All Was Quiet on New Year's Day

A nameless friend writes:

Tom, What is the most common day of the week for New Year's Day to fall on?

The answer might surprise you.

First, let me explain for the uninitiated out there how the Gregorian Calendar works.
From the Wikipedia article:

The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar.^[1][2][3] It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening wordsInter gravissimas.^[4] The reformed calendar was adopted later that year by a handful of countries, with other countries adopting it over the following centuries. The motivation for the Gregorian reform was that the Julian calendar assumes that the time between vernal equinoxes is 365.25 days, when in fact it is presently almost exactly 11 minutes shorter. The error between these values accumulated at the rate of about three days every four centuries, resulting in the equinox occurring on March 11 (an accumulated error of about 10 days) and moving steadily earlier in the Julian calendar at the time of the Gregorian reform. Since the Spring equinox was tied to the celebration of Easter, the Roman Catholic Church considered that this steady movement in the date of the equinox was undesirable.
[snip]

The Gregorian calendar modified the Julian calendar's regular cycle ofleap years, years exactly divisible by four,^[6]including all centurial years, as follows:

Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100; the centurial years that are exactly divisible by 400 are still leap years. For example, the year 1900 is not a leap year; the year 2000 is a leap year.^[7]

In addition to the change in the mean length of the calendar year from 365.25 days (365 days 6 hours) to 365.2425 days (365 days 5 hours 49 minutes 12 seconds), a reduction of 10 minutes 48 seconds per year, the Gregorian calendar reform also dealt with the past accumulated difference between these lengths. Between AD 325 (when the Roman Catholic Church thought the First Council of Nicaea had fixed the vernal equinox on 21 March), and the time of Pope Gregory's bull in 1582, the vernal equinox had moved backward in the calendar, until it was occurring on about 11 March, 10 days earlier. The Gregorian calendar therefore began by skipping 10 calendar days, to restore March 21 as the date of the vernal equinox.

Cite: Gregorian calendar. (2012, January 11). In Wikipedia, The Free Encyclopedia. Retrieved 23:45, January 12, 2012, from http://en.wikipedia.org/w/index.php?title=Gregorian_calendar&oldid=470859503

Here's an interesting trick: What day of the week will Jan 1, 2400 fall on?  Answer: Saturday.  Every four hundred years the calendar resets itself and begins on a Saturday.  The reason is simple:


365.2425 × 400 = 20871, which is evenly divisible by 7. 


That means that New Year's Day of any year divisible by 400 will always be Saturday.  January 1, 2000, was on a Saturday, and January 1, 4000, will be on a Saturday.


That also means that years 400 years apart will begin on the same day: January 1, 2012, for example, was on a Sunday, and so will January 1, 2412.


This all means that in order to find out what the most common day of the week New Year's Day falls on, we need to analyze 400 years worth of New Year's Days.

Here is the table:

Days of week	Count
Sun	58
Mon	56
Tue	58
Wed	57
Thu	57
Fri	58
Sat	56
Grand Total	400

So the answer to your question is a three-way tie: Sunday, Tuesday, and Friday are the most common days for New Year's Day to fall on, but not by much.