Balmer's question seems fair for the complexity of the answer he was expecting.
As the interviewee you would presumably provide the (mathematically) wrong answer, but you'd show your thinking along the way, including a small demonstration of CS principles.
Keep in mind that Balmer had a long career, so if he ever asked this question, it was probably back in the 80s when no one expected you to come up with the complex solution outlined in the post.
If you did outline the correct answer, that would be amazing, and you'd be an instant hire. But the question doesn't fundamentally seem broken to me because either answer (taking the bet or not) needs to be well justified.