## Unusual Calendars

SOmeone recently sent me an email that is apparently doing the rounds again with regard to August this year apparently having an extremely rare combination. Apparently this year August will have five Fridays, five Saturdays and five Sundays. According to th email this combination only occurs once every 823 years.

This didn’t sound quite right to me so I did a bit of checking and discovered that 1958 also had this exact same calendar with five Fridays, five Saturdays and five Sundays. Now August 1958 was about two years before I was bron so if this combination only occurs every 823 years then I must be 821 years old. I knew I was getting older but I didn’t think I was anywhere near that old yet

A little further investigation shows that August 1969, 1975, 1980, 1986, 1997, 2003 and 2008 all also had (you guessed it) five Fridays, five Saturdays and five Sundays. So if this occurs only once every 823 years then this year I will be 6582.

How ridiculous. As anyone who has ever seen a perpetual calendar knows, there are only 14 possible calendars to cover an entire year with seven of these being for leap years and the other seven covering all years that are not leap years. With the exception of February the arrangements for the particular months are duplicated exactly once between these two sets of calendars and so for every month except February a given monthly calendar repeats on average once every seven years. Leap years mean that it never repeats after exactly seven years and will instead follow a cycle of 11, 6, 5, 6, 11, 6, 5, 6 and so on (with a minor adjustment for when a year ending in 00 is not a leap year – eg. 1900 and 2100). Even February has its non-leap year calendar occur three times every 28 years and the leap year calendar appear once every 28 years (again except when the year ends in 00 and is not a leap year). So the longest that any monthly calendar can go between repeating itself is where one of the February leap year calendars lands on either side of a year ending in 00 that is not a leap year. In that situation exactly forty years pass between one occurrence of one of the February leap year calendars and the next. With the way the current calendar works it is impossible for any given monthly calendar to appear any less frequently than that. Having 40 years pass between 29th February falling on a Monday (for example) and when 29th February again falls on a Monday can be considered a rare event because as this 40 year gap only occurs three times every four hundred years the occurrence of a 40 year gap for February 29th on a given day of the week itself follows a 1100, 600, 500, 600 pattern (or at least it would if it were not that there is an extra leap year every 3300 years – the next being in 4882). So while having a 40 year gap betweew years in which 29th February lands on a Monday may be as much as 1100 before that particular gap occurs again, there will be lots of years during that period where 29th February lands on a Monday, just most of them will only be 28 years apart.