An interesting question. Of course there is some ambiguity in the formulation "when can we tell".

Certainly in explicit examples, one may be able to apply ad-hoc methods. For example, things are easier for the Sierpinski gasket and carpet, since these have identifiable features in terms of their complementary regions.

For example, if we wish to write the Sierpinski gasket as a union of smaller copies, it should be fairly easy to see that each complementary region must be mapped to another complementary region. But this means that each contraction must correspond to one of the "smaller triangles" that appear in the usual gasket contraction, and we need at least three of these to make up the whole gasket.

The same type of argument should work for the Sierpinski carpet.

EDIT: Let me provide a few additional arguments to illustrate what I mean in the case of the Sierpinski gasket and carpet.

## Lemma

Any equilateral triangle contained in the Sierpinski gasket is the triangle surrounding one of the "children" in the usual construction.

## Proof

An exercise for the reader. (Hint: Note that the only way for a straight line in the gasket to begin in one of the standard triangles but not end there is to pass through the two points by which it is connected to the rest of the gasket.)

## Corollary

If $A$ is an affine similarity that maps the Sierpinski gasket into itself, then $A$ can be written as a finite composition of the three maps from the "standard" IFS that generates the gasket.

A similar argument works for the carpet. Here it is not enough to consider the outer square, but if we add the first generation squares to it, the same works. To state this, let A be the union of the boundaries of nine squares that are joined together to form a larger square; e.g.
$$A=\{(x,y)\in[0,3]^2: x\in\{0,1,2,3\} \text{ or } y\in\{0,1,2,3\}\}$$.
Let us call any image of A under an affine similarity a "3-by-3 grid".

## Lemma

The outer boundary of any nine-by-nine grid contained in the Sierpinski gasquet is the boundary of one of the squares occuring in the usual construction.

Again, I will leave the proof as an exercise. The claim that the usual system is optimal then follows immediately once more.