25 T = 360
26 T = 370
27 T = 380
28 T = 390
67 T = 780
As the x number rises by 1 the total rises by 10. This is again due to the fact that there are 10 numbers in the 4-step square.
E.g.
- 32
21 22 22 23
11 12 13 12 13 14
1 2 3 4 2 3 4 5
Each of the 10 numbers in this case has increased by one.
Therefore;
1 x 10 = 10
So far the equation stands like so; X= 10X???
We can discover the rest of the equation in the same way as before:
X 10X Total (Tx)
1 10 +110 120
26 260 +110 370
27 270 +110 380
28 280 +110 390
67 670 +110 780
Or algebraically;
x + 30
x + 20 x + 21
x + 10 x + 11 x + 12
x x +1 x + 2 x + 3
The formula for the 4-step stair on a 10 x 10 grid is Tx = 10x + 110
Using this formula I will predict that T = (10 x 7) + 110
T = 70 + 110
T = 180
37
27 28
17 18 19
7 8 9 10 37+27+28+17+18+19+7+8+9+10 = 180
I will also predict that T = (10 x 61) + 110
T = 610 +110
T = 720
91
81 82
71 72 73
61 62 63 64 91+81+71+61+82+72+62+73+63+64 = 720
Both my predictions and formula were correct, I can now produce a graph to show the relationship between x and Tx with 4-step stairs.
I am now investigating the formula for 5-step stairs on a 10 x 10 grid.
Results Table:
X T (total)
1 T = 235
23 T = 340
24 T = 355
25 T = 370
26 T = 385
As the x number rises by one the total number rises by 15. This is because there are 15 numbers in a 5-step stair, and as x increases by one so do the rest of the 15 numbers.
E.g.
- 42
31 32 32 33
21 22 23 22 23 24
11 12 13 14 12 13 14 15
1 2 3 4 5 2 3 4 5 6
As we can see each of the 15 numbers has increased by one. Therefore;
1 x 15 = 15
So far the equation stands like so; X = 15X???
We can find out the remaining of the equation by using example.
X 15X Total (Tx)
1 15 +220 235
3 45 +220 265
4 60 +220 280
23 345 +220 565
24 360 +220 580
25 375 +220 595
26 390 +220 610
56 840 +220 1060
The formula for the 5-step stair on a 10 x 10 grid is Tx = 15x + 220
Using this formula I will predict that T = (15 x 2) + 220
T = 30 + 220
T = 250
42
32 33
22 23 24
12 13 14 15
2 3 4 5 6
42+32+33+22+23+24+12+13+14+15+2+3+4+5+6 = 250
My prediction and formula was correct. I can now produce a graph showing the relationship between x and Tx with 5-step stairs.
I am now going to investigate further by finding a formula for 3, 4 and 5 step stairs on an 8 x 8 grid.
3-step stair:
Results Table:
X T (total)
- 42
- 48
- 54
- 60
45 306
46 312
So far the equation stands like so; x= 6X???
We can find the remaining of the equation in the same way as before:
X 6X Total (Tx)
1 6 +36 T = 42
2 12 +36 T = 48
3 18 +36 T = 54
4 24 +36 T = 60
46 276 +36 T = 312
The formula for a 3-step stair on any 8 x 8 grid is Tx = 6x + 36
I am going to predict the totals of 3-step stairs on an 8x8 grid
I predict that T = (6 x 6) + 36
T = 36 + 36
T = 72
22
14 15
6 7 8 22+14+15+6+7+8 = 72
My prediction was correct. I can now produce a graph for this formula.
I am now going to find the formula for a 4-step stair on an 8 x 8 grid.
Results Table:
X T (total)
- 100
- 110
- 120
- 130
- 140
So far the equation stands like so; X = 10X???
We can find out the remaining of the equation like before:
X 10X Total (Tx)
1 10 +0 T = 100
2 20 +0 T = 110
3 30 +90 T = 120
4 40 +90 T = 130
5 50 +90 T = 140
The formula for any 4-step stair on an 8 x 8 grid is Tx = 10x + 90
I will use this formula to yet again make a prediction. I predict that T = (10 x 17) + 90
T = 170 + 90
T = 260
41
33 34
25 26 27
17 18 19 20 41+33+34+25+26+27+17+18+19+20 = 260
My prediction was correct. I will now produce another graph to show the relationship between x and Tx on an 8 x 8 grid for any 4-step stair.
I am now going to find the formula for a 5-step stair on an 8 x 8 grid.
Results Table:
X T (total)
- 195
- 210
- 225
- 240
- 255
So far the equation stands like so; X = 15X???
