N=7
7 + 8 + 9 + 17 + 18 + 27 = 86
T=86
I worked out all the totals of the 3 step stair by adding all the number within the individual step stair.
For each position I have added up the numbers to get a total. Here is a table of my results:
3 Step Stairs
I have worked out the difference between all the numbers and the number for all of them is ‘6’. As I have tried many positions I am sure that the difference is always 6 if you keep moving it to the right one position at a time. Now that I have all the totals and the consistent difference I know have to work out a equation to work the total out without adding all the numbers within the step stair.
Formula
From previous working out I know that the Position ‘N’ and the number 6 will be in the equation. As for every time I investigated further I moved the Position. This was a variable which I controlled. Whatever total I got it always had the same difference of 6. So this would definitely be involved in the equation. As I am trying to find a formula to work out the total the expression ‘T’ would be in the equation.
The equation would start of with:
‘T=…’
The letter ‘T’ means the total and this is what I want to work out.
I would also have to times the position number by 6, because I got 6 from the difference of the total for the position. This so far is the first part of the equation.
‘T = 6n…’
If I times the number 6 by the position number 1 this doesn’t give me the total 50. So I have to add an additional number to this equation to reach 50.
6 n = 6 × 1
6 × 1 = 6
50 – 6 = 44
I got 44 from subtracting 6 from 50. So now I have my equation for the 3 step stair.
This is the formula I have got from my working out. Now I will test it too see whether it is correct or not. Firstly I will test each position to see if it is correct. Second I will be using the equation to work out the total if the position is ‘8’. Then I will work it out using the 1st method I used.
Testing
T = 6n + 44
Position 2
T = 6n + 44 = 6 × 2 + 44
6 × 2 = 12
12 + 44 = 56
The formula has worked for this position. I know this because when I had worked this out the first time using the slower method I had got the same number.
T = 6n + 44
Position 3
T = 6n + 44 = 6 × 3 + 44
6 × 3 = 18
18 + 44 = 62
The formula has given the right total number. I have checked this with the total I got from previous working out. Another way to check if this is right is to see if there is a difference of six.
T = 6n + 44
Position 4
T = 6n + 44 = 6 × 4 + 44
6 × 4 = 24
24 + 44 = 68
The formula has proven to be successful again although I should check all of the positions to make sure that this is the equation.
T = 6n + 44
Position 5
T = 6n + 44 = 6 × 5 + 44
6 × 5 = 30
30 + 44 = 74
T = 6n + 44
Position 6
T = 6n + 44 = 6 × 6 + 44
6 × 6 = 36
36 + 44 = 80
T = 6n + 44
Position 7
T = 6n + 44 = 6 × 7 + 44
6 × 7 =42
42 + 44 = 86
Now I will be testing the formula on the position number 8.
T = 6n + 44
Position 8
T = 6n + 44 = 6 × 8 + 44
6 × 8 = 48
48 + 44 = 92
Now I will work this out using the first method I used to see whether this is the correct answer. I have only one indication that it is right so far which is that the fact that it has the difference of six.
8 + 9 + 10 + 18 + 19 + 28 = 92
T = 92
The total is the same in no matter which method I used so the formula is a success. Now that I have figured out the formula for 3-step stair I am going to find the equation for 4 step stair.
4 Step Stairs
Look at the stair shape drawn on the 10 by 10 number Grid below.
This is a 4 Step Stair
The total of the numbers inside the stair shape is:
1 + 2 + 3 + 4 + 11 + 12 + 13 + 21 + 22 + 31 = 120
‘T = 120’
I got this number by adding all the numbers within the 4 Step Stairs together.
The second position is 2. Like the 3 step stair you move the position one over to the right.
The total for the position 2 is:
2 + 3 + 4 + 5 + 12 + 13 + 14 + 22 + 23 + 32 = 130
‘T = 130’
Now I’ll be finding the totals if the positions were 3, 4 and 5.
3rd 5th
Position Position
4th
Position
N = 3
3 + 4 + 5 + 6 + 13 + 14 + 15 + 23 + 24 + 33 = 140
T = 140
N = 4
4 + 5 + 6 + 7 + 14 + 15 + 16 + 24 + 25 + 34 = 150
T = 150
N = 5
5 + 6 + 7 + 8 + 15 + 16 + 17 + 25 + 26 + 35 = 160
T = 160
All of these positions all had one steady difference which was the number ten.
