If by the end of the days there’s more cups in the zinc than in the shelf, this is a fun solution. Stick some magnets to a plastic or paper cup. That way you can put it in your fridge door. Your kids will love it and they will always use the same one.
Don’t panic. Everything will be ok. I swear. If your kid grabbed a permanent marker and wrote all over your wooden furniture clean them with toothpaste. Yes, really. Try it and then tell me.