let

a = the length of one leg, a is integer, a >0

b = the length of the base, b is integer, b >0

using the triangle inequality we get:

2a > b > 0

the perimeter is 18:

2a + b = 18

b = 18 - 2a

b > 0

18 - 2a > 0

18 > 2a

9 > a

plug b = 18 - 2a into 2a>b:

2a > 18 - 2a

4a > 18

a > 18/4

a > 4

now we know that 4 < a < 9:

for a = 5, b = 18 - 2*5 = 8

for a = 6, b = 18 - 2*6 = 6

for a = 7, b = 18 - 2*7 = 3

for a = 8, b = 18 - 2*8 = 2

there are 4 different isosceles triangles.