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Repeating dates – answers

November 7, 2012

Some thoughts on my previous post. The question – in two variations – is…

What is the largest possible gap (in the last 200 years anyway) between repeating dates?

a. Same Day + Date + Month

The rarest date is obviously 29 February, so repetitions of this date will be much less common.

For each 4 year cycle, each date – including 29 February – moves forward five days of the week (four days plus the leap day). The last seven leap days (from 1988 to 2012) have been successively on a Monday, Saturday, Thursday, Tuesday, Sunday, Friday and Wednesday. So the Wednesday 29 February this year (2012) was the first instance of that date since 1984, a gap of 28 years, which is typical for 29 February.

We can do even better than that. Our Gregorian calendar has a leap every four years, except when the year is a multiple of 100 but not 400. So 2000 was a leap year, but 1800 and 1900 were not and 2100 will also not be a leap year. Such exceptions are a good place to look for extreme values.

The day of the week for the years either side of 1900 are as follows:

In the year of… 29 February was on a…
1872 Thursday
1876 Tuesday
1880 Sunday
1884 Friday
1888 Wednesday
1892 Monday
1896 Saturday
1900 (none)
1904 Monday
1908 Saturday
1912 Thursday
1916 Tuesday
1920 Sunday
1924 Friday
1928 Wednesday

So, the five leap days (29 February) from 1872 to 1888 had a day of the week which was not repeated on that day for another 40 years (1912 to 1928). Some younger readers may see the similar such dates 200 years later (i.e. starting 2072), if they manage to become centenarians.

b. Same Day + Date

There are only seven 31st of the month dates each year, in January, March, May, July, August, October and December.

The gaps here between repeated dates depend upon the irregular succession of lengths of calendar months. As it happens, leap years again play their part.

For time periods which do span a leap day, repeating day-dates are up to 20 months apart, for periods starting on 31st December and 31 July. For example…

Friday, 31 December 2010
Monday, 31 January 2011
Thursday, 31 March 2011
Tuesday, 31 May 2011
Sunday, 31 July 2011
Wednesday, 31 August 2011
Monday, 31 October 2011
Saturday, 31 December 2011
Tuesday, 31 January 2012
Saturday, 31 March 2012
Thursday, 31 May 2012
Tuesday, 31 July 2012
Friday, 31 August 2012
Wednesday, 31 October 2012
Monday, 31 December 2012
Thursday, 31 January 2013
Sunday, 31 March 2013
Friday, 31 May 2013
Wednesday, 31 July 2013
Saturday, 31 August 2013
Thursday, 31 October 2013
Tuesday, 31 December 2013
Friday, 31 January 2014
Monday, 31 March 2014
Saturday, 31 May 2014
Thursday, 31 July 2014
Sunday, 31 August 2014

In fact, after March 2013, Sunday 31st is not repeated till August 2014. That is, there is only one Sunday 31st in just over three years, starting from July 2011. On the other hand there are two further instances Friday 31st in that time.

For time periods not spanning a leap day, the longest gap starts on any date in July, where day-date combinations (not the 31st) are not repeated for 14 months, in September of the following year. For example…

Saturday, 7 July 2012
Tuesday, 7 August 2012
Friday, 7 September 2012
Sunday, 7 October 2012
Wednesday, 7 November 2012
Friday, 7 December 2012
Monday, 7 January 2013
Thursday, 7 February 2013
Thursday, 7 March 2013
Sunday, 7 April 2013
Tuesday, 7 May 2013
Friday, 7 June 2013
Sunday, 7 July 2013
Wednesday, 7 August 2013
Saturday, 7 September 2013

For the 31st of the month and  time periods not spanning a leap day, repeating day-dates are up to 17 months apart, for periods starting on 31st March and 31st July. For example…

Saturday, 31 March 2012
Thursday, 31 May 2012
Tuesday, 31 July 2012
Friday, 31 August 2012
Wednesday, 31 October 2012
Monday, 31 December 2012
Thursday, 31 January 2013
Sunday, 31 March 2013
Friday, 31 May 2013
Wednesday, 31 July 2013
Saturday, 31 August 2013
Thursday, 31 October 2013
Tuesday, 31 December 2013
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