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Distinct Triangle Determination (Posted on 2015-12-16) Difficulty: 3 of 5
Each of A, B and C is a distinct positive divisor of 2016 (including 1 and itself).

Determine the total number of triangles having sides A, B and C.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 3
Here are the 524 triangles. If it counts as to how many ways of labeling them A, B and C, there are six times as many, or 3144.

7 21 24
7 9 14
7 9 12
7 14 18
7 14 16
7 42 48
7 18 21
7 18 24
7 28 32
7 12 14
7 12 18
7 12 16
7 36 42
7 8 9
7 8 14
7 8 12
7 24 28
7 16 21
7 16 18
7 32 36
3 7 9
3 7 8
3 14 16
3 6 7
3 6 8
3 4 6
3 12 14
3 8 9
3 16 18
21 63 72
21 42 56
21 42 48
21 126 144
21 28 42
21 28 36
21 28 48
21 28 32
21 84 96
21 36 42
21 36 56
21 36 48
21 56 63
21 56 72
21 24 42
21 24 28
21 24 36
21 24 32
21 72 84
21 112 126
21 48 63
21 48 56
21 32 42
21 32 36
21 32 48
21 96 112
9 21 28
9 21 24
9 14 21
9 14 18
9 14 16
9 42 48
9 18 21
9 18 24
9 28 36
9 28 32
9 12 14
9 12 18
9 12 16
9 36 42
9 56 63
9 24 28
9 24 32
9 16 21
9 16 18
9 16 24
9 48 56
9 32 36
63 126 168
63 126 144
63 84 126
63 84 112
63 84 144
63 84 96
63 252 288
63 168 224
63 72 126
63 72 84
63 72 112
63 72 96
63 112 126
63 112 168
63 112 144
63 144 168
63 224 252
63 96 126
63 96 112
63 96 144
63 288 336
2 7 8
2 3 4
2 6 7
2 8 9
14 21 28
14 21 24
14 21 32
14 63 72
14 42 48
14 18 21
14 18 28
14 18 24
14 28 36
14 28 32
14 84 96
14 36 42
14 36 48
14 56 63
14 24 28
14 24 36
14 24 32
14 72 84
14 16 21
14 16 18
14 16 28
14 16 24
14 48 56
14 32 42
14 32 36
6 7 9
6 7 12
6 7 8
6 21 24
6 9 14
6 9 12
6 14 18
6 14 16
6 18 21
6 28 32
6 12 14
6 12 16
6 8 9
6 8 12
6 24 28
6 16 21
6 16 18
6 32 36
42 63 84
42 63 72
42 63 96
42 126 144
42 84 112
42 84 96
42 252 288
42 56 63
42 56 84
42 56 72
42 56 96
42 72 84
42 72 112
42 72 96
42 112 126
42 112 144
42 48 63
42 48 84
42 48 56
42 48 72
42 144 168
42 224 252
42 96 126
42 96 112
18 21 28
18 21 36
18 21 24
18 21 32
18 63 72
18 42 56
18 42 48
18 28 42
18 28 36
18 28 32
18 84 96
18 36 42
18 36 48
18 56 63
18 56 72
18 24 28
18 24 36
18 24 32
18 72 84
18 112 126
18 48 63
18 48 56
18 32 42
18 32 36
18 32 48
18 96 112
126 252 336
126 252 288
126 168 252
126 168 224
126 168 288
126 144 252
126 144 168
126 144 224
126 224 252
126 224 336
126 224 288
126 288 336
4 7 9
4 7 8
4 21 24
4 9 12
4 14 16
4 6 7
4 6 9
4 6 8
4 18 21
4 12 14
4 8 9
4 16 18
28 63 84
28 63 72
28 42 63
28 42 56
28 42 48
28 126 144
28 84 96
28 36 63
28 36 42
28 36 56
28 36 48
28 56 63
28 56 72
28 72 84
28 72 96
28 112 126
28 48 63
28 48 56
28 48 72
28 144 168
28 32 42
28 32 36
28 32 56
28 32 48
28 96 112
12 21 28
12 21 24
12 21 32
12 63 72
12 14 21
12 14 18
12 14 24
12 14 16
12 42 48
12 18 21
12 18 28
12 18 24
12 28 36
12 28 32
12 36 42
12 56 63
12 24 28
12 24 32
12 16 21
12 16 18
12 16 24
12 48 56
12 32 42
12 32 36
84 126 168
84 126 144
84 252 288
84 168 224
84 112 126
84 112 168
84 112 144
84 144 168
84 144 224
84 224 252
84 224 288
84 96 126
84 96 168
84 96 112
84 96 144
84 288 336
36 63 84
36 63 72
36 63 96
36 42 63
36 42 56
36 42 72
36 42 48
36 126 144
36 84 112
36 84 96
36 56 63
36 56 84
36 56 72
36 72 84
36 72 96
36 112 126
36 112 144
36 48 63
36 48 56
36 48 72
36 144 168
36 224 252
36 96 126
36 96 112
252 504 672
252 336 504
252 288 504
252 288 336
8 21 28
8 21 24
8 9 14
8 9 12
8 9 16
8 14 21
8 14 18
8 14 16
8 42 48
8 18 21
8 18 24
8 28 32
8 12 14
8 12 18
8 12 16
8 36 42
8 56 63
8 24 28
8 16 21
8 16 18
8 32 36
56 63 84
56 63 72
56 63 112
56 63 96
56 126 168
56 126 144
56 84 126
56 84 112
56 84 96
56 252 288
56 72 126
56 72 84
56 72 112
56 72 96
56 112 126
56 112 144
56 144 168
56 224 252
56 96 126
56 96 112
56 96 144
56 288 336
24 63 84
24 63 72
24 42 63
24 42 56
24 42 48
24 126 144
24 28 42
24 28 36
24 28 48
24 28 32
24 84 96
24 36 42
24 36 56
24 36 48
24 56 63
24 56 72
24 72 84
24 112 126
24 48 63
24 48 56
24 32 42
24 32 36
24 32 48
24 96 112
168 252 336
168 252 288
168 224 252
168 224 336
168 224 288
168 288 336
72 126 168
72 126 144
72 84 126
72 84 112
72 84 144
72 84 96
72 252 288
72 168 224
72 112 126
72 112 168
72 112 144
72 144 168
72 224 252
72 224 288
72 96 126
72 96 112
72 96 144
72 288 336
504 672 1008
16 21 28
16 21 36
16 21 24
16 21 32
16 63 72
16 42 56
16 42 48
16 18 21
16 18 28
16 18 24
16 18 32
16 28 42
16 28 36
16 28 32
16 84 96
16 36 42
16 36 48
16 56 63
16 24 28
16 24 36
16 24 32
16 72 84
16 112 126
16 48 63
16 48 56
16 32 42
16 32 36
112 126 168
112 126 144
112 126 224
112 252 336
112 252 288
112 168 252
112 168 224
112 144 252
112 144 168
112 144 224
112 224 252
112 224 288
112 288 336
48 63 84
48 63 72
48 63 96
48 126 168
48 126 144
48 84 126
48 84 112
48 84 96
48 252 288
48 56 63
48 56 84
48 56 72
48 56 96
48 72 84
48 72 112
48 72 96
48 112 126
48 112 144
48 144 168
48 224 252
48 96 126
48 96 112
336 504 672
144 252 336
144 252 288
144 168 252
144 168 224
144 168 288
144 224 252
144 224 336
144 224 288
144 288 336
32 63 84
32 63 72
32 42 63
32 42 56
32 42 72
32 42 48
32 126 144
32 84 112
32 84 96
32 36 63
32 36 42
32 36 56
32 36 48
32 56 63
32 56 84
32 56 72
32 72 84
32 72 96
32 112 126
32 48 63
32 48 56
32 48 72
32 144 168
32 224 252
32 96 126
32 96 112
224 252 336
224 252 288
224 504 672
224 336 504
224 288 504
224 288 336
96 126 168
96 126 144
96 252 336
96 252 288
96 168 252
96 168 224
96 112 126
96 112 168
96 112 144
96 144 168
96 144 224
96 224 252
96 224 288
96 288 336
288 504 672
288 336 504

