Number Stairs.

The stair total for this stair shape is 21 + 22 + 23 + 31 + 32 + 41 = 170.

Now we are going to find the stair total for stair31 (a translation of square21 one square up).

The stair total for this stair shape is 31 + 32 + 33 + 41 + 42 + 51 = 230.

We can compare the stair totals for stairs 1, 11, 21, and 31 in order to find a pattern. If we look at their stair totals it is easy to see that as the stair shapes move one square up on the grid the stair total increases by 60. This is shown in the diagram below.

Stair Number:  1                         11                        21                        31

Stair total:      50                          110                        170                        230                

Difference:                    60                         60                               60

I can now predict that the stair total for stair 41 is going to be 230 + 60=290.

Here is the stair to prove my prediction is correct.

The stair total for this shape is 41 + 42 + 43 + 51 + 52 + 61 = 290.

We can therefore say that every time you move the shape one square down the total decreases by 60 and when you move one square up the grid then total increases by 60. Also when you move one square to the right of the grid the total increases by 6 and when you move one square to the left the total decreases by 6.

The reason the total increases by 60 when you move the shape one square up on the grid is because since there are 6 squares in a 3-step stair and each separate number increases by 10. 6 multiplied by 10 equals 60.

The reason the total increases by 6 when you move the shape one square to the right on the grid is because there are 6 squares in a 3-step stair and each separate number increases by 1. 1 multiplied by 6 equals 6.

We can now introduce algebra in order to find a common formula to find the stair total for and 3-step stair on a 10 by 10-number grid.

                                                     

If this stair is stairx then the numbers in the stair will be

(x) + (x+2) + (x+3) + (x+10) + (x+11) + (x+20)

This can also be written as 6x + 44 = stair total

Using stair7 we are now going to use my formula to find out its stair total and see if my formula is correct.  x is equal to 7 as 7 is the number in the bottom left hand corner of stair7 and x is the number in the bottom left hand corner of stair x.

      6x + 44 = stair total

(6 x 7) + 44 = 86

So my formula says that the stair total of this stair is 86.

Now we can test this.

7 + 8 + 9 + 17 + 18 + 27 = 86

This is true. Therefore we can now say that the general formula for any 3-step stair on a 10 by 10-number grid is 6x + 44 = stair total.

We are now going to investigate further the relationship between stair totals and other step stairs on other number grids.

The numbers inside the stair and the formula for finding the stair total varies depending on the grid size:

For a 10 by 10 Grid :

But for a 9 by 9 Grid :

We can see that the formula for a 9 by 9 grid is 6x + 40 = stair total. This is 4 less than the formula for the 10 by 10 grid.

In order to prove this I will use stair20 on a 9by 9 grid.

               6x + 40 = stair total

      (6 x 20) + 40 = 160

20 + 21 + 22 + 29 + 30 + 38 = 160

I can therefore conclude that the general equation for a 3-step stair on a 9 by 9 number grid is 6x + 40 = stair total where x is the number in the bottom left hand corner of the stair shape.

We can now say that in order to find the equation for an 11 by 11 number grid all we have to do is add 4 to the equation of a 10 by 10 grid. If we do this then the equation will be 6x + 48 = stair total.

I am now going to check this equation with stair 12 on an 11 by 11 number grid.

        6x + 48  = stair total

(6 x 12) + 48 = 120

12 + 13 + 14 + 23 + 24 + 34 = 120

This is true so I conclude that the stair total for any 3-step stair on an 11 by 11 number grid is 6x + 48= stair total.

We can now say that every time you increase the size of the grid by one (going from a 9by9 number grid to a 10by10 number grid) you have to add four more to the equation. Therefore if you decrease the size of the grid by one you must also decrease the equation by 4.

With this knowledge I can now find a general equation for any 3-step stair on any size grid.

Grid Size                Formula

7 by 7                         6x + 32

8 by 8                         6x + 36

9 by 9                         6x + 40

10 by 10                 6x + 44

11 by 11                 6x + 48

So from the results table I can see a linear pattern which confirms my theory. Every time you increase the grid size by 1 you increase the formula by 4.

Here is the diagram that illustrates the equation for any 3-step stair on any size grid:

g = grid size                x = stair number

x+x+x+x+x+x = 6x

g+g+2g = 4g

1+1+2 = 4

stair total = 6x + 4g + 4

In order to prove that my formula is correct I am going to test it by using it to find the stair total for stair17 on a 10 by 10 size grid.

X=17                  g=10

6x + 4g + 4 = stair total

6(17) + 4(10) + 4 = 146

17 + 18 + 19 + 27 + 28 + 37 = 146

This is correct. Therefore I conclude that the general equation for finding a 3-step stair on any sized grid is 6x + 4g + 4 = stair total.

I have 6x as there are always 6 x in any 3 step stair

I add 4g as this is the number found when we increase the grid size by one.

Finally I add 4 to these two numbers because 1 + 1 + 2 equals 4.

Also every time you move one square to the right you increase the stair total by 1 and every time you move one square down up you increase the total by g.

