The Days of the Week
An Exercise In Theory of Mind
[NOTE: This essay is edited frequently as I learn new information about this subject.]
In the autism spectrum, there are people that have been called "savants."
What is a savant? A savant is someone who is very skilled at an area that is believed to be useless for any practical purposes. Those areas are called “splinter skills,” and savants are often punished for boring people about the thing that they obsess in.
Ironically, most people know very little as to how a savant is able to do what he or she can do, yet many people make fun of them for their abilities. This is yet another example of when a normal person lacks a theory of mind toward an autistic person, not the other way around.
One well-known savant skill is the ability to tell you what the day of the week any calendar date is on if you ask them (in a matter of seconds). For example, if you tell them you were born on August 20, 1988, they could tell you that it was a Saturday. But if you ask them how they did it, very few savants could explain to you how—just as you might have a hard time explaining why you know social skills from your heart and an autistic person has a hard time learning.
I’m not a savant, and I couldn’t do that in a matter of seconds. But I sat down one day, in the summer of 2000, and analyzed the weekdays in calendar dates and found that they followed specific patterns. I realized that there is a method that can explain how this is done. It requires memorizing a system of patterns and mathematical algorithms.
I wrote them down and mailed them to several autistic friends, who told me that I was accurate in my calculations. Since 2000, when I originally wrote my ideas, they have changed as I have learned more about our calendar (the Gregorian calendar).
There are likely a lot more ways to solve this problem, and numerous variations on each way. However, to most people, how to find out what day of the week a calendar date fell on is a mystery. Here I will show you one way to solve this problem, with a combination of patterns and mathematics.
So, without further ado, in an exercise regarding theory of mind, you are about to see a mathematical explanation to the art of finding weekdays on dates.
So, how do you find out what day of the week a specific date was on?
Dates are split up into three parts—-months, days, and years. For example:
Now, we need to find another calendar date where we have memorized what weekday it was on. From there we can find out what weekday 12/12/1965 fell on.
The simplest way to do that is if we know what weekday December 12th fell on in another year. Then we can simply count down from that year to 1965. But since that is not always an option, let’s pick a random date:
Although every day of the week is given at least one opportunity to be the 1st of a specific month, there are months that share days of the week. Because of this, we do not have to count each month from January to December.
However, those months differ depending on whether or not the year you are working in is a leap year or not. The following are the rules regarding which months share days of the week throughout the year—whether or not it is a leap year or a non-leap year:
ON LEAP YEARS:
January, April, and July share the same weekdays.
March and November share the same weekdays.
February and August share the same weekdays.
October does not share a weekday with any other month.
ON NON-LEAP YEARS:
January and October share the same weekdays.
April and July share the same weekdays.
February, March, and November share the same weekdays.
August does not share a weekday with any other month.
ON ALL YEARS:
September and December share the same weekdays.
May and June do not share a weekday with any other month.
So the first question is—is 2003 a leap year?
Leap years must follow two criteria. They must be divisible by 4, but they cannot be divisible by 100 unless they are also divisible by 400.
For example, even though 2100, 2200, and 2300 are divisible by 4, they are not leap years because they are also divisible by 100. But 2000 and 2400 are leap years since they are not only divisible by 4 and 100, they are also divisible by 400. However, all years that are divisible by 4 that are not divisible by 100 or 400 are leap years. 2364 is a leap year.
So, is 2003 a leap year? Let’s divide 2003 by 4.
2003 / 4 = 500.75
2003 is not divisible by 4. It is also not divisible by 100 or 400 since all numbers divisible by 100 or 400 must end in 0, and 2003 ends in 3.
Thus, it is not a leap year. With this information we can conclude:
January 2003 and October 2003 share the same weekdays.
Thus, October 23, 2003 starts on a Thursday.
We can use this information to find out when December 23, 2003 starts.
How do we find this information? The next month of the year starts on a predictable weekday based on the weekday the current month started and how many days are in the month. Here are the following formulas:
If the current month has 31 days (January, March, May, July, August, October, December), the next month starts three weekdays after the current month.
If the current month has 30 days (September, April, June, November), the next month starts two weekdays after the current month.
If the current month has 29 days (February on leap years), the next month starts one weekday after the current month.
If the current month has 28 days (February on non-leap years), the next month starts the same weekday as the current month.
This is because 28 days is equal to 4 full weeks, and 29, 30, and 31-month days have extra days along with the 7-day 4 weeks.
October is a 31-day month. Thus November starts three weekdays after October.
10/23/2003 is on a Thursday.
Thus, 11/23/2003 starts on a Sunday.
November is a 30-day month. Thus December starts two weekdays after November.
11/23/2003 is on a Sunday.
Thus, 12/23/2003 starts on a Tuesday.
Now we know that December 23, 2003 falls on a Tuesday. We can now use this information to find out what day December 1, 2003 was on. Then we can find out what December 12, 2003 started on.
