[athom]

If you DO start in March, your days/month fall into a neat little pattern:

Mar 31 - Aug 31 - Jan 31

Apr 30 - Sep 30 - Feb 28/29

May 31 - Oct 31

Jun 30 - Nov 30

Jul 31 - Dec 31

That highlights a few interesting cycles you can use to calculate dates from a simple count of days from the start of the year:

153 days every 5 months

61 days every 2

31 days per month

An "early reset" occurs every second month, jumping to the next 2-month cycle after the second day 30. Another occurs after every fifth month, jumping into a new 2-month cycle halfway through the last one of the 5-month. And of course, end of the year breaks the third "5-month" cycle WAY early, just before even its first 2-month is finished.

I won't try to detail the process of generating dates from this here, but I'm sure most of us here can work it out with just a little effort. Instead, here's a couple more fun facts to consider:

If you DO start the calendar from March, counting it as month 1, September (7) through December (10) map rather nicely to their own numeric positions. That seems a pretty strong hint, to me.

And I REALLY love this one:

The Gregorian cycle consists of four centuries. The first three are 36,524 days each: 100 years x 365 days + 24 days for the leap years. The hundredth year (ending in 00) is NOT considered a leap year, EXCEPT for every FOURTH hundredth. So that's 4 centuries * 36,524 days = 146,096, plus 1 more for the leap century, for 146,097.

That number is EXACTLY divisible by 7, which means the week cycle repeats WITH the Gregorian one. Good thing! Otherwise, we'd have to wait 2800 years!

[madcaptenor]

Zeller's congruence for the day of the week (https://en.wikipedia.org/wiki/Zeller's_congruence) exploits this:

- months are counted as 3, 4, 5, ..., 14, with 13 and 14 being January and February of the following year

- the contribution of the month to the day of the week is floor(2.6 * (m+1)) - the 2.6 comes from the 13 "extra" days (over the approximation 1 month = 4 weeks) in every 5 months.