The Days of the Week:
An Exercise In Theory of Mind

James Williams

 

 

Originally written in 2000 at age 12, revised multiple times afterwards.

 

Throughout history, there are people—some of whom are on the autism spectrum—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 1, 1989, they could tell you that it was a Tuesday. 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.

 

Since 2000, when I originally wrote my ideas, they have changed as I have learned more about our calendar. I learned that the calendar used in the United States and many other countries is technically called the Gregorian calendar, and have periodically edited the algorithms after being notified of inaccuracies by other autistic people or noticing them myself.

 

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 in the Gregorian calendar.

 

 

Introduction

 

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:

 

December 12, 1965

 

In this date, December is the month, 12 is the day, and 1965 is the year. 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:

 

January 23, 2003

 

January 23, 2003 fell on a Thursday. We can now use this information to find out what weekday December 12, 2003 fell on. We can then count backwards from 2003 to 1965. We can start this process by counting from January to December, to find out what day of the week December 23, 2003 started on.

 

Although every day of the week is given at least one opportunity to be the 1st of a specific month within each calendar year, there are months within each year 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 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 common year:

 

 

IN 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.

 

IN COMMON 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.

 

IN 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 1700, 1800, and 1900 are divisible by 4, they are not leap years because they are also divisible by 100. But 1600 and 2000 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. Thus, 1936 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 a common 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? All dates on the next month of the year start on a predictable weekday based on the weekday of the date of the current month being analyzed and how many days are in that same month, utilizing 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, as follows:

 

 

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 23rd starts three weekdays after October 23rd.

 

10/23/2003 is on a Thursday.

Thus, 11/23/2003 starts on a Sunday.

 

November is a 30-day month. Thus, December 23rd starts two weekdays after November 23rd.

 

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. For example, If December 23rd 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

 

Most dates on the Gregorian calendar follow a distinct weekday cycle, year by year. Although the Gregorian calendar completes its full cycles of weekdays and months every 400 years, a typical weekday cycle usually takes 28 years to complete, since there are 7 distinct weekdays, and because leap years typically occur every 4 years (except when, as mentioned above, the year is divisible by 4 and 100, wherein a leap year does not occur for 8 years, such as from 1896 to 1904). As a result, the 28 year weekday cycle happens uniformly unless it is disrupted by the absence of a leap year, which occurs three times during the typical 400-year cycle of the Gregorian calendar. Indeed, although February 29th only occurs on leap years and has a different counting pattern compared other dates on the calendar, it is still subject to the same 28 year weekday cycle and subsequent disruptions caused by the leap year discrepancy mentioned above.

 

Therefore—if you are counting forward or back the weekdays of any given date by year to figure out a specific calendar date, and you are not going past any of the common years that are divisible by 4 and 100, you are free to jump from the weekday you are starting from up or down every 28 years, since it will be the same weekday. If, however, you are going past one of those common years, you will need to count up or down weekdays, one year by one year, until the weekday cycle has resumed itself (which will vary, based on what year or date you are looking for). Likewise, if the year you are counting up or down to is less than 28 years before or after the year you are starting form, you will also need to count up or down weekdays one year by one year as well. For February 29th, this would mean counting up or down each leap year, with the same rules of the weekday cycle as mentioned above.

 

Example 1: The year 1967 started on a Sunday, which was a common year. 1995 is 28 years after 1967. Since no common years that are divisible by 4 and 100 occur between 1967 and 1995, the year 1995 also started on a Sunday, which was also a common year. Here is the cycle from 1967 to 1995:

 

 

Example 2:  February 29th, 1948 occurred on a Sunday. 1976 is 28 years after 1948. Since no common years that are divisible by 4 and 100 occur between 1948 and 1976, February 29th, 1976 also occurred on a Sunday. Here is the cycle from 1948 to 1976:

 

 

 

This information can be utilized to find out what weekday a specific date falls on in each year compared to another year and can be used to determine our target date—December 12, 1965—from December 12, 2003. As mentioned above, December 12, 2003, fell on a Friday.

 

Since no common years that are divisible by 4 and 100 occur between 2003 and 1965 (2000 was a leap year since it is divisible by 4, 100, and 400, and completed the 400-year cycle between 1600 and 2000), the 28 year weekday cycle is not disrupted between 2003 and 1965. Therefore, we can use this cycle to simplify the process of counting down weekdays from 2003 to 1965. First, we must subtract 1965 from 2003 to find out how many years ago 1965 was from 2003.

 

2003 – 1965 = 38

 

Therefore, 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, December 12, 1975 also started on a Friday, which is also a common 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 now can count down years from 1975 to 1965. But how do we do that?

 

To count up or down weekdays, one by one, with a specific dates between years, one of three counting patterns must be used, depending on the date.

 

If you are counting down or up with a date before February 29th, each date occurs one weekday earlier (if counting down) or one weekday later (if counting up) between common years, and between a leap year and a preceding common year (if counting down) or between a common year and a following leap year (if counting up). Each date occurs two weekdays earlier between a common year and the preceding leap year (if counting down), or two weekdays later between a leap year and the following common year (if counting up). For example:

 

 

 

If you are counting down or up with a date after February 29th, each date occurs one one weekday earlier (if counting down) or weekday later (if counting up) between common years, and between a common year and a preceding leap year (if counting down) or between a leap year and a following common year (if counting up). Each date occurs two weekdays earlier between a leap year and the preceding common year (if counting down), or two weekdays later between a common year and the following leap year (if counting up). For example:

 

 

If you are determining a weekday for February 29th (which date occurs every leap year), the following counting patterns apply. If the leap years are 4 years apart, February 29th occurs two weekdays after each preceding February 29th (if counting down), or two weekdays before each following February 29th (if counting up). If the leap years are 8 years apart (such as between 1896 and 1904), February 29th occurs two weekdays before each preceding February 29th (if counting down), or two weekdays after each following February 29th (if counting up). For example:

 

 

  

Now we can get back to our original quest of looking for the day of the week that December 12th, 1965 occurred on. Since December 12th is after February 29th, when counting down, we follow the counting pattern of dates after February 29th mentioned above. And we already know that December 12th, 1975 fell on the same weekday that December 12th, 2003 fell on.

 

We also know 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 also likely to be other methods that exist as well. 

 

 

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