6n = 290 – 44
6n = 246
n = 41
STAIR 25:
: The total for this stair is:
25 + 26 + 27 + 35 + 36 + 45 = 194
Therefore this can be calculated in a similar way:
6 x 25 + 44 = 194
If reversed it would be:
6n + 44 = 194
6n = 194 – 44
6n = 150
n = 25
STAIR 38:
: The total for this stair is:
38 + 39 + 40 + 48 + 49 + 58 = 272
Therefore this can be calculated in a similar way:
6 x 38 + 44 = 272
If reversed it would be:
6n + 44 = 272
6n = 272 – 44
6n = 228
n = 38
STAIR 1:
: The total for this stair is:
1 + 2 + 3 + 11 + 12 + 21 = 50
Therefore this can be calculated in a similar way:
6 x 1 + 44 = 50
If reversed it would be:
6n + 44 = 50
6n = 50 – 44
6n = 6
n = 1
In conclusion, I can now calculate the stair total of a three level stair on any position on the grid 10 by 10 through the general formula 6n + 44.
PART 2:
Investigate further the relationship between the stair total and other step stairs on other number grids.
My objective in this section will be to further investigate the relationship between the stair total and other step stairs on other number grids. To do that I will need to find a general formula through which I can calculate any stair total or stair number on any number grid at any position.
To find the formula I will first and foremost try the same formula in part 1 on a stair shape on an 8 by 8 number grid.
The stair total for this stair will be:
1 + 2 + 9 = 12
If I now use the formula 6n + 44 it will be:
6 x 1 + 44 = 50
This therefore clearly indicates that if we use the
formula 6n + 44 on other number grids then it
will not work.
The main reasons for this are:
- The difference between the upper square and the stair number is of 8.
-
If I had used the same method i.e. by adding up all the numbers and the nth terms then I could have reached a formula for that particular kind of stair but not as a general formula. For this it would have been: n + n + 1 + n + 8 = 3n + 9. Similarly if I had done this on a different stair on the same grid then it would have had a different formula. Therefore I am now going to look for a general formula through which I can calculate any stair total on any grid.
Below is a table showing the different formulae of different stairs on a 10 by 10 grid.
Now I will be experimenting a 5 level stair on a 10 by 10 stair.
The total for this stair is:
: 1 + 2 +3 + 4 + 11 + 12 + 13
+ 21 + 22 + 31 = 120
The following things can be observed from this stair:
- The number stair i.e. 1 is increased by 1 every time moved to the right.
- Every time we go up by 1 square the number increases by 10. For example the difference between 1 and 21 is of 20 which mean that it is 10 x 2 +1. Similarly the difference between 1 and 31 is of 30 which mean that it is 10 x 3 +1.
To calculate the formula the following things can be done:
-
The value of the stair number can be made x i.e. 1.
-
The value of 2 can similarly be written as x + 1. ( 1 +1)
-
The value of 3 can be written as x + 2. ( 1 + 2)
-
The value of 4 can be written as x +3. ( 1 +3)
As mentioned in my observations, the value of x increases by 10 every time we move up by 1. Therefore if it were to be written in terms x it could be written as:
x + 10 (1 +10 = 11)
x + 2(10) (x + 2 x 10 = 21)
x + 3(10) (x + 3 x 10 = 31)
Since there is a fixed value for moving up by 1 i.e. 10 we can give it a specific name which is g. The significance of g will be the change of grids such as 8 by 8 or 9 by 9.
Therefore, I have now come to the conclusion that:
Therefore, I will now be testing the above stair on this formula to see whether it works:
x + x +1 + x +2 + x +3 + x + g + x + g + 1 + x + g + 2 + x + 2g + x + 2g + 1 + x + 3g
= 10x + 10 + 10g
10 x 1 + 10 + 10 x 10 = 120
I will now be testing the same formula on a 9 by 9 stair. In this stair the value of g will become 9 due to the change in grid size.
: The total for this stair is:
5 + 6 + 7 + 8 + 9 + 14 + 15 + 16 + 17 + 23 +
24 + 25 + 32 + 33 + 41 = 275
Therefore:
x + x + 1 + x + 2 + x + 3 + x + 4 + x + g + x +
g + 1 + x + g +2 + x + g + 3 + x + 2g + x + 2g
1 + x + 2g +2 + x + 3g + x + 3g +1 + x + 4g
= 15x + 20 + 20g
15 x 5 + 20 + 20 x 9
= 275
Below is stair 79 on a 12 by 12 grid. The value of g on this grid will become 12.
: The total for this stair is:
79 + 80 + 81 + 91 + 92 +103 = 526
If I calculate this stair total with the same formula
then it will be:
x + x +1 + x + 2 + x + g + x + g +1 + x + 2g
= 6x + 4 + 4g
= 6 x 79 + 4 + 4 x 12
= 526
Below is 24 on a 6 by 6 grid and the value of g will become 6.
The total for this stair is:
24 + 25 + 30 = 79
The stair total through the formula will be:
x + x +1 + x g
= 3x + 1 + g
= 3 x 24 + 1 + 6
= 79
CONCLUSION:
Altogether, I have found 2 formulas. One was for calculating the stair total of a three level stair on a 10 by 10 grid and the other was to calculate the stair total of any stair on any number grid and on any position.
The two formulas were:
6n + 44 = stair total
x(stair number) + numbers added to x + g(the value of the grid) = stair total
If I had more time then I could have done the following changes:
- Test it on more grids such as 20 by 20 or 30 by 30. By doing that I could have gone into more depth regarding the formula.
- Find a more precise formula which would have fixed numbers such as the one in Part 1.
- I could have also tried putting the stairs on upside down positions or any other positions.
- I would have also tested stairs with the width of 6 squares, which would have made my task more difficult.
