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Many dates in the Gregorian calendar repeat the day of the week after 180 and/or 220 years. In fact, this was already the case for October 15, 1582 (1582, 1762, and 1802 all had October 15 on a Friday).

But not all dates do. Adding 180 or 220 years could either go back or go forward one day.

To be precise, here's what happens:

When adding 180 years

  • October 15, 1582 to February 28, 1620 — the day of the week remains the same after adding 180 years
  • March 1, 1620 to February 28, 1700 — the day of the week goes back one day after adding 180 years
  • March 1, 1700 to February 28, 1720 — the day of the week remains the same after adding 180 years
  • March 1, 1720 to February 28, 1800 — the day of the week goes back one day after adding 180 years
  • March 1, 1800 to February 28, 1900 — the day of the week remains the same after adding 180 years
  • March 1, 1900 to February 28, 1920 — the day of the week goes forward one day after adding 180 years
  • March 1, 1920 to February 28, 2020 — the day of the week remains the same after adding 180 years

When adding 220 years

  • October 15, 1582 to February 28, 1680 — the day of the week remains the same after adding 220 years
  • March 1, 1680 to February 28, 1700 — the day of the week goes back one day after adding 220 years
  • March 1, 1700 to February 28, 1800 — the day of the week remains the same after adding 220 years
  • March 1, 1800 to February 28, 1880 — the day of the week goes forward one day after adding 220 years
  • March 1, 1880 to February 28, 1900 — the day of the week remains the same after adding 220 years
  • March 1, 1900 to February 28, 1980 — the day of the week goes forward one day after adding 220 years
  • March 1, 1980 to February 28, 2080 — the day of the week remains the same after adding 220 years

With the above in mind, here's the question:

Given any date (other than February 29) in the Gregorian calendar, will that date always repeat the day of the week after either 180 or 220 years (with the exception of March 1, 1680 to February 28, 1700; March 1, 1900 to February 28, 1920; March 1, 2080 to February 28, 2100; March 1, 2300 to February 28, 2320; etc.)?

Geoffrey Trang
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    Note that weekdays in the Georgian calendar repeat exactly with a period of 400 years. – hmakholm left over Monica Aug 12 '19 at 23:50
  • It has to do with how many century years (i.e. divisible by 100) are in the 180 year interval and whether they are years divisible by 400 (leap years) or not divisible by 400 (not leap years). With the 180, if you have two of the non-leap century years, it goes back by one day, if you have only one century year which is non-leap it stays the same, if you have only one century year which is leap, it goes forward one day, and with two century years with one leap and one non-leap, it stays the same. I leave the 220 version to you :) – Ned Aug 13 '19 at 03:01

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