Introduction

I have a fascination with the day of the week for any date feat. I've posted a detailed tutorial for it here in the Mental Gym, and I've also released a commercially available version called Day One.

My fascination is two-fold: How simple can it be made? How powerful can it be made? After researching several methods, many obscure even to those who enjoy calendar calculating, I've managed to merge two astounding versions that bring together the best combination of power and simplicity I've ever run across.

Effect

You start by bringing out a business card with this calendar design:



You then ask someone for the year, and then the month, of their birthday. With no further questions, you're instantly able to fill in the calendar for that month and year. For example, if the person said they were born in 1979 in September, you could quickly fill in the calendar from memory like this:



They can verify your calendar with any outside source, such as Wolfram|Alpha, and see that the calendar you created from memory is 100% accurate!

In addition, you'll even learn to handle Julian calendar dates all the way back to 45 B.C. almost as easily as our modern Gregorian calendar dates!

Even if you've become frustrated trying to master other day of the day of the week for any date formulas, I think you'll like the simplicity and flexibility of this version. You can start learning the method in the next tab!

Year Key

With this method, you ask for the year first, and you're going to calculate a date from it, ranging from 0 to 6.

In many methods, calculating a number from the year is the biggest hurdle, both because of the number of calculations involved, and the fact that many calculations involve increasing the amount represented by the year. In this method, however, the steps are few, and the method always involves making the year number smaller.

When given a year, the first thing you need to do is to break it up into the last 2 digits, which will be referred to as the year from here on out, and the remaining digits, which will be referred to as the century. For exam, if you're given the year 1975, you would think of 75 as the year and 19 as the century.

First, I'll explain the calculations to perform on the 2-digit year, and then I'll explain how to handle the century.

Step 1 - Partition: Split the 2-digit year into a sum, consisting of the largest multiple of 4 equal to or less than the given year, plus a remainder of 1, 2, or 3, if needed.

Example: 75 is split into 72 + 3. 72 is the largest number equal to or less than 75, and 3 is the remainder needed to make it a sum that totals 75.

Step 2 - Modify & solve: Divide the multiple of 4 by 2, and change the plus sign to a minus sign. Once you've modified the problem, the next step is to solve it.

Example: 72 (from 72 + 3 above) ÷ 2 = 36, so 72 + 3 becomes 36 - 3. Solving this, we get 36 - 3 = 33.

Step 3 - Cast out 7s: From the total in the previous step, subtract the largest multiple of 7 equal to or less than the total. For the reasoning behind this step, see the Modular Arithmetic section of my Day for any Date Online Toolbox post.

Example: The largest multiple of 7 equal to or less than 33 is 28, so we subtract 33 - 28 to get 5.

This may seem like a lot of work at first, but it can be done surprisingly quickly, once you become familiar with it.

Below is a simple widget to show you how this is done with various years. Simply select the first digit of the 2-digit year with the left menu, and the second digit with the right menu. You'll automatically see how the calculations below are affected by each change.

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Note that years which are evenly divisible by 4 are easy to calculate. You'll quickly learn that multiples of 4 allow you to skip the partitioning step, and go straight to dividing a single number by 2.

Also, pay particular attention to the years 01, 02, 03, and 07. When these are partitioned, divided, and subtracted, they all result in negative numbers. In these few cases, you need to add 7 to make them positive. If you can remember that -1 + 7 = 6, -2 + 7 = 5, and -3 + 7 = 4, you can handle all of these situations.

You can practice calculating this key in the Quick Calendar Month Creation Quiz by clicking on the 2-Digit Year Key button.

Century Key

The number you calculated from the year above will now be modified to take the century into account. Adjusting for the century only requires a little memory and a single addition.

