infrared - Is there a sensor for counting the number of cards in a flipping scorecard?

I have a stack of cards and want to know how many cards are in the stack at any given time. Any thoughts on how I might be able to do this? Would it be possible for me to put a tag on every card and then use some type of sensor below the cards which could count the tags above it? The cards are suspended from rings like this:

While there was a prior question that was similar, it does not appear there were any answers that came out of that question.