To create a formula, I first called the step-number n. I noticed that the step-number is always the biggest, so I created a formula to relate the step-total to the step-number.
Since all the numbers in the stair-shape adds up to the step-total, if I simplify all the values in the 3-step stair, I should get a formula for the step-total
n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20) = 6n + 44
Therefore, the step-total should equal to 6n + 44
I called step-total t
For stair-step 1, n = 1
t = 6n + 44
= 6(1) + 44
= 6 + 44
= 50
I have proven that the formula works for the first 3-step stair. To verify that this formula works for every other 3-step stair, I will test the first example given, stair-step 25. The step-total should equal 194.
t = 6n + 44
= 6(25) + 44
= 150 + 44
= 194
The formula works for all the step-numbers on the 10 by 10 grid. The relationship between the stair-totals and the position of the stair-shape on the grid is that if you multiply the step-number by 6 and the add 44, you should find the step-total.
Part 2: Investigate further the relationship between the stair totals and other step stairs on other number grids.
I have worked out the formula for the 10 by 10 grid to be t = 6n +44 but this formula only works for a 10 by 10 grid. I will investigate the relationship between the step-total, step-number and the grid size.
In the 10 by 10 grid I noticed that the numbers 1, 11 & 21 etc increases by 10. If you subtracted n from the number above n, you would get the grid size number Therefore, the step-total and the step-number relate with the grid size.
In this example, n is 25 and the number above n is 35. If I subtract n from 35, I would get 10. This is the grid size number.
The relationship between the step-number and the rest of the numbers are that;
The number 35 has a difference of 10 from the step-number. The number 45 has a different of 20 from the step-number, which is 2 multiplied by 10. The number 26 has a difference of 1 from the step number, which is 1 multiplied by 10, minus 9.The number 27 has a difference of 2 from the step number, which is 1 multiplied by 10, minus 8.
The number 36 has a difference of 11 from the step number, which is 1 multiplied by 10, plus 1
With this information, I will create a formula for each of the numbers in the 3 step-stair shape. The Grid size will be called g.
Therefore, the step-total is the total of these formulas. I will now simplify the formulas.
n + n + ( g – 9) + n + ( g – 8 ) + n + g + n + ( g + 1 ) + n + 2g
= 6n + 6g -16
In the previous formula, we say it was t = 6n + 44.
Now in this formula, t = 6n + 6g - 16
t = 6n + 6g - 16
if you multiply the 6 by g (which in this case is 10) then minus 16, you would get 44, which is the same number as the last figure in the previous formula
I will now prove this formula works with a 9 by 9 grid.
The sum of all the numbers in the 3-step stair shown in this example is 280.
40 + 41 + 42 + 49 + 50 + 58 = 280
The first 3-step stair is made up of
1 + 2 + 3 + 10 + 11 + 19 = 46
If I move the 3-step stair 1 unit to the right, the 3-step stair would be made up of
2 + 3 + 4 + 11 + 12 + 20 = 52
If I move the 3-step stair another unit to the right, the 3-step stair would be made up of
3 + 4 + 5 + 12 + 13 + 21 = 58
I have created a table of a few results of the 3-step stairs
Just like in the 10 by 10 grid, the ratio is 1:6 because the step-number increases by 1 and the step-total increases by 6 for each stair. Therefore, the formula for the 10 by 10 grid should work for the 9 by 9 grid.
If the step-number is 1 and the grid size is 9
t = 6n + 6g – 16
= (6 x 1) + (6 x 9) – 16
= 6 + 54 – 16
= 44
The answer I should have gotten was 46. The reason why there is a -2 error is because when I applied the 10 by 10 grid to the 9 by 9 grid I forgot that the numbers above n were increasing by 9 instead of 10.
So in the equation we have g – 9 but it needs to be g – 8
And where we have g – 8 and it needs to be g – 7
From this information, we can assume that for an 11 by 11 grid, we have to subtract 2 from the equation and for a 9 by 9 grid, we have to add 2 to the equation.
So if we did an 11 by 11 grid it would be 6n+6g – 18 and if we did a 9 by 9 grid the equation would be 6n+6g – 14
From this information, I have come up with another equation that will work with a grid of any size;
6n + 6g – ( 16 ± 2d )
Where d is the difference of the grid from a 10 by 10 grid
I will test this new formula with the 10 by 10 grid and the 9 by 9 grid.
Step-number = 1
Grid size = 10 by 10
t = 6n + 6g – ( 16 ± 2d )
= 6(1) + (6 x 10) – [16 ± (2 x 0)]
= 66 – 16
= 50
The new equation has worked for the 10 by 10 grid
Step-number = 1
Grid size = 9 by 9
t = 6n + 6g – ( 16 ± 2d )
= 6(1) + (6 x 9) – [16 – (2 x 1)]
= 60 – 14
= 46
The new equation has also worked for the 9 by 9 grid
This formula should even work for a smaller scale, such as a 5 by 5 scale
The answer we should get is;
1 + 2 + 3 + 6 + 7 + 11 = 30
t = 6n + 6g – ( 16 ± 2d )
= 6(1) + (6 x 5) – [16 – (2 x 5)]
= 36 – 6
= 30
I have proven that my formula works with any grid
Conclusion:
Here I have put all the formula I have come up with. This formula will apply to a grid of any size;
t = 6n + 6g – ( 16 ± 2d )
Where;
t : step- total
n : step-number
g : grid size
d : difference of grid size from a 10 by 10 grid
In this project I have found out many ways in which to solve the problem I have with the stair-shape being in various positions with different sizes of grids. The way I have made the calculations less difficult is by creating a main formula that works for all the different circumstances.
