July 2016: 5 Fridays, 5 Saturdays, 5 Sundays

As a curiosity or with claimed mystical significance, people have noted that July 2016 has five Fridays, five Saturdays, and five Sundays. Various claims have asserted that this is extremely rare or extremely common. It turns out there’s an average of one such combination per year. Most years have one of these combinations per year. Some have none, and the rest have two.

The Gregorian Calendar has 14 possible calendars: a common year starting on each day of the week, and a leap year starting on each day of the week. Let’s call them A through N: A starts on Sunday in a common year, B on Monday in a common year, and so on through N, starting on Saturday in a leap year.

The Gregorian Calendar repeats on a 400-year cycle: a leap year in every year divisible by 4, except for century years that aren’t divisible by 400. In a 400-year cycle, there are 97 leap years, which means there are 400*365 + 97 = 146,907 days. That’s also an exact multiple of 7, which means that every 400-year cycle starts on the same weekday. January 1, 2001, was a Monday (Calendar B), so January 1, 2401, will also be a Monday (Calendar B). Whatever pattern we find, it will repeat every 400 years.

A month with five Fridays, Saturdays, and Sundays can happen only in a 31-day month that starts on a Friday. If the month starts on Sunday through Thursday, there won’t be five Sundays. If it starts on Saturday, there won’t be five Fridays. If the month starts on a Friday but it doesn’t have 31 days, there won’t be five Sundays.

A 31-day month that starts on a Friday happens only in the following cases:

  • Calendar A, December
  • Calendar C, March
  • Calendar D, August
  • Calendar E, May
  • Calendar F, January and October
  • Calendar G, July
  • Calendar I, March
  • Calendar J, August
  • Calendar K, May
  • Calendar L, October
  • Calendar M, January, July
  • Calendar N, December

In other words, in any given year, there might be zero, one, or two months that have five Fridays, five Saturdays, and five Sundays.

How often does each of these calendars show up? Calendars A, B, D, F, and G come up 43 times each in a 400-year cycle. Calendars C and E, 44 times each. Calendars H and M, 15 times. I, L, and N, 13 times. J and K, 14 times.

Calendars B and H have no 5-Friday/Saturday/Sunday combos, and they make up 58 years in a 400-year cycle. Calendars F and M have two in the same year. They also make up 58 years. The remaining 284 years have one combo each. In other words, in 400 years, there’ll be 400 months that include this particular combination. That’s an average of one per year, but 14.5% of the time, there’ll be two in the same year, and 14.5% of the time, there’ll be none in a given year.

The verdict: Months with five Fridays, five Saturdays, and five Sundays are pretty common, averaging once per year.

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