It's a cross between two puzzles!

Here are the patterns to notice:

When the scale factor changes by a factor of \(k\), the hypervolume (generalization of length, area, volume) changes by a factor of \(f = k^d\), where $d$ is the number of dimensions. So the number of dimensions is \(\log_k f\).

When a mole is whacked, all neighboring holes toggle whether they have a mole or not.

The real deal

Now we calculate the number of dimensions of these fractals.
Now we find out the optimal number of moves needed. (Moves indicated are with respect to all legal moves, not all spaces)

A SNAKE

Number of dimensions = \(\log_4 8\) = 1.5

DOWN ARROW

2 moves (middle, bottom)

DOTTED X

Number of dimensions = \(\log_3 5\) = 1.46497…

TRIAMOND

4 moves (only, bottom, left, left)

TETRA HOLE

Number of dimensions = \(\log_5 21\) = 1.891668149…

LONG LINE

5 moves (only, right, right, right, 4th-from-left)

BIG STARFLAKE

Number of dimensions = \(\log_3 4\) = 1.26185…

NUMBER SIX

7 moves (only, only, left, bottom, bottom, left-middle, bottom)

FRACTAL TRIGON

Number of dimensions: \(\log_2 3\) = 1.58496…

CURVED ART

8 moves (left, only, middle, bottom-right, middle, top, top-left, top)

We may notice that the names of the puzzles are weird. Perhaps we should index the numbers into the names. But the flavor text says something about “cross”. So we cross-index into the puzzle name on the opposite side (e.g. index 1 into DOWN ARROW and 2 into A SNAKE). This gives START on the left column and DIGIT on the right column. So the answer is START DIGIT.