Number Stairs Investigation
91
92
93
94
95
96
97
98
99
100
81
82
83
84
85
86
87
88
89
90
71
72
73
74
75
76
77
78
79
80
61
62
63
64
65
66
67
68
69
70
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
31
32
33
34
35
36
37
38
39
40
21
22
23
24
25
26
27
28
29
30
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
lowest number (n)
Long operation
Total (t)
1st Difference (D0)
1
1 + 2 + 3 + 11 + 12 + 21
50
6
2
2 + 3 + 4 + 12 + 13 + 22
56
6
3
3 + 4 + 5 + 13 + 14 + 23
62
6
4
4 + 5 + 6 + 14 + 15 + 24
68
6
5
5 + 6 + 7 + 15 + 16 + 25
74
6
Judging by this, the first part of the overall equation for a 3 step stair is 6n +?
n
6n
T
D0
1
6
50
44
2
12
56
44
3
18
62
44
4
24
68
44
5
30
74
44
This shows the difference between 6n and T as + 44
Therefore the equation for a 3 step stair on a 10×10 grid is 6n + 44
The same sequence can be used for a 4 step stair
lowest number (n)
Long operation
Total (t)
1st Difference (D0)
1
1 + 2 + 3 + 4 + 11 + 12 + 13 + 21 + 22 + 31
120
10
2
2 + 3 + 4 + 5 + 12 + 13 + 14 + 22 + 23 + 32
130
10
3
3 + 4 + 5 + 6 + 13 + 14 + 15 + 23 + 24 + 33
140
10
4
4 + 5 + 6 + 7 + 14 + 15 + 16 + 24 + 25 + 34
150
10
5
5 + 6 + 7 + 8 + 15 + 16 + 17 + 25 + 26 + 35
160
10
n
10n
T
D0
1
10
120
110
2
20
130
110
3
30
140
110
4
40
150
110
5
50
160
110
Therefore the equation for a 4 step stair on a 10×10 grid is 10n + 110
Following the same sequence you find that the equation for a 5 step stair is 15n + 220, 6 stair is 21n + 365 etc.
S
T
0
0
1
0
2
1
3
4
4
10
T=sa�+sb�+sc+d
s=0, T=0, d=0
s=1, T=0, a + b + c=0
s=2, T=1, 8a + 4b + 2c = 1
s=3, T=4, 27a + 9b + 3c = 4
s=4, T=10, 64a + 16b + 4c =10
T= 0.1666(s�) + 0(s�)-0.1666s-0
T= 0.1666(s�) – 0.1666s
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