compressed calendars and the Doomsday algorithm

Infodesign challenge -- how to fit a calendar for a year onto a business card.

There are many solutions; basically it seems that in order to do this well one has to exploit the fact that our calendar has some sort of structure. In short, all months look the same if you just rename the days of the week.

#2, for example, takes this into account: in a non-leap year January and October "look the same", as do February, March and November; September and December; April and July.

Incidentally, this is not a problem I particularly worry about, because I know the Doomsday algorithm. In short:

• It is relatively easy to determine what day of the week the last day of February falls on in a given year. (Because of leap years, a lot of calendar algorithms focus on this day. There are some cases where for the purposes of formulas it is convenient to think of January and February as the 13th and 14th months of the previous year.) This day of the week is called "Doomsday". Doomsdays run in a 400-year cycle, as does the entire Gregorian calendar. Doomsday in 1900 was a Wednesday, and in 2000 it was a Tuesday. For every twelve years after a multiple of 100 Doomsday moves forward one day; this is because twelve years include three leap years, so Doomsday actually moves forward 15 days. (So Doomsday in 2012 is a Wednesday, in 2024 a Thursday, and so on.) So take the last two digits of the year and divide by 12; call the quotient q and the remainder r. Move Doomsday forward from its day in the last century year by q+r days (each dozen years moves Doomsday forward fifteen (= one) days, then each year after that moves it forward one more day) plus r/4 days rounded down (for leap years). So, for example, Doomsday in 2000 was Tuesday; 7 divided by 12 is 0, with remainder 7. So Doomsday in 2007 is Tuesday, plus zero days, plus seven days, plus one day (7/4 rounded down is 1); that's Wednesday. Indeed, February 28 was a Wednesday this year.


• certain days will always fall on the same day of the week as that day; the easiest set to remember is 4/4, 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, 11/7 [note that for the set named so far! it doesn't matter whether you put the month first or the day first!], 3/7, 2/(28 or 29), 1/(3 or 4). (A few others that stick in my head are 2/14 (Valentine's Day) in ordinary years, 7/4 (Independence Day), 10/31 (Halloween) and 12/26 (the day after Christmas).) All of these days were Wednesdays this year as well.


• once you have a set that includes at least one day in each month, it's not hard to work within each month, probably because we have a decent amount of practice doing so. November 7 was a Wednesday; thus November 21 (fourteen days later) was also a Wednesday; thus November 25 is a Sunday. And my television agrees with me; it is showing that execrable show that only survives because of its cushy post-Simpsons time slot, King of the Hill.

A real challenge would be to figure out rules that let someone calculate the day of the week of a given date in the Hebrew calendar in their head. I suspect it's not possible.