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Commemorative triangles (Posted on 2013-02-20) Difficulty: 3 of 5
...and now the reveal.

Determine the total number of integer sided triangles with perimeters 1 through 10.

Compare this sequence with the sequence from the problem Commemorations 2.

Prove these sequences are actually the same with a slight offset.

Note: an integer sided triangle is a triangle whose 3 sides are integers. They may or may not be equal to each other.

No Solution Yet Submitted by Jer    
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Solution computer solution but no proof Comment 1 of 1

DEFDBL A-Z
CLS
FOR perim = 1 TO 35
  ct = 0
  FOR a = 1 TO perim / 3
    FOR b = a TO (perim - a) / 2
     c = perim - a - b
     IF a + b > c THEN
       ct = ct + 1
     END IF
    NEXT b
  NEXT a
  PRINT perim, ct
NEXT

 

1             0
2             0
3             1
4             0
5             1
6             1
7             2
8             1
9             3
10            2
11            4
12            3
13            5
14            4
15            7
16            5
17            8
18            7
19            10
20            8
21            12
22            10
23            14
24            12
25            16
26            14
27            19
28            16
29            21
30            19
31            24
32            21
33            27
34            24
35            30

But it's also interesting if we disallow equal sides in the triangle (isosceles or equilateral):

DEFDBL A-Z
CLS
FOR perim = 1 TO 35
  ct = 0
  FOR a = 1 TO perim / 3
    FOR b = a TO (perim - a) / 2
     c = perim - a - b
     IF a + b > c AND c > b AND b > a THEN
       ct = ct + 1
     END IF
    NEXT b
  NEXT a
  PRINT perim, ct
NEXT

1             0
2             0
3             0
4             0
5             0
6             0
7             0
8             0
9             1
10            0
11            1
12            1
13            2
14            1
15            3
16            2
17            4
18            3
19            5
20            4
21            7
22            5
23            8
24            7
25            10
26            8
27            12
28            10
29            14
30            12
31            16
32            14
33            19
34            16
35            21

The same numbers appear, with a different offset this time.

As for a proof that these match the sequence in Commemorations 2, I don't really have the slightest idea of why the connection exists with the mathematical structure of that puzzle.


  Posted by Charlie on 2013-02-20 18:13:33
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