + (4x11) + 4= 198 (We check this by replacing the algebra again: 25+26+27+36+37+47 = 198 therefore OK!)

Now, we can try changing the step stair size. Starting from 1 and ending at 6, we change the grid size and see whether there is a relationship between them.

1x1 step stair:

The formula is quite simply S + 0G for whatever grid size you use

2x2 step stair

For 10x10 grid:                        For 11x11 grid:                For 12x12 grid        

                

3S + 11                        3S + 12                        3S + 13

So Formula is 3S + G + 1

Test: (3x25) + 10 + 1 = 86, 25+26+35 = 86 So OK!

4x4 step stair:

For 10x10                        For 11x11:                        For 12x12:

Formula is 10S +10G + 10

Test: (10x25) + (10x10) + 10= 360, 25+26+27+28+35+36+37+45+46+55=360 So OK!

5x5 step stair:

For 10x10                        For 11x11                          For 12x12

Formula is 15S + 20G + 15

Test: (15x25) + (20x10) +20 = 595

25+26+27+28+29+35+36+37+38+45+46+47+55+56+65=595 so OK!

6x6 step stair:

For 10x10                               For 11x11

Formula: 21S + 35G + 35

Test: (21 x 25) + (35x10) + 35 = 910 25+26+27+28+29+30+35+36+37+38+39+45+46+47+48+55+56+57+65+66+75=910

Now, we can form the following table showing the formula:

So now, we can find a Universal Formula™ to find the general formula for any sized step for any sized grid given we have the stair number.

The stair number seems to increase in the following fashion:

1 (+2) 3 (+3) 6 (+4) 10 (+5) 15 (+6) 21

From this information, we know that it is a triangle number. We can use the triangle number formula here: (S2 + S)/2  

For the Grid Number, it is somewhat related to the Stair number and Grid number before it. For example: The Grid number of stair size 5 is 20, which is also stair size 4’s stair number (10) and grid number (10) added together.

With the table above, we find the differences between each number, so that we can find the formula easier (since it involves many calculations, it will be shown on the next page).

As you can see, the Stair number and the grid number before the current grid number that you want to find.

That rule works for every other grid number. Somehow, the extra added on number seems to work this way also. We try to find a formula:

G1 = n

G2 = n + n

G3 = n + n + [n+(n+1]

G4 = n + n + [n+(n+1)] + [n+(n+1)+(n+2)]

G5 = n + n + [n+(n+1)] + [n+(n+1)+(n+2)] + [n+(n+1)+(n+2)+(n+3)]

.           .

.           .

.           .

Gn = Gn-1 +[n+(n+1)+…+(n+(n-2)]

So therefore, the equation is:

[(S2 + S)/2] + 2[ Gn-1 +[n+(n+1)+…+(n+(n-2)] ]