Treebeard's Stumper Answer 4 October 2002

Stumper Birthdays

Today is Friday, October 4. It is our Dunn Middle School Newsnote (and my stumper) day. Today also happens to be my birthday, so this is one of my stumper birthdays! When will this happen again, and when did it last happen, that October 4 falls on a Friday? You can easily find the answers with any computer calendar program. Then you will notice the real stumper. My next Friday birthday is almost twice as many years away as my last one. It seems any date should fall on all the days of the week equally over the years, but is that really true, and what is the pattern?

I swiped this Perpetual Gregorian Calendar Javascript code from Stephen R. Schmitt on the Web. Thanks Stephen! You can lookup any date in any year to see which days of the week it falls on in different years. I made a few changes and haven't validated it very far. If you don't have Javascript ennabled, I'm not sure what you'll see!

Sun Mon Tue Wed Thu Fri Sat

What is the pattern over the years for my stumper birthdays or any other date?

My next Friday, Oct. 4 stumper birthdays will happen in 2013, 2019, 2024, and 2030, with a spacing of 11, 6, 5, and 6 years. In those 28 years, my birthday (and any date) will fall on each week day four times. A year of 365 days is 52 weeks and a day, or two days in a leap year. So every date falls a day (or two) later every year. But the leap years make it complicated. It's even more surprising in the long haul because of the changes made with our Gregorian Calendar, by which 2000 was a leap year but 2100 won't be. Keep reading for the details.

Notes:

Starting with year 2002, you will find that pattern of 11, 6, 5, and 6 years holds for any date after February 28. Christmas Day is a Wednesday this year, and it will be again in 2013. Dates that fall before February 28 follow the pattern of 6, 5, 6, 11 years. New Year's Day was a Tuesday this year, as it will be again in 2008. Of course this is the same cycle starting in a different place. Like I said, it's the leap years that make it complicated.

The sum of 11, 6, 5, and 6 is 28. After a period of 28 years (sometimes called a Solar Cycle), the pattern repeats, or at least it would if every fourth year was a leap year. That's how it was in the original Julian Calendar adopted in Roman times. This is still the leap year rule most of us know, that a year is a leap year if it is evenly divisible by four.

But over the centuries, this rule proved to be a bit too much, and the calendar slowly grew out of sync with the traditional date for the Vernal Equinox. Finally in 1582, the Gregorian Calendar was proposed. The new calendar eliminated the ten extra days that had accumulated, and changed the leap year rule to eliminate three leap years every 400 years.

According to the new rule, century years (divisible by 100) are not leap years unless they are evenly divisible by 400. Therefore the century years 1800 and 1900 were not leap years, but the year 2000 was. The century years 2100, 2200, and 2300 will not be leap years, but the year 2400 will be. It's easy to forget this rule since the century year 2000 in our lifetime was a leap year by both rules.

The Gregorian Calendar reform also messed up the 28 year Solar Cycle whenever a cycle crosses a century that is not a leap year. Days reckoned by the Gregorian Calendar exactly repeat over a 400 year cycle.

I wrote a small (DOS) BASIC program to count how many times each date of the year falls on each weekday. I used the period January 1, 2000 to December 31, 2399, but the count would be the same for any 400 year period. Here's the first month of the result and a bit more. (You can view the complete list for the year as a separate text file.)