Changing The Size Of The Stair

2-Step Stair

3x +1+g = stair total is the equation for any 2–step stair on any size grid.

3-Step Stair

stair total = 6x + 4g + 4 is the equation for any 3-step stair on any size grid.

4-Step Stair

Stair total = 10x + 10g + 10 is the equation for any 4-step stair on any size grid.

5-Step Stair

The equation for any 5-step stair on any size grid is

15x + 20g + 20 = stair total.

6-Step Stair

21x + 35g + 35 = stair total is therefore the equation for a 6-step stair on any size grid.

Now that I have found these equations I can see three sets of sequences forming that will help to form the final equation for any size stair on any size grid.

It is clear that these three sets of sequences are the co-efficient in front of x sequence, the coefficient in front of g sequence and the constant sequence. I am now going to investigate each separate sequence in order to find the general equation for the coefficient of x sequence, the general equation for the co-efficient of g sequence and the general equation for the constant sequence. The final product will be an equation in the following form :

(Section 1)x + (Section2)g + (Section3)

Section 1: The Coefficient In Front Of x Sequence

Main Sequence=  3        ,        6        ,        10        ,        15        ,        21

1st Difference=                   3        ,        4        ,        5        ,        6

2nd Difference=                        1        ,        1        ,        1

 As this sequence has a constant number at the 2nd difference we can easily say that this is a quadratic sequence. Therefore the first part of the equation is going to be :

2nd difference / 2 n2 =  1/2 n2

We can now use this value with different values of n :

n=                1        ,        2        ,        3        ,        4        ,        5

1/2 n2 =        0.5        ,        2        ,        4.5        ,        8        ,        12.5

Now that we know the values of n for 1/2 n2  all we have to do is subtract it from the original values of  n in the quadratic sequence.

Original Sequence=  3        ,        6        ,        10        ,        15        ,        21

1/2 n2 =                0.5        ,        2        ,        4.5        ,        8        ,        12.5

                        2.5        ,        4        ,        5.5        ,        7        ,        8.5

                                1.5        ,        1.5        ,        1.5        ,        1.5

        

We can now conclude that the co-efficient in front of n will be 1.5n or 3/2n.

Since we now know now the beginning of this sequence formula to be                 1/2 n2  + 3/2n we can now find the constant for this sequence.        

Original Sequence=  3        ,        6        ,        10        ,        15        ,        21

1/2 n2  + 3/2n =                    2     ,        5        ,        9        ,        14        ,        20

Result         =                1        ,        1        ,        1        ,        1        ,        1

Looking at these results we can therefore say that the constant for this sequence is 1.

We can now say that the nth term for this quadratic sequence is :

1/2 n2  + 3/2n + 1 = co-efficient of x.

Section 2: The Co-Efficient In Front of g Sequence

        The coefficient in front of g has the following sequence:

Main Sequence=  1        ,        4        ,        10        ,        20        ,        35

1st Difference=                   3        ,        6        ,        10        ,        15

2nd Difference=                        3        ,        4        ,        5

3rd Difference=                                1                1

Because this sequence has a constant number at the 3rd difference it is easy to see that it is a cubic sequence. If this sequence is cubic it will follow this formula :

                                an3 + bn2 + cn + d

If n = 1        ,        2        ,        3        ,        4        ,            5

     

 a+b+c+d        ,    8a+4b+2c+d        , 27a+9b+3c+d     ,   64a+164+4c+d ,        125a+25b+5c+d

               

 7a+3b+c        ,     9a+5b+c        ,   37a+7b+c        ,         61a+9b+c

           12a+2b        ,      18a+2b        ,    24a+2b

                        

          6a        ,        6a        

I am now going to match the numbers in the cubic sequence formula above to the numbers that correspond to their positions in the co-efficient in front of g sequence.

6a = 1

a   = 1/6        

12a+2b = 3

12(1/6) + 2b = 3

2 + 2b = 3

2b = 1

b = 0.5

7a+3b+c= 3

7(1/6)+3(1/2)+c = 3

c= 1/3

a+ b + c + d = 1

1/6 + ½ + 21/3 + d = 1

d = 0

Therefore the equation for this part of the sequence will be :

(1/6 n3 + 1/2n2 + 1/3n) = coefficient of g

Section 3 : The Sequence For The Constant

The sequence for the constant will go exactly the same as the sequence for the co-efficient of g as all these values are the same. Therefore the equation for this part of the sequence will be :

                 (1/6 n3 + 1/2n2 + 1/3n) = the constant

Conclusion

 Now that we have finished the three parts of the sequence we can put them together in order to make one equation for any size step-stair on any size grid.

(1/2 n2  + 3/2n + 1)x + (1/6 n3 + 1/2n2 + 1/3n)g + (1/6 n3 + 1/2n2 + 1/3n)= stair total

        x means the number of squares in the stair shape.

g stands for grid size.

n is the equation number. ( If you don’t know the equation number it is the same number as the number of squares in the 2nd column from the left of the stair shape).