Observations

# Patterns in the calendar – how often can we reuse old calendars?

Calendar wheel from the Hayden Planetarium at the American Museum of Natural History in New York. Placing the year under the correct month indicates the calendar for that month. As can be seen the calendar for August 1993 starts with a Sunday. Picture and copyright Nick Lomb ©, all rights reserved

Z asks: Is this true? This month of August 2010 has 5 Sundays, 5 Mondays and 5 Tuesdays, which hasn’t happened for 800 years?

Answer: Any month of 31 days must have five of three consecutive days of the week. Which three depends on the day of the week with which the month begins. For example, if the month starts with a Wednesday then it has five Wednesdays, Thursdays and Fridays. Now as there are seven days of the week with which the month could begin, we would expect that every seven years August will begin with a Sunday as in 2010. However, the leap years every four years provide some complications.

The last time August had five Sundays, Mondays and Tuesdays was in 2004, six years ago as there was a leap year in between.

The previous August with five Sundays, Mondays and Tuesdays was in 1999, five years earlier as there were two leap years – 2000 and 2004 itself – in between. Then 1993 and then, surprisingly there was no such August until 1982. And 1982 is, of course, 28 years prior to 2010.

We have now established the patterns in the calendar. The calendar can repeat at five or six year intervals, but does not always do so. The only simple pattern is that it repeats every 28 years. It repeats every 28 years as 28 is the lowest common multiple of the 4 year cycle of leap years and the 7 weekdays.

The 28-year calendar cycle works well as long as the 4-year cycle of leap years is maintained. However, in the Gregorian Calendar that we use century years are not leap years unless they are divisible by 400. So 1900 was not a leap year, 2000 was a leap year while 2100 will not be a leap year. We are now in the middle of a long unbroken sequence of 4-yearly leap years from 1904 to 2096, but then in 2100 the 28-year calendar cycle will be broken.

That gives us many years until 2100 to recycle yearly calendars. You could store your 2010 calendar for use in 2038 and, as we saw above, you may even get a chance to use it earlier!

## 3 responses to “Patterns in the calendar – how often can we reuse old calendars?”

There are 14 calendars. In a calender cycle they follow the patterns (1,1), (2,2),(3,3), (4,5) , (6,6), (7,7), (1,1), (2,3), (4,4), (5,5), (6,6), (7,1), (2,2), (3,3), (4,4), (5,6), (7,7), (1,1), (2,2), (3,4), (5,5),(6,6), (7,7), (1,2), (3,3), (4,4), (5,5), (6,7) in a cycle. In each of the above ordered pair of coordinates the first coordinate represents the day the year begins and the second coordinate represents the day the year ends. Thus the years 1905 – 1932; 1933 – 1960; 1961 – 1988; 1989 – 2016; 2017 – 2044 etc, would follow the pattern above. Thus the years 1905, 1933, 1961, 1989, 2017 begin on Sunday and end on Sunday. There are many observations that can be noted in the pattern above but I leave the rest to the reader

• Robert says:

One pattern I have noticed. If Christmas Day, for example, falls on a Sunday in the year after a leap year (like it did in 2005) and it is on the same day six years later (in 2011), the weekday for Christmas in the leap year halfway between the two repeating odd-numbered years will always be a Thursday.

Does anyone know why it is always on Thursday in the leap between the two odd numbered years with it on Sunday (2008 between 2005 and 2011)?

• Andrew Jacob says:

Robert, while the pattern occurs, there is nothing unusual or special about it. It is simply a consequence of how the date of Christmas (or any other date) cycles through the days of the week. The cycling occurs because in a standard Calendar year there are 7*52+1, or 365 days – the +1 causes Christmas Day (or any other date) to fall on the following day each year.