Number Stairs Investigation – Course Work

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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

·        One place above it will be n +10

·        One place above it then one place to the right will be n + 10 + 1 = n + 11

·        Two places above it will be n + 20

So to get the total you multiply ‘n’ by six then add 1, 2, 10, 11 and 20 which

gives you a general formula of 6n + 44.

I can test my theory by making doing the following:

First I need to make a table giving a sample of results found by adding each

stair up, square by square:

Stair Number (n)        1        2        3        4        5        6        7        8

Stair Total (Tn)        50        56        62        68        74        80        86        92

I then need find the difference between the totals:

Total        50        56        62        68        74        80        86        92

First difference              6               6               6               6               6               6               6

Join now!

Because there is an instant pattern in the first difference – the totals go up

in 6, I need find out what 6n would come to:

Stair Number (n)        1        2        3        4        5        6        7        8

6n        6        12        18        24        30        36        42        48

6n doesn’t give me the correct answer so I add 44 to 6n and get the Stair Total:

Stair Number (n)        1        2        3        4        5        6        7        8

Stair Total (Tn)        50        56        62        68        74        80        86        92

6n        6        12        18        24        30        36        42        48

+ 44        44        44        44        44        44        44        44        44

6n + 44        50        56        62        68        74        80        86        92

Testing the Formula

To ensure the formula works in every case, I tested it in eight situations where

‘n’ was different each time.

Using the formula 6n + 44

If n = 1 then

·        6 x ...

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