How many different isosceles triangle have perimeter of 18 and side lengths of natural numbers?

There are three or there are four, depending which definition you have been taught. If you are of the school that is taught that an isosceles triangle has 2 equal sides and 2 equal angles, there are three. If, however, you were taught that an isosceles triangle has at least 2 equal sides and at least 2 equal angles, then there are four: 5+5+8 6+6+6 Note: Strictly, this one is equalateral but conforms to an isosceles by the second theory. 7+7+4 8+8+2 The two equal sides, when added together, must be equal to or greater than the third side or they will not join up.