One version of the Mother Goose rhyme...
Thirty days have September,
April, June, and November;
February has 28 alone,
All the rest have 31;
Except leap year, that's the time,
When February's days are 29.
You can also remember which months have 31 by counting on and between your knuckles. Beginning with January, you run out of knuckles on one hand (at July) and start over (with August). December and January are also two months in a row requiring 31 days.
As a mathematician, I like simplicity, not that the above is two awful complicated. From an information-theory perspective, comparing the content of the information to the formula used to define it, it's complicated.
So, given that our years have 365 days (except for leap years, which have 366 days, unless the leap year happens to fall on a century mark, etc....), can this be simplified?
My first instinct is to take one day from each of, say, July and January, and add those days to February. That makes February a 30-day month, except when it has 31 days (as in leap year, a non-century leap year, mind you). Then we have the following pattern:
January 30 February 30 March 31 April 30 May 31 June 30 July 30 August 31 September 30 October 31 November 30 December 31
That pattern gets upset, of course, in leap year, when we might add an extra day, say, to June, yielding a distinctly different pattern. Alternatively, if we're prepared for a radical change, we might try the following:
January 30 February 30 March 31 April 30 May 31 June 30 July 31 August 30 September 31 October 30 November 31 December 30
Then, in leap year, we add 1 day to January.
Or, again, as another radical alternative, we could do the following:
January 30 February 31 March 30 April 31 May 30 June 31 July 30 August 31 September 30 October 31 November 30 December 30
and, in leap year, add the extra day at the end of the year to December. Either way, counting the days of the year is simplified...
7x30 + 5x31 = 210 + 155 = 365 6x30 + 6x31 = 180 + 186 = 366 in leap year