This table shows the total results of each position and there differences.
The Equation will have the following:
‘T = 10n + ...’
To work out the rest of the equation I will have to minus ‘10n’ from a position.
Position 1
10n = 10
120-10 = 110
The formula for 4 step stairs is:
Testing
Position 2
T = 10n + 110 = 10n × 2 + 110
10 × 2 = 20
20 + 110 = 130
Position 3
T = 10n + 110 = 10n × 3 + 110
10 × 3 = 30
30 + 110 = 140
Position 4
T = 10n + 110 = 10n × 4 + 110
10 + 4 = 40
40 + 110 = 150
Position 5
T = 10n + 110 = 10n × 5 + 110
10 + 5 = 50
50 + 110 = 160
All of the totals are correct. So the formula proved successful.
5 Step Stairs
Look at the stair shape drawn on the 10 by 10 number Grid below.
This is a 5 Step Stair
The total of the numbers inside the stair shape is:
1 + 2 + 3 + 4 + 5 + 11 + 12 + 13 + 14 + 21 + 22 + 23 + 31 + 32 + 41 = 235
'T = 235’
The second position is 2. Here is the total for it:
2 + 3 + 4 + 5 + 6 + 12 + 13 + 14 + 15 + 22 + 23 + 24 + 32 + 33 + 42 = 250
So far the difference I have spotted is 15. I am not sure if this is steady so I will investigate further so I can be sure.
3rd 5th
Position Position
4th
Position
N=3
3 + 4 + 5 + 6 + 7 + 13 + 14 + 15 + 16 + 23 + 24 + 25 + 33 + 34 + 43 = 265
T=265
N=4
4 + 5 + 6 + 7 + 8 + 14 + 15 + 16 + 17 + 24 + 25 + 26 + 34 + 35 + 44 = 280
T=280
N=5
5 + 6 + 7 + 8 + 9 + 15 + 16 + 17 + 18 + 25 + 26 + 27 + 35 + 36 + 45 = 295
T=295
The difference between all these numbers is 15.
The table shows the totals and the difference which is 15.
Now that I know the totals and the difference I now know the first part of the equation which is:
‘T = 15n …’
For me to work out the rest of the equation I will have to subtract have to do the first part of the formula and then I would find the difference.
Position 1
15n = 15 × 1
15 × 1 = 15
235 – 15 = 220
So the formula for the 5 step stairs is:
Testing
Position 2
15 × 2 = 30
30 + 220 = 250
Position 3
15 × 3 = 45
45 + 220 = 265
Position 4
15 × 4 = 60
60 + 220 = 280
Position 5
15 × 5 = 75
75 + 220 = 295
This formula has proved successful.
6 Step Stairs
Look at the stair shape drawn on the 10 by 10 number Grid below.
This is a 6 Step Stair
The total of the numbers inside the stair shape is:
1 + 2 + 3 + 4 + 5 + 6 + 11 + 12 + 13 + 14 + 15 + 21 + 22 + 23 + 24 + 31 + 32 + 33 + 41 + 42 + 51 = 406
‘T = 406’
This number came from adding all the numbers within the 6 step stairs.
The second position is 2. The total is:
2 + 3 + 4 + 5 + 6 + 7 + 12 + 13 + 14 + 15 + 16 + 22 + 23 + 24 + 25 + 32 + 33 + 34 + 42 + 43 + 52 = 427
‘T = 427’
The difference I can see at this early stage is 21.
3rd 5th
Position Position
4th
Position
N=3
3 + 4 + 5 + 6 + 7 + 8 + 13 + 14 + 15 + 16 + 17 + 23 + 24 + 25 + 26 + 33 + 34 + 35 + 43 + 44 + 53 = 448
T= 448
N=4
4 + 5 + 6 + 7 + 8 + 9 + 14 + 15 + 16 + 17 + 18 + 24 + 25 + 26 + 27 + 34 + 35 + 36 + 44 + 45 + 54 = 469
T= 469
N=5
5 + 6 + 7 + 8 + 9 + 10 + 15 + 16 + 17 + 18 + 19 + 25 + 26 + 27 + 28 + 35 + 36 + 37 + 45 + 46 + 55 = 490
T= 490
The difference between all these numbers is 21
The table shows the totals and the difference.