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For a2 = 0 To 5
 For a3 = 0 To 2
 For a7 = 0 To 1
  DoEvents
  a = Int(2 ^ a2 * 3 ^ a3 * 7 ^ a7 + 0.5)
  For b2 = 0 To 5
  For b3 = 0 To 2
  For b7 = 0 To 1
   b = Int(2 ^ b2 * 3 ^ b3 * 7 ^ b7 + 0.5)
 For c2 = 0 To 5
 For c3 = 0 To 2
 For c7 = 0 To 1
  c = Int(2 ^ c2 * 3 ^ c3 * 7 ^ c7 + 0.5)
  
  small = a: large = a
  If b < small Then small = b
  If c < small Then small = c
  If b > large Then large = b
  If c > large Then large = c
  middle = a + b + c - small - large
  
  If small + middle > large And small < middle And middle < large Then
    If a < b And b < c Then
      Text1.Text = Text1.Text & a & Str(b) & Str(c) & crlf
    End If
    ct = ct + 1
  End If
  
 Next
 Next
 Next
  Next
  Next
  Next
 Next
 Next
 Next


 Text1.Text = Text1.Text & crlf & ct & " done"
  
End Sub

Edited on December 17, 2015, 11:32 am
  Posted by Charlie on 2015-12-16 15:13:21

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