Let the sides be a,b, and c, (scalene, means no sides equal)
a + b + c < 18, and with the triangle inequality a + b > c, or -a - b < -c
a + b + c < 18
-a - b < -c
Adding the inequalities c < 18 - c
2c < 18
c < 9
By symmetry, a < 9, b < 9, so no side can be over 8.
Add any two sides, their sum must be greater than the 3rd side.
I'll just list them (a,b,c), shortest side to longest.
Shortest side cannot be 1, because (1,2,3) is not a triangle.
With shortest side 2:
(2,3,4), (2,4,5), (2,5,6), (2,6,7), (2,7,8).
With shortest side 3:
(3,4,5), (3,4,6), (3,5,6), (3,5,7), (3,6,7), (3,6,8).
With shortest side 4:
(4,5,6), (4,5,7), (4,5,8), (4,6,7).
Shortest side cannot be 5 or greater, because (5,6,7) has perimeter 18.
So there are 15.
Edwin
