Number Stairs

        2

Fig. 2

For example for a 2-step stair the formula is always 3x plus something, therefore to find the co-efficient of x, in this case 3, I would substitute the step size, in this case 2, into the triangular formula:

2(2+1)           =        4+2        =        6        =        3

              2                  2                2

Therefore this equation works for a 2-step stair as the end result was 3. I then went on to check if this formula worked with other sized stairs. Another example I used was a 3-step stair for which the formula is always 6x plus something. Once again I substituted in the step size into the triangular formula to check if it was correct:

        3(3+1)          =        9+3        =        12        =        6

        2        2        2

Therefore this formula is correct for calculating the co-efficient of x so the first part of my equation shall be:

[ n(n+1) ] x

        [            2     ]

Where n = step size

        x = number in the bottom left corner of the stair

Now I have found out how to calculate the co-efficient of x using a formula I shall look into finding a formula to calculate the end part of my original formulas:

eg. for a 3-step stair on a 10×10 grid: 6x + 44 or 6x + 4g + 4

I will keep the equation in the simpler form where there is an addition of just a number and put this information into tables to make it clearer to notice a pattern (Tables 3 and 4). I shall call this number I am trying to find y. Firstly I created the following tables:

Table 3

As grid size increases by 1, y increases by 0

As grid size increases by 1, y increases by 1

As grid size increases by 1, y increases by 4

As grid size increases by 1, y increases by 10

As grid size increases by 1, y increases by 20

Table 4

Multiples of 9

Multiples of 10

Multiples of 11

Multiples of 12

(Both tables show the same information, just laid out differently.)

I then went on to work out the equation by finding the nth term, using my y values for an 8×8 size grid. I took all the y values for an 8×8 size grid and set them out as follows to find the nth term of the number pattern.

1st difference             +9         +27        +54        +90

2nd difference        +18                     +27                 +36

3rd difference        +9        +9

From this I can see that the nth term will be cubic as it is the 3rd difference that is the same. This means my nth term will use the formula:

an³ + bn² + cn + d

I then carried on to find out the values for a, b, c and d for my cubic equation:

When n = 1

a + b + c + d = 0                1

When n = 2

8a + 4b + 2c + d = 9                2

When n = 3

27a + 9b + 3c + d = 36        3

When n = 4

64a + 16b + 4c + d = 90        4

Using these equations I then subtracted equation 1 from equation 2:

7a + 3b + c = 9                1

I then subtracted equation 2 from equation 3:

19a + 5b + c = 27                2

I then subtracted equation 3 from equation 4:

37a + 7b + c = 54                3

Now using these 3 new equations:

2 – 1:

12a + 2b = 18                        1

3 – 2:

18a + 2b = 27                        2

I then subtracted equation 1 from equation 2 to find the value of a:

6a = 9

Therefore a = 9

          6

Now I can substitute a into my previous equations to find the value of b:

Sub. a in 1:

9 + b + c + d = 0

6

        b + c + d = -9                1

                        6

Sub. a in 2:

12 + 4b + 2c + d = 9

        4b + 2c + d = -3                2

Sub. a in 3:

 243 + 9b + 3c + d = 36

   6

        9b + 3c + d = -27        3

        6

Sub. a in 4:

96 + 16b + 4c + d = 90

        16b + 4c + d = -6        4

Then I did:

2 – 1

3b + c = -9                        1

         6        

3 – 2

5b + c = -9                        2                

        6

4 – 3 

7b + c = -9                        3        

           6

Then I subtracted equation 1 from equation 2:

2b = 0

Therefore b = 0

Now I can substitute b into my previous equations to find the value of c:

Sub. b in 1:

0 + c + d = 0

      c + d = 0                        1

Sub. b in 2:

0 + 2c + d = -3

      2c + d = -3                        2

Sub. b in 3:

0 + 3c + d = -27                3

        6

I then did:

2 – 1:

c = -9                        

       6

Now I can substitute c into my previous equations to find the value of d:

Sub. c in 1:

-9 + d = -9

 6        6

Therefore d = 0

From all of my workings to find the nth term I found out that in my formula;

an³ + bn² + cn + d

a = 9        b = 0                c = -9                 d = 0

          6                                        6

leaving me with a final cubic formula of:

  9 n³ - 9 n

        6              6

This formula was to find the y value in my original formulas therefore I need to combine it with my formula for finding the co-efficient of x:

[ n(n+1) ] x + [( 9 ) n³ - ( 9 ) n ]

                                  [      2     ]       [( 6 )        ( 6 )     ]       

Where n = step size and x = number in the bottom left corner of the stair

This formula is for any size stair but on an 8×8 grid only. I know this because from previous work I have done on finding the nth term for cubic equations I realised that in the cubic equation, an³ + bn² + cn + d, a always equals the 3rd difference divided by 6., so in this case the 3rd difference is 9, making a equal 9 over 6. As my y values are different for each sized grid, they would all have a different number for their 3rd difference, meaning a different number divided by 6. From Table 4, which I created earlier, I noticed that in the example I used to find the nth term of, the 3rd difference for my y values was 9 and that the y values for an 8×8 grid are always a multiple of 9. The y values on other sized grids were also always multiples of the same number. On a 9×9 grid, the y values were always a multiple of 10, 10×10 grid were multiples of 11, and 11×11 grid were multiples of 12. I can see from this that they are always multiples of the grid size plus 1 (g + 1). As I noticed earlier that for my y values on an 8×8 grid the 3rd difference was 9 which was also the number that all the y values are multiples of, I went on to see if the 3rd difference for the y values on a different sized grid was the same as the number they were all multiples of. I checked if the 3rd difference for the y values on a 9×9 grid was 10 as the y values for a 9×9 grid are all multiples of 10:

1st difference        +10        +30        +60        +100        

2nd difference        +20        +30        +40

3rd difference        +10        +10

This was the case, therefore the 3rd difference, which is also the number divided by 6 in my equation is always grid size plus 1, therefore in my equation would be:

[ n(n+1) ] x + [ ( g + 1 ) n³ - ( g + 1 )n ]      

                            [    2      ]        [ (            6    )        (    6     )  ]

Where n = step size, x = number in the bottom corner of the stair and g = grid size

I then simplified this equation:

[ n(n+1) ] x + [ g + 1( n³ - n ) ]

                                   [     2      ]       [          6                   ]

This is the formula for any sized step stair on any size grid.

I then went on to try this formula on a different size step stair on a different size grid to prove and check that it works. The example I used was a 3-step stair on a 10×10 grid with the number in the bottom left of the step shape being 25. The stair total for this step stair should equal 194. I substituted my example into my formula to calculate the stair total to see if this answer was given:

[ n(n+1) ] x + [ g + 1( n³ - n ) ] = Stair Total

                        [      2     ]       [    6                          ]

[ 3(3+1) ] 25 + [ 10 + 1( 3³ - 3 ) ] = Stair Total

                      [       2    ]         [      6                          ]

[ 9+3 ] 25 + [ 11 (27 - 3 ) ] = Stair Total

                            [   2   ]         [  6                        ]

[ 12 ] 25 + [ ( 11 )24 ] = Stair Total

                               [  2  ]          [ (  6  )     ]

150 + 44 = Stair Total

194 = Stair Total

This proves that my formula works as it produced the correct stair total for a 3-step stair on a 10×10 grid.