6(1)+n = 50
n = 50-6
n = 44
So the formula for this 3Step Stair seems to be 6x+44=t To make sure the formula works in every case, I tested it in eight situations where ‘x’ was different each time.
.
Using The formulae(6(x)+44=n)
(x) Workings Total Workings (t)
35 35+36+37+45+46+55= (254) (35)+44 254 Correct
47 47+48+49+57+58+67= (326) 6(47)+44 326 Correct
58 58+59+60+68+69+78= (392) 6(58)+44 392 Correct
65 65+66+67+75+76+85= (434) 6(65)+44 434 Correct
The result when using the formula 6(x)+44=(t) is the same as when not using the formula which proves that the formula 6(x)+44=(t) gives you the Stair Total of any 3step stair which travels from bottom right to top left in any position on the 10 by 10 number grid.
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Task 2.
In this part I want Investigate further the relationship between the stair total and other step stairs on the number grids.
Method:
I will start by investigating each group of the same numbered stairs and try to make a formula for them. Since1- Step stair has nothing to investigate because it has only 1 number, i will start by the smallest stair possible (in this case 2-Step Stair) and then try to make a formula that works for any 2-Step Stair. Then make a formula for 4Step Stair, Then 5, 6, 7, 8, 9and lastly 10-Step Stair.
I will then investigate the formulas to see if I can generate a universal formula for any step stair in any position on the 10 by 10 Grid.
In this section the symbols for the inputs will be;
(s) Number of Steps Of the Stair for example 3-Step Stair(s)=3
(x) The number in the bottom left of the stair grid.
(t) Stair Total of the stair
1+2+11=14
The Stair Total of this 2-Step Stair is 14
2+3+12=17
The Stair Total of this 2-Step Stair is 17
3+4+5=20
The Stair Total of this 2-Step Stair 20
4+5+14=23
The Stair Total of this 2-Step Stair 23
5+6+75=26
The Stair Total of this 2-Step Stair 26
6+7+16=29
The Stair Total of this 2-Step Stair 29
At this point I can notice that the difference of the Stair Total inside these 2Step Stairs is 3. and it is constant. To see the difference more clearly I will put in diagram.
(x) (t) Diff
1 14 0
2 17 3
3 20 3
4 23 3
5 26 3
6 29 3
So I think the formula for 2-Step Stairs will be (3) Multiplied by the bottom left corner of the stair(x) Added or subtracted by number(n), is equal to, The 2-Step Stair Total (t).
3x+ y =n. so if ‘x’=1 then ‘n’=14,
so 3(1)+(n)=14
(n)=14-3
(n)=11
So I think the formula for 2-Step Stair in the 10 by 10 is. 3x+11=n. to make sure if this formula work for any 2-stepStair, I will test it in Four situations where ‘x’ is different each time.
.
26+27+36 89
77+78+87 242
28+29+38 95
55+56+57 168
(Using formula 3(x)+11=n)
(x) Workings (t) Workings (t) Correct
26 26+27+36 89 3(26)+11= 89 Correct
77 77+78+87 242 3(77)+11= 242 Correct
28 28+29+38 95 3(28)+11= 95 Correct
55 55+56+57 168 3(55)+11= 168 Correct
All the results using the formula are correct, so I can conclude that this formula3(x)+11=(t) gives you the stair total of any 2-Step Stair in the 10 by 10 Grid which travels from bottom right to top left.
At this point I also notice that the 1st difference of the stair totals is equal to the number of cells inside the Stair. For example the difference of Stair Totals of 3-Step Stair is 6. and the number of cells inside the 3-Step Stair is 6. I will draw a 3-Step Stair to show my finding more clear.
As I can see from this 3-Step Stair it has 6 Cells or Blocks inside its grid. And the stair totals of 3-Step Stair have 1st common difference which is 6 This Because the fact that the shape of the stair is actually triangular.
As you can see from this diagram the stair is triangular blocks with it is
cells just Shifted to the left
3-Step Stair
2tep
This means that if I find the formula for the number of cells inside the triangle shape, the first part of a general formula for all stair sizes will be found. And referring to my previous math lessons I know the formula for the triangle number is. The height of the triangular shape (h), plus 1, divided by 2, multiplied by (h). h(h+1)/2 . but instead of height (h). I will be using (s) as, Number of Steps of Stair. For example if the number of steps (s)=3, then 3((3+1)/2)gives you 6, which is exactly the number of cell inside 3-Step Stair. I will then multiply by the bottom left number of the stair (x). if x=1 1(3((3+1)/2)) gives me 6. which is correct. So the first part of a universal formula will be x(s(s+1/2)). And the second part of the formula will be, the first formula added by number or numbers(n) . Thus the two formulas when added together will give you, The Stair Totals (t) of any Stair which travels from bottom right to top left in any position on the 10 by 10 number grid.