Day Distributions from 2000 to 2399 (400 years): Date Sun Mon Tue Wed Thu Fri Sat Total January 1 58 56 58 57 57 58 56 400 January 2 56 58 56 58 57 57 58 400 January 3 58 56 58 56 58 57 57 400 January 4 57 58 56 58 56 58 57 400 January 5 57 57 58 56 58 56 58 400 January 6 58 57 57 58 56 58 56 400 January 7 56 58 57 57 58 56 58 400 January 8 58 56 58 57 57 58 56 400 January 9 56 58 56 58 57 57 58 400 January 10 58 56 58 56 58 57 57 400 January 11 57 58 56 58 56 58 57 400 January 12 57 57 58 56 58 56 58 400 January 13 58 57 57 58 56 58 56 400 January 14 56 58 57 57 58 56 58 400 January 15 58 56 58 57 57 58 56 400 January 16 56 58 56 58 57 57 58 400 January 17 58 56 58 56 58 57 57 400 January 18 57 58 56 58 56 58 57 400 January 19 57 57 58 56 58 56 58 400 January 20 58 57 57 58 56 58 56 400 January 21 56 58 57 57 58 56 58 400 January 22 58 56 58 57 57 58 56 400 January 23 56 58 56 58 57 57 58 400 January 24 58 56 58 56 58 57 57 400 January 25 57 58 56 58 56 58 57 400 January 26 57 57 58 56 58 56 58 400 January 27 58 57 57 58 56 58 56 400 January 28 56 58 57 57 58 56 58 400 January 29 58 56 58 57 57 58 56 400 January 30 56 58 56 58 57 57 58 400 January 31 58 56 58 56 58 57 57 400   February 28 58 56 58 56 58 57 57 400 February 29 13 15 13 15 13 14 14 97 March 1 58 56 58 56 58 57 57 400   October 4 58 56 58 56 58 57 57 400   December 25 58 56 58 57 57 58 56 400   December 31 56 58 57 57 58 56 58 400   Sun Mon Tue Wed Thu Fri Sat Total Total Week Days: 20872 20870 20873 20869 20872 20870 20871 146097

What a remarkable pattern! Note the diagonal lines of 57s, and how the pattern repeats every seven days except at the February 29 leap day. Over any 400 year span, every date occurs 400 times, except there are just 97 leap days (instead of 100) because of the three missing century year leapdays. In any 400 year period, there are exactly 400 x 365 + 97 = 146,097 days. This number is evenly divisible by 7 for an even number of weeks, but it is not evenly divisible by 28, so it's not an even number of solar cycles. (It's interesting that the remainder is 7.) Because of that, there are more Tuesdays and fewer Wednesdays over the long haul. Christmas is slightly less likely to occur on a Monday or Saturday. My October 4 birthdays are more likely to occur on a Sunday, Tuesday, or Thursday, and less likely to occur on a Wednesday. My Friday stumper birthdays are in-between.

I'm sure this unexpected result is absolutely unimportant. But I love these trivial questions that expose deep patterns in things we take for granted and use every day. That's what stumpers are for!

Here are some links for more research:

• I've gone over some of this ground before in my Friday the 13th (13 Nov 1998) stumper which shows that the 13th of any month is slightly more likely to occur on a Friday than any other day of the week. Drew Larson's Calendar Trivia page is useful. The Calendar FAQ, Calendar Zone, and Calendars and their History have lots more calendar info and links. Edward M. Reingold's book Calendrical Calculations has comprehensive algorithms for many different world calendars.

• The Gregorian Calendar reform was adopted in Catholic countries in the year 1582, but it was not adopted in Britain (or it's then colony America) until 1752. One odd result is that the year 1700 was a leap year in America, but not in Spain. See my When is Washington's Birthday? stumper for another.

• The new leap year rule creates an odd pattern that I first found as problem 516 in Henry Ernest Dudeney's 536 Puzzles & Curious Problems (edited by Martin Gardner, 1967): The first day of a century year can never occur on a Sunday, Tuesday, or a Thursday, and the first day of a leap year century will always be a Saturday. Now I know it's not that remarkable. My birthday in a leap year century century will always be a Wednesday. Same principle.

• I've got more tricky calendar stumpers, including Palindromic Dates (5 Oct 2001), Five Fridays (12 Nov 1999), Friday the 13th (13 Nov 1998), Full Moon Birthday (2 Oct 1998), When is Washington's Birthday? (28 Feb 1997), and my Birthday Stumper (4 Oct 96). Here's another: we all know about the sixties and the nineties, so what should we call this decade? The oh-ohs sounds just right so far.

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