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Solutions to "Calendar Oddities"
By Solomon Golomb '51

A. The Gregorian Calendar
1. The Gregorian calendar was named for Pope Gregory XIII.
2. It was introduced in 1582.
3. It replaced the Julian calendar (named for Julius Caesar), in which every fourth year was a leap year.
4. In the Gregorian calendar, a year is ordinary unless the year number is a multiple of 4, in which case it is a leap year unless the year number is a multiple of 100, in which case it is ordinary unless the year number is a multiple of 400, in which case it is a leap year. (Got that?) So the year 2000 (500 times 4) was a leap year, but the year 1900 was not, and the year 2100 will not be.

B. Friday the 13th
1. In an ordinary year, February, March, and November all start on the same day of the week. If they start on Sunday, then all three will have a Friday the 13th. In a leap year, January, April, and July all start on the same day of the week. If this day is Sunday, then all three will have a Friday the 13th. There is no way to have four Fridays the 13th in a single calendar year.
2. In an ordinary year, February and March can both have a Friday the 13th, only four weeks apart.
3. In a leap year that starts on Sunday, January, April, and July will have Fridays the 13th in a span of 183 days (counting both end-days), or half a year. In an ordinary year that precedes a leap year, September, December, and March can have Fridays the 13th. This also has a span of 183 days, counting both end-days.
4. Each of the seven months from May through November begins on a different day of the week; so every year, one of these months will have a Friday the 13th and no year can be Friday-the-13th free. In two consecutive ordinary years, if July of the first year has a Friday the 13th, this will not recur until September of the following year, 14 months, or 61 weeks (427 days) later. (On average, in 14 months, we would expect each starting day of the month to occur twice on each day of the week.) The result is the same when a leap year is followed by an ordinary year (since there is no leap day in the 14-month interval in question). However, in an ordinary year followed by a leap year, if August of the first year has a Friday the 13th, this will not recur until October of the following year, again an interval of 14 months (or 61 weeks, or 427 days).

C. Miscellany
1. Every month has 11 ambiguous days, from the 1st to the 12th, excluding the day with the same number as the month, since dates like 4/4 have the same meaning in both systems. Thus each year has 132 ambiguous dates. (If a European saw a sign "Open 24/7" on an American store, might she think that the store would not be open until July 24?)
2. "OCT. 31 = DEC. 25" could mean "octal 31 = decimal 25"; that is, the number we would write as 25 in standard decimal notation becomes 31 when written in base 8.
3. The holidays in question occur on 1/1, 2/2, 5/5, 6/6, and 11/11.
4. When George Washington was born, England and its New World colonies were still on the Julian calendar, and reckoned February as near the end of the year. Hence, when George Washington was born, the "local date" was February 11, 1731. That corresponds to what we now call February 22, 1732, in the Gregorian calendar.

D. Subtleties
1. In an ordinary year there are 365 days, which is 52 weeks plus one day. In a leap year, there are 366 days, which is 52 weeks plus two days. In a 400-year cycle of the Gregorian calendar, there are 97 leap days (every fourth year except for three of the 100-years). Hence in 400 years, we have 400 x 52 weeks, plus 400 + 97 days. Now, 400 + 97 = 497 = 7 x 71, so we have exactly (400 x 52) + 71 weeks = 20,871 weeks. That is, every 400-year cycle of the Gregorian calendar starts on the same day of the week (1600, 2000, 2400 all begin on Sunday); and everything repeats, including the day of the week, every 400 years. Since 400 is not a multiple of 7, no date falls on a given day of the week exactly one-seventh of the time! (It would have to occur on that day of the week 400/7 times in 400 years, but 400/7 is not a whole number.)
2. In 400 years, there are 12 x 400 = 4,800 months. This number is also not a multiple of 7, so no day of the month can occur on a particular day of the week exactly one-seventh of the time. By actual count, the 13th of the month is (slightly) more likely to fall on Friday than on any other day of the week!
3. In 1616, Catholic Spain was already on the Gregorian calendar, while Protestant England (under Elizabeth I) was still on the Julian calendar. If we restated the date of Shakespeare's death as we did with Washington's date of birth, we would find that Shakespeare died on May 3 (Gregorian), but for some reason it is not customary to restate 17th-century English dates.
4. Russia under the Tsars, and in the Russian Orthodox Church even today, the Julian calendar prevails; but after the Bolshevik Revolution, Russia abandoned the Julian calendar for civil purposes, and the Julian date in October became a date in November. (I was careful not to say that Lenin and Co. adopted the Gregorian calendar. The official Soviet calendar had a slightly different rule for which century years should be leap years, but this would not diverge from the Gregorian calendar for several more centuries. I assume that with the collapse of the Soviet Union, Russia will simply use the Gregorian calendar for civil purposes.) The various Eastern Orthodox churches today do not agree on the date of Christmas. For Russia, it is still December 25 Julian, which (since 1900) has been January 7, Gregorian. The Armenians, for reasons related to the ancient date of the Epiphany, celebrate Christmas on January 6, Gregorian. And the Greeks, who were never part of the Tsarist empire, and are in both NATO and the European Union, celebrate Christmas on December 25, Gregorian (except for the Greek Orthodox Church in the Holy Land).
If this is not complicated enough for you, we could mention the intricacies of the Jewish, Muslim, and Chinese calendars, which are used to determine holiday dates such as Passover, the Eid al-Fitr (end of Ramadan), and Chinese New Year, respectively.

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