To test my formula I will use it on a different sized stairs but keep their (x) value the same.
Testing the formula: x(s(s+1)/2)+n=t.
(x) Workings (t)
1 If (s)=1 then, (n)= 0 then (t)= 1 1(1(1+1)/2)+0 = 1
1 =2, =11 =14 1(2(2+1)/2)+11 = 14
1 =3 = 44 =50 1(3(3+1)/2)+44 = 50
The formula when tested on Stair grids 1, 2 and 3 seems to be working correctly, and I think it will work on the other stairs, but to make sure it does, I will try on other step stairs that I haven’t done before.
This time I will change the method of investigation. Instead of looking the stair total of each individual Step Stair, I will now try find the value of (n),of each individual stair shape. This is because If i get the value of (n) of a stair, I can work out the totals of the rest of those same sized stairs. Notice that (n) always has the same value in each set of stairs. And changes in other sets of stairs For example, the value of (n) in any 3-Step Stair is always 44 .And the value of (n) in any 2-Step Stair is always 11. so adding (n) to the first part of the formula x(s(s+1)/2), will always give me the stair total.
So this time I will try to find the value of (n) in 4, 5, 6, 7, 8, 9 and 10-Step stairs. By investigating the totals of only stairs with x equals 1. After doing that I will try to put them in diagram and try to find there is a pattern that links them (n)s altogether. And if there is one, I will try to create a formula for the value of (n) which will be the second part of my formula, thus completing my universal formula for all the Stair Totals in the 10 by 10 grid.
(s)4-Step Stair:
(x)
(t)
1+2+3+4+11+12+13+21+22+31=120
The Stair Total of this 4-Step Stair is 120
(x)(s(s+1)/2)+n= t
(1)(4(4+1)/2)+n=120
10+n=120
n=120-10
n=110
So the value of (n) in 4-Step Stair is 110
And the formula is (x(s(s+1)/2))+110 to check if the formula is correct I will test it on another 4-Step Stair this time (x) = 10
(x)(s(s+1)/2) +n =t
10(4(4+1)/2)+110=t
100+110=210
by looking at the 4-Step Stair with it is x=10, you can see the answer is correct.
10+11+12+13+20+21+22+30+30+40=210
The Stair Total of this 4-Step Stair is 210
Notice that
- One place right side of x it will be x+1
- Two places right of it will be x+2 three places x+3
- One place above it will be x+10 two places,x+20
- Diagonally one place above it will be x+11 two places x+22 etc:
When I noticed this at first I didn’t feel it makes any difference but when I looked closely I found that if you Add all the (x) together you get 10x. which is the value of the first part of the formula. and if you Add all the numbers. you get 110. which the value of (n).
=(x)+(x+1)+(x+2)+(x+3)+(x+10)+(x+11)+(x+12)+(x+20)+(x+21)+(x+30)
=10x+110.
But this doesn’t give me any formula but it will help me calculate the value of (n) in the stair quickly. by just adding the numbers.
In this 5-Step stair I know the first part of it is formula is x(5(5+1)/2) =15x
So to get the value of (n) I will add the numbers together. 1+2+3+4+10+11+12+13+20+21+22+30+31+40 =220
So I think the Stair Total for any 5-Step Stair is 15x+220=t
And I predict that if x= 1 then, 15(1)+220= 235.
And if x=4 then,15(4)+220 =280
To see if I am right I will draw the two stairs I predicted and input their original numbers.
1+2+3+4+5+11+12+13+14+21+22+23+31+32+41=235
The Stair Total of this 5-Step Stair is 235
4+5+6+7+8+14+15+16+17+24+25+26+34+35+44=280
The Stair Total of this 5-Step Stair is 280
As I have predicted the answers are correct . So the formula for any 5-Step Stair in the 10 by 10 Grid which travels from bottom right to top left is. x(s(s+1)/2)+220 =t.
6-Step Stair:
X(s(s+1)/2)+y=t
X(21)+1+2+3+4+5+10+11+12+13+14+20+21+22+23+30+31+32+40+41+50=385
=x21+385=t
So I am confident that the formula of Stair Total for 6-Step Stair is x(s(s+1)/2)+385=t
And i predict that if x=1 then 21(1) +385=406
And if x=3 then 21(2) +385=427
To see if I am right I will draw the two stairs I predicted and input their original numbers.
