This game is solvable even if there are 4 pairs, 5 pairs or more pairs of penguins, the point is to get the pattern right.

First, move a small penguin to the ice, and then its parent, move them to the opposite shore:


Move the small penguin back:


Move another small penguin to the ice, and move them to the opposite shore:


Move the first small penguin to the land, and move the ice back:


Move the small penguin's parent to the ice, and then move to the opposite shore:


Leave the parent behind and move back:


Follow the same pattern and you can move all the penguins to the opposite shore: