Number Stairs Investigation – Course Work

Aim

The aim of this coursework is to find relationships and patterns in the total of

all the numbers in ‘Number Stairs’ such as the one below. For example, the total

of the number stair shaded in black is:

25+26+27+35+36+45 = 194

I have to investigate any relationships that might occur if this stair was in a

different place on the grid.

Part One

For other 3-stepped stairs, investigate the relationship between the stair total

and the position of the stair shape on the grid.

By looking at this grid, the stair total is:

25 + (25+1) + (25+2) + (25+3) + (25+10) + (25+11) + (25+20)

If we call 25, the number in the corner of the stair, ‘n’ then we get:

n + (n + 1) + (n + 2) + (n + 3) + (n + 10) + (n + 11) + (n + 20) =

6n + 1 + 2 + 3 + 10 + 11 + 20 =

6n + 44

This formula should work with every number stair that can fit onto a 10 by 10

grid. I can say the total for square n (Tn) is, “n + n + 1 + n + 2 + n + 3 + n +

10 + n + 11 + n + 20”, because where ever n is on the grid, the number of the

square:

·        One place right of it will be n +1

·        Two places right of it will be n +2

·        6 x 23 + 44 = 138 + 44 = 182

If n = 34 then

·        6 x 45 + 44 = 270 + 44 = 314

If n = 56 then

·        6 x 56 + 44 = 336 + 44 = 380

If n = 67 then

·        6 x 67 + 44 = 402 + 44 = 446

If n = 78 then

·        6 x 78 + 44 = 468 + 44 = 512

Stair Number (n)        1        12        23        34        45        56        67        78

Stair Total (Tn)        50        116        182        248        314        380        446        512

6n + 44        50        116        182        248        314        380        446        512

All the results using the formula are correct, so I can come to the conclusion

that the formula for the total of a stair:

·        On a 10 by 10 grid

·        Which travels downwards from left to right

·        With a height of 3 squares

Is:

6n + 44

Part Two (Extension)

Investigate further the relationship between the stair total and other step

stairs on the number grids.

I am going to investigate patterns and relationships in:

·        The position of the stairs

·        The height of the stairs

·        The width of the grid

I will then put everything together and produce a universal formula.

Throughout this section, the symbols for the variable inputs will be as follows:

Tn = Total of stair

n = Stair number (the number in the bottom left had corner of the stair)

h = Number of squares high the stair is

w = Width of grid (number of squares)

Relationships between different stair heights on a 10 by 10 grid

3rd triangle would multiply n). The formula for triangle numbers is:

h2 + h

2

So the first part of the formula will be:

n (h2 + h)

   2

If I include this in a table, I may get some better results:

Height (h)        1        2        3        4        5        6

Stair Number (n)        1        1        1        1        1        1

Triangle Number of h        1        3        6        10        15        21

Triangle Number of h x n        1        3        6        10        15        21

11h3 – 11h        0        66        264        660        1320        2310

(11h3 11h)ק6        0        11        44        110        220        385

(Triangle Number of h x n) + (11h3 11h)ק6        1        14        50        120        235        406

Total (T1)        1        14        50        120        235        406

I included 11h3 11h from my last table. I divided 11h3 11h by 6. This is

because 6 was the number that got 11h3 11h closest to the required total. I

noticed that if you add the Triangle Number of h x n and (11h3 11h)ק6

together you get the Total. So my formula for any height Number stair on a 10 by

10 grid is:

n (h2 + h) + (11h3 11h)

2                 6

Testing the formula

Tn = Total of stair

n = Stair number (the number in the bottom left had corner of the stair)

h = Number of squares high the stair is

w = Width of grid (number of squares)

Test One – Square 25, Height 3

Total = 25 + 26 + 27 + 35 + 36 + 45 = 194

T25 = 25 x ((32 + 3) ק 2) + ((11 x 33 11 x 3) ק 6)

T25 = 25 x (11 ק 3) + (297 33) ק 6

T25 = 25 x 6 + 44

T25 = 194

Test Two Square 68, Height 3

Total = 68 + 69 + 70 + 78 + 79 + 88 = 194

... thats all ive don