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# Treebeard's Stumper Answer4 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):
DateSunMonTueWedThuFriSatTotal

January 1 58565857575856400
January 2 56585658575758400
January 3 58565856585757400
January 4 57585658565857400
January 5 57575856585658400
January 6 58575758565856400
January 7 56585757585658400
January 8 58565857575856400
January 9 56585658575758400
January 10 58565856585757400
January 11 57585658565857400
January 12 57575856585658400
January 13 58575758565856400
January 14 56585757585658400
January 15 58565857575856400
January 16 56585658575758400
January 17 58565856585757400
January 18 57585658565857400
January 19 57575856585658400
January 20 58575758565856400
January 21 56585757585658400
January 22 58565857575856400
January 23 56585658575758400
January 24 58565856585757400
January 25 57585658565857400
January 26 57575856585658400
January 27 58575758565856400
January 28 56585757585658400
January 29 58565857575856400
January 30 56585658575758400
January 31 58565856585757400

February 28 58565856585757400
February 29 1315131513141497
March 1 58565856585757400

October 4 58565856585757400

December 25 58565857575856400

December 31 56585757585658400

SunMonTueWedThuFriSatTotal
Total Week Days: 20872208702087320869208722087020871146097

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:

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