Yep, it counts all the way through, perpetually.
You could potentially make a really thick one of these with 365/2 tiles so that every month has its perfect amount of tiles, but then "what about leap years?"
At this point, February is a short month and you get bonus flips, easy enough.
This is just a proof of concept after all, there really ought to be a way to keep track of what month it is too.
In fact, the manufactured souvenir ones do.
If you're lucky enough to have a 3D printer, you can even print one of these that is pretty slick looking AND has a month tracker: https://www.thingiverse.com/thing:1785261. »