@johnwasser - yeah that's exactly the kind of thing I'm looking for. How to mount the magnet (if the hall effect sensor is the way to go) so that it only triggers the home state at the proper time.
You could put a magnet on the back of the 0 card and use a Hall Effect switch to sense it.
The problem I found after thinking about this suggestion is that the back of every card is actually the front of the next card, so I can't mount something there or it will cover my display. As an example, on the back of the card that has the top half of the numeral "0" will be the bottom half of the numeral "1". So when that flap flips over, you see and display that reverse side. Hopefully that makes sense. So attaching a magnet somewhere could still work, but it can't be on any face of the cards.
Maybe the magnet needs to be mounted in a stationary position and a notched wheel "reveals" the magnet at the right time to the sensor?
Another thought I had while reading over my post was to use a micro switch. Maybe the wheel holding the flaps has a notch or bump on it at the home position that presses a microswitch as it comes around.