How Often Are There 5 Sundays in February?

Date: 02/27/2004 at 09:51:54
From: Don
Subject: How often are there 5 Sundays in February

How often are there 5 Sundays during the month of February?

Date: 02/27/2004 at 15:22:56
From: Doctor Rob
Subject: Re: How often are there 5 Sundays in February

Thanks for writing to Ask Dr. Math, Don!

In order for there to be 5 Sundays in February, two things must
happen.  First, it must be a leap year.  Second, February 1st must
fall on a Sunday, so that the remaining four Sundays will fall on the
8th, 15th, 22nd and 29th.  This just happened in 2004, which I assume
prompted your question!

Now, if February 1st is a Sunday, then January 1st must have been on a
Thursday (you can confirm that with any calendar).  So an equivalent
question is asking how often January 1 occurs on a Thursday in a leap
year.

At first glance one would think that, since leap years are 1/4 of the
years, and Thursdays are 1/7 of the days, that the answer should be
1/28 of the years have that property.  In the Julian calendar, this is
the correct answer, and those years come exactly 28 years apart.

However, things are slightly more complicated in the Gregorian
calendar, because not every year divisible by 4 is a leap year.  For
example, 1800 and 1900 were not leap years, and 2100 will not be, either.

The Gregorian calendar pattern of years repeats with a cycle of 400
years.  This is equal to 146,097 days, or 20,871 weeks.  Within those
400 years there are 97 leap years.  Since 97 is not divisible by
7, the day of the week for 1 January cannot be the same for all the
days of the week.

If we look at the 97 leap years in a 400-year cycle, and count the
number with January 1 on a Thursday, we find just the following
years:

   2004, 2032, 2060, 2088, 2128, 2156, 2184,
   2224, 2252, 2280, 2320, 2348, 2376.

Notice that between century years, there are gaps of 28 years, but
across a century year, there is a gap of 40 years, and
10*28 + 3*40 = 400, and 10 + 3 = 13.  Thus there are 13 leap years
containing five Sundays in every 400 years.

The situation for other days of the week that January 1 falls on in
leap years in any cycle of 400 years is:

   Sunday     15
   Monday     13
   Tuesday    14
   Wednesday  14
   Thursday   13
   Friday     15
   Saturday   13
             ----
              97

Now between century years there are gaps of 28 years, but across
a century year, there are gaps of either 12 or 40 years.  There
can be 0, 1, or 2 12-year gaps.

   10*28 + 3*40 + 0*12 = 400,   10 + 3 + 0 = 13,
   11*28 + 2*40 + 1*12 = 400,   11 + 2 + 1 = 14,
   12*28 + 1*40 + 2*12 = 400,   12 + 1 + 2 = 15.

Finally, the answer to your question is, 13 times every 400 years, or,
on average, about once every 30.77 years.  The gaps between can be 28
or 40 years.

Feel free to reply if I can help further with this question.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/