apropos of nothing.... there are 850 lunar months in 25101 days. (more accurately than Gregorian calendar tracks the solar year).

It's difficult to design a geared mechanism (a mechanical clock with a 'phase of the moon' hand) that gives exactly that ratio because any straightforward design will need at least one gear with 2789 teeth (an uncomfortably large prime which is a factor of 25101).

But, after some analysis, I discovered some more tractable identities. The gearing design is no longer exactly 'straightforward' because it will require some differential gearing, but it's buildable.

one day is also (1/183 + 1/154) x 17/6 lunar months. That's a ratio you can get pretty easily with an input gear that moves 17 teeth every 6 days. Honest to god, it only takes three gears. 183,154, and your differential gear.

But what if you want the smallest possible number of gear teeth? Even if it makes your mechanism more complicated? In that case, one day is also (1/61 x 17/18) + (1/77 x 17/12) lunar months. Now you need 5 gears, but only 4 axles (one of them is fixed directly to the drive gear and just has a different radius/number of teeth). The biggest gear is 77 teeth. You could carve that out of _wood_.

I have a nice little diagram of it on a sheet of typing paper in front of me.

This is the sort of obsessive distraction thing my brain does when I'm avoiding thinking about something.