In a calendar year, how many months, maximum, can start on a Sunday? What will be the next (very unlucky?) year when this maximum occurs?

Must every calendar year have at least one month that starts on a Sunday?

In a given year the day of the week of succeeding months can be obtained from that of the first month by adding the length of that preceding month mod 7.  In a non-leap year, therefore, the following are the days of the week offset from the day of the week of the first month's beginning:

      0  3  3  6  1  4  6  2  5  0  3  5

     
Note that every possibility is mentioned at least once, so any given day of the week, including Sunday, will occur as the first day of the month in any given non-leap year.

For leap years, the numbers are:

      0  3  4  0  2  6  0  3  6  1  4  6

     
and again, all possibilities are mentioned at least once, and any given day of the week occurs as the first of the month in any given year.

The most any given offset occurs is three. In non-leap years this is three 3's: February, March and November.  For leap years it's January, April and July.

Every day of the week gets to occupy the favored positions, as the first day of January progresses one day of the week per year, except from leap year to the following year, one day of the week is leapt over.  That's a 4-year cycle of leap years, against a 7-day week.  Since these are relatively prime numbers, the repetition cycle is 28 years.  That's within our lifetimes anyway.  The Gregorian correction last took place in 1900 and will next take place in 2100, but between such anomalous non-leap years, the 28-year cycle holds.

Here is a complete cycle of 28 years, in which each type of leap year is present once and each type of non-leap year is present three times.  It shows 2009 to be the next occurrence of three months starting on a Sunday and therefore having a Friday the 13th.  The actual day of the week is shown for the 1st of each month.

2001  2  5  5  1  3  6  1  4  7  2  5  7  2
2002  3  6  6  2  4  7  2  5  1  3  6  1  2
2003  4  7  7  3  5  1  3  6  2  4  7  2  1
2004  5  1  2  5  7  3  5  1  4  6  2  4  2
2005  7  3  3  6  1  4  6  2  5  7  3  5  1
2006  1  4  4  7  2  5  7  3  6  1  4  6  2
2007  2  5  5  1  3  6  1  4  7  2  5  7  2
2008  3  6  7  3  5  1  3  6  2  4  7  2  1
2009  5  1  1  4  6  2  4  7  3  5  1  3  3
2010  6  2  2  5  7  3  5  1  4  6  2  4  1
2011  7  3  3  6  1  4  6  2  5  7  3  5  1
2012  1  4  5  1  3  6  1  4  7  2  5  7  3
2013  3  6  6  2  4  7  2  5  1  3  6  1  2
2014  4  7  7  3  5  1  3  6  2  4  7  2  1
2015  5  1  1  4  6  2  4  7  3  5  1  3  3
2016  6  2  3  6  1  4  6  2  5  7  3  5  1
2017  1  4  4  7  2  5  7  3  6  1  4  6  2
2018  2  5  5  1  3  6  1  4  7  2  5  7  2
2019  3  6  6  2  4  7  2  5  1  3  6  1  2
2020  4  7  1  4  6  2  4  7  3  5  1  3  2
2021  6  2  2  5  7  3  5  1  4  6  2  4  1
2022  7  3  3  6  1  4  6  2  5  7  3  5  1
2023  1  4  4  7  2  5  7  3  6  1  4  6  2
2024  2  5  6  2  4  7  2  5  1  3  6  1  2
2025  4  7  7  3  5  1  3  6  2  4  7  2  1
2026  5  1  1  4  6  2  4  7  3  5  1  3  3
2027  6  2  2  5  7  3  5  1  4  6  2  4  1
2028  7  3  4  7  2  5  7  3  6  1  4  6  1

The first twelve columns show the days of the week of the first of each month, with Sunday being 1.  The last column is a count of how many of the months began on a Sunday.  The most can be seen to be three.  Every year has at least one month beginning on a Sunday.

Edited on April 18, 2005, 1:45 pm

Posted by Charlie on 2005-04-18 13:40:00