1+2+3+4+5+6+11+12+13+14+15+21+22+23+24+31+32+33+41+42+51=406
The Stair Total of this 6-Step Stair is 406
2+3+4+5+6+7+12+13+14+15+16+22+23+24+25+32+33+34+42+43+52=427
The Stair Total of this 6-Step Stair is 427
As I have predicted the answers are correct. So the formula for any 6-Step Stair in the 10 by 10 Grid which travels from bottom right to top left is. x(s(s+1)/2)+385.
By looking at the Stair Totals of different sized stairs that I have investigated so far, I can see that, first part of a universal formula for different sized Stair Totals is x(s(s+1)/2, Plus number(n). but to complete the formula I need to find a general expression for the value of (n). in order to do that I need to find the differences between (n) values of Stairs I have investigated already.
I made a table to see if there were any patterns. And on this table I included a single step stair because it has only single cell it has one number so it’s (n) value will be 0. because of it is small number it will be my starting value:
The third difference for all of them is 11, which tells me the formula may haven something to do with 11 and (s)3. I made a diagram to see if there were any patterns:
Stair Size (s)1 2 3 4 5 6
11(s3) 11 88 297 704 1375 2376
The answer is too big so I will subtract by 11, because the n value of 1-Step Stair is 0, and 11 subtracted by 11 gives me 0. I will try to see if it works for other step stairs.
Stair Size(s) 1 2 3 4 5 6
11s3 -11= 0 77 286 696 1364 2365
It seems when I subtracted 11 from 11s3 it worked for the 1-Step Stair but clearly doesn’t work for the other step stairs. So this time I will try to subtract 11s because the n value of 1-Step Stair is 0, and 11s subtracted by 11 also gives me 0. so I will try if it works for other step stairs.
Stair Size(s) 1 2 3 4 5 6
11s3-11s= 0 66 264 660 1320 2310
My answer is still not correct and I need to reduce the numbers to the correct answers. so since dividing 0 by any number will give me 0 I will not concentrate much on 1-step Stair. By looking at my previous Step Stair investigation, I know the (n) value of 2-Step Stair is 11. And 3-Step Stair is 44. and by looking at the diagram above(11s3-11s) the workings gives you 66 and 264. I notice if I divide 66 by 6.I get 11, and if I divide 264 by 6. I also get 44.W hich is correct. So I will try and see if it works for the rest of the Stairs.
Stair Size(s) 1 2 3 4 5 6
11s3-11s/6 0 11 44 110 220 385
The answers are correct and at this point I think I have found a universal formula for all the stair totals in the 10 by 10 grid. Which is x(s(s+1)/2)+ 11s3-11s/6= t.
But to make sure it works for the other step stair totals I will test my formula on number of different sized step stairs in the 10 by 10 grid.
7-Step Stair
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
The Stair Total of this 7-Step Stair is =644
Using the formula= x(s(s+1)) + (11s3 - 11s) = t.
2 6
(1)(7(7+1)) + (11(73))-(s11)=t
2 6
(1) (28) + (616) = 644
this 8-Step Stair
1+2+3+4+5+6+7+8+11+12+13+14+15+16+17+21+22+23+24+25+26+31+32+33+34+35+41+42+43+44+51+52+
53+61+62+71=960
The Stair Total of this 8-Step Stair is 960
Using the formula= x(s(s+1)) + (11(s3) – 11s) = t.
2 6
1(8(8+1) + (11(83)-11s) =t
2 6
1 (36) + 924 =960
1+2+3+4+5+6+7+8+9+11+12+13+14+15+16+17+18+21+22+23+24+25+26+27+31+32+33+34+35+36+41+42+43+44+45+51+52+53+54+61+62+63+71+72+81=1365
The Stair Total of this 9-Step Stair is=1365
Using the formula=
x(s(s+1)) + (11(s3) – 11s) =t.
2 6
1(9(9+1)) + (11(s3)-11s) =t
2 6
45 1320 =1365
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1870
Using the formula=
x(s(s+1)) + (11(s3) – 11s) =t.
2 6
1(10(10+1)) + (11(s3)-11s) =t
2 6
55 + 1815 =1870
As I have proved from my test the formula works for every Step Stair in the 10 by 10 grid, and I can confidently conclude that x(s(s+1)) + (11(s3) – 11s) =t. works for
2 6
any step stair that travels from bottom right to top left in the 10 by 10 grid.
The formula for 44 = h(height of the grid) (11h3 – 11h)/6
The whole formula= (x(x+1)/2)n + (11x3 – 11x)/6
And now I am going I am going to investigate the 4-Step Stair