I am going to repeat myself again to find out the remainder of the equation:
X 15X Total (Tx)
1 15 +180 T = 195
2 30 +180 T = 210
3 45 +180 T = 225
4 60 +180 T = 240
5 75 +180 T = 255
The formula for any 5-step stair on an 8 x 8 grid
Tx = 15x + 180
I will use this formula to predict a total from a 5-step stair on an 8 x 8 grid. I predict that: T = (15 x 29) + 180
T = 435 + 180
T = 615
61
53 54
45 46 47
37 38 39 40
29 30 31 32 33
61+53+54+45+46+47+37+38+39+40+29+30+31+32+33 = 615
My prediction was correct so I am going to produce another graph.
I am now going to find out the formulas for 3,4 and 5-step stairs on a 9 x 9 grid. This is the 3-step stair formula.
Results Table:
X T (total)
- 46
- 52
- 58
- 64
- 70
So far the equation stands like so; X = 6X???
X 6X Total (Tx)
1 6 +40 T = 46
2 12 +40 T = 52
3 18 +40 T = 58
4 24 +40 T = 64
5 30 +40 T = 70
The formula for any 3-step stair on a 9 x 9 grid is Tx = 6x + 40
I will now use this prediction to predict sum totals. I predict that
T = (6 x 40) + 40
T = 240 + 40
T = 280
58
49 50
40 41 42 58+49+50+40+41+42 = 280
My prediction was correct so I am now able to draw up another graph of the relationship between x and Tx with a 3-step stair on a 9 x 9 grid.
I am now going to find the formula for a 4-step stair on a 9 x 9 grid.
Results Table:
X T (total)
- 110
- 120
- 130
- 140
- 150
So the equation stands like so; X = 10X???
X 10X Total (Tx)
1 10 +100 T = 110
2 20 +100 T = 120
3 30 +100 T = 130
4 40 +100 T = 140
5 50 +100 T = 150
The formula for any 4-step stair on a 9 x 9 grid is Tx = 10x + 100
I am going to predict that T = (10 x 50) + 100
T = 500 + 100
T = 600
77
68 69
59 60 61
50 51 52 53 77+68+69+59+60+61+50+51+52+53 = 600
My prediction was correct. I am now able to produce yet another graph showing the relationship between x and Tx with 4-step stairs on 9 x 9 grids.
I am now going to find the formula for 5-step stairs on a 9 x 9 grid.
Results Table:
X T (total)
- 215
- 230
- 245
- 260
- 275
So the equation stands like so; X = 15X???
X 15X Total (Tx)
1 15 +200 T = 215
2 30 +200 T = 230
3 45 +200 T = 245
4 60 +200 T = 260
5 75 +200 T = 275
The formula for any 5-step stair on a 9 x 9 grid is
Tx = 15x + 200
I am going to predict that T = (15 x 30) + 200
T = 1500 + 200
T = 1700
66
57 58
48 49 50
39 40 41 42
30 31 32 33 34
66+57+58+48+49+50+39+40+41+42+30+31+32+33+34 = 1700
My prediction was correct, I can now draw up another graph, similar to the others showing the relationship between x and Tx with any 5-step stair on a 9 x 9 grid.
By drawing all my formulas together I can see a pattern occurring. The first set of numbers are in a CUBIC SEQUENCE; 6, 10, 15 etc.
3-step 4-step 5-step
8 x 8 6x + 36 10x + 90 15x + 220
9 x 9 6x + 40 10x + 100 15x + 200
10 x 10 6x + 44 10x + 110 15x + 180
I can predict that for a 3-step stair on an 11 x 11 grid that the formula will be 6x + 48.
T = (6 x 1) + 48
T = 6 + 48
T = 54
23
12 13
1 2 3 23+12+13+1+2+3 = 54
My prediction was correct.
I will now predict that the next formula on an 11 x 11 grid for a 4-step stair will be Tx = 10x + 120
T = (10 x 3) + 120
T = 30 + 120
T = 150
36
25 26
14 15 16
3 4 5 6 36+25+26+14+15+16+3+4+5+6 = 150
My prediction was correct.
I now predict the next formula for a 5-step stair on an 11 x 11 grid will be Tx = 15x + 160
T = (15 x 1) + 160
T = 15 + 160
T = 175
45
34 35
23 24 25
12 13 14 15
1 2 3 4 5
45+34+35+23+24+25+12+13+14+15+1+2+3+4+5 = 255
Unfortunately, my prediction was not correct. This is something that maybe if I extended my investigation further I could find out.
In conclusion to my investigating I have discovered the formulas for any 3, 4 and 5- step stairs on 8 x 8, 9 x 9, and 10 x 10 grids. I have also made a prediction about 11 x 11 formulas too. I think of this to be a success.
