...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.
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.
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Posted by Charlie
on 2013-02-20 18:13:33 |
computer solution but no proof
