Two people are playing a game. The first player announces "January 1". From then on, each player announces a date that's later in the year, subject to the proviso that only the month name or the date of the month can change, not both. The object is to be the one to announce "December 31".
If you're the second player, what date do you announce after the first player has announced "January 1", and what is your strategy from there on?
I would try January 20. That leaves 11 months and 11 days to go. From that point on, if the first player advances the months by n, I advance the days by n. if the first player advances the days by n, I advance the months by n.
In other words, as long as I keep the months remaining equal to the days remaining, the other player is constrained to have them always be different, and I am the only one who can win. I don't think that leap years will present a problem, but maybe I am missing something.
a) after Jan 20, if the first player says April 20 then I say April 23.
a) after Jan 22, if the first player says Jan 24 then I say March 24.
This strategy would have a problem, I think, if December had only 30 days, or if October or November was the leap month (instead of February), but the calendar was apparently crafted to make this game easy.