Patterns in the Perpetual CalendarDate: 03/17/2000 at 03:09:16 From: Ella Subject: Patterns in the perpetual calendar I'm trying to find an answer to my question: Why do some years (like 1717, 1923, 1934, 1945, 1951, 9162, 1973 etc) that have the same calendar, have this calendar repeated every 6, then 11, then 11 again, and then again 6, 11, 11, 6 years? And why are leap years calendar repeated every 28 years? Please help. Date: 03/17/2000 at 09:12:03 From: Doctor Rob Subject: Re: Patterns in the perpetual calendar Thanks for writing to Ask Dr. Math, Ella. The reason for the 28-year cycle is that 28 is the least common multiple of 4 and 7. 7 is the number of days in a week, and 4 is the gap between leap years. If a certain year had 1 January as Sunday, and it was a leap year, then the following would be the day of the week on which 1 January fell during the 28-year cycle: Year No. Day Year No. Day Year No. Day Year No. Day 1 L Sun 8 Tue 15 Thu 22 Sat 2 Tue 9 L Wed 16 Fri 23 Sun 3 Wed 10 Fri 17 L Sat 24 Mon 4 Thu 11 Sat 18 Mon 25 L Tue 5 L Fri 12 Sun 19 Tue 26 Thu 6 Sun 13 L Mon 20 Wed 27 Fri 7 Mon 14 Wed 21 L Thu 28 Sat This happens because 365 = 52*7 + 1, and 366 = 52*7 + 2, so that over a normal year, the day of the week goes up by 1, and over a leap year, it goes up by 2. You see that the gaps between Sundays are 5, 6, 11, 6, for Mondays they are 6, 5, 6, 11, for Tuesdays they are 6, 11, 6, 5, and so on. The reason 6 appears is that one leap year occurs between the first and second occurrence, 5*1 + 1*2 = 7, and 5 + 1 = 6. The reason 5 appears is that two leap years intervene, one being the year of the first occurrence, 3*1 + 2*2 = 7, and 3 + 2 = 5. The reason 11 appears is that three leap years intervene, one of them causes that day of the week to be skipped, 8*1 + 3*2 = 2*7, and 8 + 3 = 11. If we drop the leap year calendars, which do repeat on a cycle of 28, then the normal year gaps between identical calendars follow the 11, 11, 6, 11, 11, 6, ... pattern, and note that 11 + 11 + 6 = 28. Each day of the week will appear four times during this cycle, once on a leap year and three times on normal years. There will be 7 leap years in the cycle and 21 normal years. Of course this 28-year cycle can be disrupted in a year like 1900 which is a year divisible by 100 but not by 400, and so is *not* a leap year, but the pattern holds from 1901 through 2100, the next exceptional year. If you take exceptional years into account, there is a 2800-year cycle, since 2800 is the least common multiple of 4, 7, 100, and 400. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 03/17/2000 at 12:44:06 From: Doctor TWE Subject: Re: Patterns in the perpetual calendar Hi Ella - thanks for writing to Dr. Math. You are on the right track by considering leap years. Non-leap years have 365 days, which is 52 weeks plus 1 day. If there were no leap years, every year would start off one weekday later than the year before it (and every other date on that year would be one weekday later than the year before as well). 1/1/2001 falls on a Monday, so 1/1/2002 will fall on a Tuesday, 1/1/2003 will fall on a Wednesday, and so on. If it weren't for leap years, we'd have a nice, simple pattern as follows: Year Weekday of 1/1 ---- -------------- 2001 MONDAY 2002 Tuesday 2003 Wednesday 2004 Thursday 2005 Friday (assuming that 2004 was NOT a leap year) 2006 Saturday 2007 Sunday 2008 MONDAY 2009 Tuesday (assuming that 2008 was NOT a leap year) : : If this were the case, isomorphic years (those years whose calendars are identical) would occur every 7 years. But leap years complicate the pattern. Because leap years have an extra day (52 weeks plus 2 days), each year following a leap year begins two weekdays later than the previous year - we "leap" over a day. (This is where the term "leap year" comes from.) With leap years, the repeating cycle for isomorphic years shrinks to 6 years (typically), because one day is "leaped over" during the cycle. For example, in 2004, Friday is leaped over, so the next year isomorphic to 2001 (starting on a Monday) moves up from 2008 (if there were no leap years) to 2007 (with leap years) - see the table below. Year Weekday of 1/1 ---- -------------- 2001 MONDAY 2002 Tuesday 2003 Wednesday 2004 Thursday (Friday leaped over) 2005 Saturday 2006 Sunday 2007 MONDAY 2008 Tuesday (Wednesday leaped over) 2009 Thursday : : Sometimes, however, the calendar we're looking at (say for example, the one that starts on a Monday) is the one that *gets* leaped over. For example: Year Weekday of 1/1 ---- -------------- 2007 MONDAY 2008 Tuesday (Wednesday leaped over) 2009 Thursday 2010 Friday 2011 Saturday 2012 Sunday (MONDAY leaped over) 2013 Tuesday 2014 Wednesday 2015 Thursday 2016 Friday (Saturday leaped over) 2017 Sunday 2018 MONDAY : : The year 2007 starts on a Monday, so the next isomorphic year should be 2013 (2007+6), but since the previous year (2012) is a leap year, Monday gets leaped over, and we have to go another 6 years from 2012 - the year which causes Monday to be skipped. Thus, we go from 2007 to 2018 before getting back to a Monday - a total of 11 years (two 6's minus the Monday that was leaped over). Sometimes, too, the year that is "supposed" to be isomorphic falls on a leap year. A leap year can't be isomorphic to a non-leap year because they don't have the same number of days. For example, 2018 starts on a Monday. The next isomorphic year is "supposed" to be 2024, but that is a leap year. Tuesday gets leaped over in 2024 AND Sunday gets leaped over in 2028 before the next isomorphic year to 2018. Thus, there are 11 years separating these isomorphic years instead of 6. Year Weekday of 1/1 ---- -------------- 2018 MONDAY 2019 Tuesday 2020 Wednesday (Thursday leaped over) 2021 Friday 2022 Saturday 2023 Sunday 2024 MONDAY (Tuesday leaped over) 2025 Wednesday 2026 Thursday 2027 Friday 2028 Saturday (Sunday leaped over) 2029 MONDAY : : These leap year complications form a 28-year cycle in which isomorphic non-leap years occur in intervals of 6 years - 11 years - 11 years, etc. Why is the cycle 28 years? Because there are 7 weekdays and leap years occur every 4 years. The smallest multiple of both 7 and 4 is 28. The days leaped over on successive leap years follow the pattern Monday-Saturday-Thursday-Tuesday-Sunday-Friday-Wednesday- Monday-Saturday etc. The year 2000 leaps over a Sunday. Based on this information, I think you can figure out why isomorphic leap years occur every 28 years. There is one more complication, however. Some years evenly divisible by 4 are NOT leap years, and therefore don't leap over a day. This causes further "irregularities" in the pattern. Years ending in 00 (for example, 1700, 1800, 1900, 2100, etc.) are not leap years UNLESS the century (the first two digits) is divisible by 4. Here's a partial chart: Year Leap year? ---- ---------- : : 1500 No 1600 Yes 1700 No 1800 No 1900 No 2000 Yes 2100 No 2200 No 2300 No 2400 Yes 2500 No : : I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/ |
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