What it teaches
The shortest honest example of a plan that contains itself. To move four discs you move three out of the way, move the big one, and move the three back — and moving three is the same problem again.
A child does not need the word recursion to feel that, and the feeling is what the game is for: the moment they notice the four-disc puzzle is the three-disc puzzle twice.
How it works
Tap a peg to lift its top disc, tap another to put it down. Dragging was tried and rejected: a disc half dropped between two pegs has nowhere to be, and this game is about the decision rather than the hand.
Three discs take seven moves, four take fifteen, five take thirty-one, and the screen says so. The Watch button plays the perfect answer, which is the recursion made visible.
What to look for
Start at three. Seven moves is short enough to hold in the head, and the pattern has to be seen there before four discs mean anything.
The useful question is "where does the biggest disc need to be, and what is in its way?" — which is the recursive step, asked in words.