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Old 2019-03-30, 06:24   #7
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Jan 2016

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Originally Posted by ATH View Post
Why is it in the 28 years (336 months) cycle the 1st of the months falls on a monday exactly 48 times, tuesday 48 times, ..., and sunday 48 times, even though the length of the months switches in the weird pattern between 28,29,30 and 31 days long ?
In the Julian calendar, the leap year pattern repeats after 4 years which is 1,461 days. Now, when you consider weekdays, with 7 days in each week, you have to consider whether or not 7 divides 1,461. It does not. Therefore, the "least common multiple" where both leap year pattern and week day pattern repeats, will be 7 times 4 years, or in other words 7 times 1,461 days (that is 10,227 days). 7 is a prime.

So you do each of the 7 possible starts of the 1,461-day cycle. So any particular date (like the 1st of January, the 13rd of February, or anything) falls equally frequently on any day of week.

Let us see what happens in the Gregorian calendar. If we ignore the weekdays at first, the Gregorian calendar repeats after 400 years. 400 years is 146,097 (the 97-part is the 97 leap days in 400 years) in that system. Now, you would presuppose that you needed 7 times that period, i.e. 2,800 years, before both leap year and day-of-week pattern repeated, but that is wrong. Because, by accident, 7 divides 146,097 (= 400*365 + 97). In the Gregorian system, 400 years is exactly 20,871 weeks. Because of this coincidence, we do not get every possible date/day-of-week combination in the "expected" frequency.

2001-Jan-01 was a Monday. 2401-Jan-01 will be a Monday. 2801-Jan-01, 3201-Jan-01, etc. are all Mondays.

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