Question 843979
I counted {{{highlight(17)}}} reasoning like this:
 
Perimeter=18:
The longest side must be less than {{{18/2=9}}}
The possible side lengths, in decreasing order are:
8,7,3
8,6,4
7,6,5
 
Perimeter=17:
The longest side must be less than {{{17/2=8.5}}}
The possible side lengths, in decreasing order are:
8,7,2
8,6,3
8,5,4
7,6,4
 
Perimeter=16:
The longest side must be less than {{{16/2=8}}}
The possible side lengths, in decreasing order are:
7,6,3
7,5,4
 
Perimeter=15:
The longest side must be less than {{{15/2=7.5}}}
The possible side lengths, in decreasing order are:
7,6,2
7,5,3
 
Perimeter=14:
The longest side must be less than {{{14/2=7}}}
The possible side lengths, in decreasing order are:
6,5,3
 
Perimeter=13:
The longest side must be less than {{{13/2=6.5}}}
The possible side lengths, in decreasing order are:
6,5,2
6,4,3
 
Perimeter=12:
The longest side must be less than {{{12/2=6}}}
The possible side lengths, in decreasing order are:
5,4,3 (That's a right triangle).
 
Perimeter=11:
The longest side must be less than {{{11/2=5.5}}}
The possible side lengths, in decreasing order are:
5,4,2
 
Perimeter=10:
The longest side must be less than {{{10/2=5}}}
No possible solution, because with a longest side of length 4, the other two sides lengths could at most be 3 and 2, adding to a perimeter of {{{4+3+7=9}}} .
 
Perimeter=9:
The longest side must be less than {{{9/2=4.5}}}
The possible side lengths, in decreasing order are:
4,3,2
 
Perimeter=8:
The longest side must be less than {{{8/2=4}}}
No possible solution, because with a longest side of length 3, we cannot have a scalene triangle with integer side lengths. Segments of lengths 3, 2, and 1 do not form a triangle because {{{1+2=3}}} .