A month consists of at least four weeks. As there are seven weekdays, and thus seven days in a week, the next day that repeats the weekday is seven days apart from a previous day. If December 23st starts on a Tuesday, then December 30th starts on a Tuesday.
We can determine what day of the week December 1st started on from December 23rd by analyzing how many weekdays we must add or subtract from the 23rd of the month’s weekday, based on this formula:
If the day is the 8th, 15th, 22nd, or 29th in the month, it falls on the same weekday as the 1st of the month.
If the day is the 2nd, 9th, 16th, 23rd, or 30th in the month, it falls on the first weekday after the 1st of the month’s weekday.
If the day is the 3rd, 10th, 17th, 24th, or 31st in the month, it falls on the second weekday after the 1st of the month’s weekday.
If the day is the 4th, 11th, 18th, and 25th in the month, it falls on the third weekday after the 1st of the month’s weekday.
If the day is the 5th, 12th, 19th, or 26th in the month, it falls on the third weekday before the 1st of the month’s weekday.
If the day is the 6th, 13th, 20th, or 27th in the month, it falls on the second weekday before the 1st of the month’s weekday.
If the day is the 7th, 14th, 21st, or 28th in the month, it falls on the weekday before the 1st of the month’s weekday.
This means that December 1st starts one weekday before December 23rd. Since December 23rd starts on a Tuesday:
Go back one weekday from Tuesday = Monday
This means December 1st, 2003 fell on a Monday. With the same formula, we can find out what December 12th, 2003 fell on.
Looking back, we see that December 12th falls on the third weekday before the 1st of the month. Since December 1st starts on a Monday:
Go down three weekdays from Monday = Sunday, Saturday, Friday
December 12, 2003, fell on a Friday. Now we can find out what day of the week December 12, 1965 fell on.
The Year Cycles
Every date on the Gregorian calendar (the present calendar in much of the Western world) follows a cycle of weekdays, year by year, that remain constant throughout decades and centuries. Although every date is given a time to be on every weekday over time, a date shares a weekday on a set pattern. Because of leap years, calendar dates move up one date four years in a row and then move up two days when a leap year occurs. The cycle takes 27-29 years to repeat itself. This variation is due to the absence of a leap year during the 3 years every 400 years that are divisible by 100 as well as 4, as I discussed earlier when I described the rules behind leap years.
Although each year starts on each weekday more than once during this cycle, a leap year that starts on a specific day only occurs every 27-29 years.
Example: The year 2000 started on a Saturday, and it was a leap year. The next leap year that starts on a Saturday will not be until 2028.
Here is the cycle from 2000 to 2028:
Like in 2000, 2028 starts on a Saturday and is a leap year.
The days of the week table shown above is needed to find out what weekday a specific date falls on in each year compared to another year. This same method can be used with our date: December 12, 2003.
First, we must subtract 1965 from 2003 to find out how many years ago 1965 was from 2003.
2003 – 1965 = 38
Because of the 28-year weekday cycle, we do not need to individually count down all 35 years to find out what weekday 1965 started on. We can jump 28 years from 2003 and then count individual years from there.
So now we must subtract 28 from 2000 to find out what year that is.
2003 – 28 = 1975
This means that like in 2003, 12/12/1975 also started on a Friday, and is also not a leap year.
However, we cannot jump any further from here because 1965 is only 10 years from 1965, and 10 is less than 28.
So we will now count down years from 1975 to 1965. But how do we do that?
You might have noticed on the cycle above that as each year passes, the day starts on the next weekday on the list until a leap year appears, and the next weekday is two weekdays after the previous one. This applies to some dates, but not all dates. Because the disruption inside the leap year takes place on February 29th, dates before February 29th follow a different schedule than dates after February 29th. For example:
Now take a look at the chart for a day after February 29th:
In the chart for 1/3, the weekday is skipped between the leap year and the next year. Dates before February 29 follow this schedule.
In the chart for 4/9, the weekday is skipped between the leap year and the previous year. Dates after February 29 follow this schedule.
So, we must determine whether December 12th is before or after February 29th.
December 12th is after February 29th. Thus, when counting down, we skip a weekday after we encounter a leap year. And we know that 1975 is not a leap year, and that 12/12/1975 fell on the same weekday that 2003 fell on.
We also know that since a leap year is divisible by 4, and the last two digits of any number divisible by 4 are divisible by 4 themselves, that as we count down from 1975 to 1965, we will pass two leap years: 1972 and 1968.
So, now let’s count down:
And this means that December 12, 1965 fell on a Sunday.
And that’s how you find out the day of the week for a specific date. There are likely to be other methods that will be written. If anyone can find any mistakes that I have made, I’ll gladly correct them.
[More information about the Gregorian calendar can be found at "The Gregorian Calendar," Wikipedia.]