How many triangles?

Clearly triangle with sides 2, 2, and 2 is a solution.

To maintain integer sides and a perimeter of 6, if one of the sides were to be increased by 1, another would have to be decreased by 1.

This would lead to 3, 1, and 2. But this is not a triangle. Since $1+2=3$ , it fails to be larger than 3.

Any change larger than 1 would have an even more drastic effect, so no triangle other than 2, 2, 2 can have integer sides and perimeter 6.

The only possibility here is an equilateral triangle of side lengths 2,2 and 2.This is the only triangle that satisfies all the above listed conditions, clearly it is the only answer to this question