#september 21

An Animal Crossing Nook calendar shows two cubes arranged horizontally, showing the day "21" above a rectangle that says "September". Each of the two cubes has one digit on each face. Behind the calendar, there's a Glaceon Nanoblock sculpture, a variety of wooden geared figures like a train and a cat, and a bookshelf containing rendering books.

I got this two-cube calendar a couple of weeks ago! There's a puzzle with a trick here: how can one assign digits so it can show any day from 01 to 31, even though there are only 12 faces?

Extra puzzle once you've solved thatCan you show why there are 10 solutions — and why every solution also allows showing days 00 and 32?

Wikipedia has the two-cube calendar solution, and some neat variants: a three-cube calendar for 3-letter month abbreviations, a 6-cube calendar that also shows the day of the week, and another two-cube calendar that shows both the 2-digit day and day of the week.