document.write( "Question 843979: The number of scalene triangles having sides with integer lengths and perimeter less than 19 is:
\n" ); document.write( "a)17, b)18, c)19, d)20, e)21\r
\n" ); document.write( "\n" ); document.write( "I have tried this over and over but the most that I can get is 12.
\n" ); document.write( "2,3,4
\n" ); document.write( "2,4,5
\n" ); document.write( "2,5,6
\n" ); document.write( "2,6,7
\n" ); document.write( "2,7,8
\n" ); document.write( "3,4,5
\n" ); document.write( "3,4,6
\n" ); document.write( "3,5,6
\n" ); document.write( "3,5,7
\n" ); document.write( "3,6,7
\n" ); document.write( "3,6,8
\n" ); document.write( "3,7,8
\n" ); document.write( "I know that any 2 sides must have a sum greater than the 3rd side. \n" ); document.write( "
Algebra.Com's Answer #508560 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
I counted \"highlight%2817%29\" reasoning like this:
\n" ); document.write( "
\n" ); document.write( "Perimeter=18:
\n" ); document.write( "The longest side must be less than \"18%2F2=9\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "8,7,3
\n" ); document.write( "8,6,4
\n" ); document.write( "7,6,5
\n" ); document.write( "
\n" ); document.write( "Perimeter=17:
\n" ); document.write( "The longest side must be less than \"17%2F2=8.5\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "8,7,2
\n" ); document.write( "8,6,3
\n" ); document.write( "8,5,4
\n" ); document.write( "7,6,4
\n" ); document.write( "
\n" ); document.write( "Perimeter=16:
\n" ); document.write( "The longest side must be less than \"16%2F2=8\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "7,6,3
\n" ); document.write( "7,5,4
\n" ); document.write( "
\n" ); document.write( "Perimeter=15:
\n" ); document.write( "The longest side must be less than \"15%2F2=7.5\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "7,6,2
\n" ); document.write( "7,5,3
\n" ); document.write( "
\n" ); document.write( "Perimeter=14:
\n" ); document.write( "The longest side must be less than \"14%2F2=7\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "6,5,3
\n" ); document.write( "
\n" ); document.write( "Perimeter=13:
\n" ); document.write( "The longest side must be less than \"13%2F2=6.5\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "6,5,2
\n" ); document.write( "6,4,3
\n" ); document.write( "
\n" ); document.write( "Perimeter=12:
\n" ); document.write( "The longest side must be less than \"12%2F2=6\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "5,4,3 (That's a right triangle).
\n" ); document.write( "
\n" ); document.write( "Perimeter=11:
\n" ); document.write( "The longest side must be less than \"11%2F2=5.5\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "5,4,2
\n" ); document.write( "
\n" ); document.write( "Perimeter=10:
\n" ); document.write( "The longest side must be less than \"10%2F2=5\"
\n" ); document.write( "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%2B3%2B7=9\" .
\n" ); document.write( "
\n" ); document.write( "Perimeter=9:
\n" ); document.write( "The longest side must be less than \"9%2F2=4.5\"
\n" ); document.write( "The possible side lengths, in decreasing order are:
\n" ); document.write( "4,3,2
\n" ); document.write( "
\n" ); document.write( "Perimeter=8:
\n" ); document.write( "The longest side must be less than \"8%2F2=4\"
\n" ); document.write( "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%2B2=3\" . \n" ); document.write( "
\n" );