From these two pieces of information I can figure out the first part of the formula. This is:
‘T = 21n…’
Now to work out the rest of the formula I will use it then find the difference.
Position 1
21n = 21 × 1
21 × 1 = 21
406 – 21 = 385
So the formula for the 6 stair is:
Testing
Position 2
21 × 2 = 42
42 + 385 = 427
Position 3
21 × 3 = 63
63 + 385 = 448
Position 4
21 × 4 = 84
84 + 385 = 469
Position 5
21 × 5 = 105
105 + 385 = 490
The formula was successful as it gave the correct answers to the totals.
7 Step Stairs
Look at the stair shape drawn on the 10 by 10 number Grid below.
This is a 7 Step Stair
The total of the numbers inside the stair shape is:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 11 + 12 + 13 + 14 + 15 + 16 + 21 + 22 + 23 + 24 + 25 + 31 + 32 + 33 + 34 + 41 + 42 + 43 + 51 + 52 + 61= 644
‘T=644’
I got this by totalling the figures found in the 7 step stairs.
The total for position 2 is:
2 + 3 + 4 + 5 + 6 + 7 + 8 + 12 + 13 + 14 + 15 + 16 + 17 + 22 + 23 + 24 + 25 + 26 + 32 + 33 + 34 + 35 + 42 + 43 + 44 + 52 + 53 + 62 = 672
The total position for 3 is:
3 + 4 + 5 + 6 + 7 + 8 + 9 + 13 + 14 + 15 + 16 + 17 + 18 + 22 + 23 + 24 + 25 + 26 + 27 + 33 + 34 + 35 + 36 + 43 + 44 + 45 + 53 + 54 + 63 = 700
The difference between all of them is the number 28.
The formula for the seven step stairs is:
Now I have all the equations. From all the formulas I’m going to find the difference between all of them. Hopefully when I will be able to create another formula which allows me to find the total no matter what size the grid is or the position of the step stair.
There is a pattern in each formula I have worked out so far. To summarize the formula it would be:
The difference between the ‘a’ is + 1. This is the first pattern I have realized. The second pattern to this formula is to ‘b’. Where the difference’s between them are 11.
Further Investigation
Now I have all the equations. From all the formulas I’m going to find the difference between all of them. Hopefully when I will be able to create another formula which allows me to find the total no matter what size the grid is or the position of the step stair.
Here are all the formulas for different sized step stairs.
3 step stair T = 6n + 44
4 step stair T = 10n + 110
5 step stair T = 15n + 220
6 step stair T = 21n + 385
7 step stair T = 28n + 616
T = an + b
I have done this so that I can work out the ultimate formula.
Now I plan to find the differences between the formulas. So that if I put the formula’s together I would get the final formula.
Stem 3 4 5 6 7
(s)
44 110 220 385 616
66 110 165 231
44 55 66
11 11
From finding the difference of the numbers I did not spot the difference at first so I continued to find the difference of the results I got. As I had to continue on finding the difference I knew this wouldn’t be a quadratic equation. This is because it had three differences not two. Now that I have the difference I realized that the formula would be cubic because it took three steps to find the difference.
The equation for this is:
11S ³ 49 ½
6
11S ³ -5 ½ - 11
- 2
-5.5 – 7 1 7 1 = 22
3 3 3
-11 - 22
2 3
33 44
6 6
3 × 11 4 × 11
6 6
-11 S
6
1 2 3 4 5 6
1 3 6 10 15 21
2 3 4 5 6
1 1 1 1
The 1st row in this diagram are all triangular numbers.
3 4 5 6 7
44 110 220 385 616
66 110 165 231
44 55 66
-5.5
+11 +11
493
11 S³ 99
6 2
11 × 3 11 S³
6 6
-
= 11
2
This is a cubic equation because it has three differences.
The ultimate equation from previous working out is:
-
T = ( 1 S² + 1 S ) N + 11 S³ - 11 S
2 2 6 6
This equation is the final equation used to calculate the total no matter what size step stair you use and the position of it.