If the requested century is in the 1700s, you don't need to add anything. For the 1800s, you'll add 2 to your running total. For the 1900s, you'll add 4, and for the 2000s, you'll add 5. For years after that, the pattern repeats every 400 years:

+ 0	+ 2	+ 4	+ 5
---	---	1500	1600
1700	1800	1900	2000
2100	2200	2300	2400

This table only goes back to 1500 because the calendar we currently use, the Gregorian calendar, was first adopted on October 15, 1582. In later lessons, you'll learn how to handle earlier dates.

To help you recall the right number, remember the nonsense mnemonic, “No tuna for Friday”. No represents +0, tuna represents +2, for represents +4, and Friday represents +5, since Friday is the 5th and last day of the traditional 5-day Monday through Friday work week.

How do you know which numbers go with which years? From right to left, they cover the 1700s, the 1800s, the 1900s, and the 2000s respectively. Bob Goddard (see Credits & Notes below) suggesting remembering that the mnemonics apply to the first 400 years of American history.

Once you can recall which modification goes with which year, you simply add the appropriate number for the century. Don't forget to cast out 7s if you get a number that's 7 or larger.

Going back to our 1975 example from above, we worked out a key number of 5 for 1975. To modify this number for the century, we add 4 for the 1900s, so 5 + 4 = 9. Casting out 7s, we subtract 9 - 7 = 2. So, 2 is our final key number for 1975.

What about, say, 1776? 76 doesn't need to be partitioned, since it's a multiple of 4. 76 ÷ 2 = 38, and casting out 7s we get 38 - 35 = 3. Since the 1700s have a century modification of +0, we simply don't add anything, and leave the total as 3. So, the year key for 1776 is 3.

Next Step

Practice calculating the full key over in the Quick Calendar Month Creation Quiz by clicking on the Full Key 1700-2099 button. Ideally, you want to be able to run through the calculation process for a given year in about 5-6 seconds.

Once you have this year key, what do you do with it? The next step is to take the key number you just calculated, and combine it with a particular month. You'll learn how to do that in the next tab.

Credits & Notes

The mathematical method for handling the 2-digit year comes directly from an article titled, “How to find the day of the week on which any particular date falls” by E. Rogent and W.W. Durbin in the August 1927 issue of The Linking Ring.

The modification of the century, as well as the “No tuna for Friday” mnemonic, is from step 2 of Bob Goddard's First Sunday Doomsday Algorithm.

Months

Now that you've calculated a key number based on the year and century, which will be treated as a date from here on out. The next step is to ask for a month. Each month is associated with a day of the week, and all you'll do to create the correct calendar for the month is to put the date you calculated from the year and century together with the day associated with the given month!

Months and Days

To help you recall with days are associated with which months, I developed 7 short rhymes to help make the connections memorable:

February, March, and November,
Three on Sunday to remember.

February, March, and November are all associated with Sunday. Sunday also happens to be the only day of the week associated with 3 different months. Each of the other days is associated with only 1 or 2 months each.

June day?
Moon-day!

This short rhyme makes it easy to remember that June is associated with Monday.

A skeptic needs clues,
but Sept./Dec. needs Tues.

Sept. is Septemeber, and Dec. is short for December, and they're both associated with Tues., or Tuesday.

Winds in April, Winds in July,
“Winds”-day’s the day, wet or dry.

This is simple enough. April and July are associated with Wednesday.

Jan’s and October’s 31,
Reminds us that Thursday won.

Jan, of course, refers to January. Not only are January and October both associated with Thursday, but they're the only set of months in a group which all happen to share the same length.

Month 5 is May,
Day 5 is Friday.

May being associate with Friday made for an easy rhyme, of course. Since May actually rhymes with every day of the week, I thought the additional association of the 5th month and the 5th and final day of the work week strengthened the association.

August’s few holidays make you yearn,
For the day off named after Saturn.

In most English-speaking countries, August tends to be the month with the fewest nation-wide holidays. This made for an interesting rhyme to help you memorize that August is associated with “Saturn-day”, or Saturday.

Leap Year Exceptions

Leap years stand out in this method because they don't need partitioning like most other numbers. If you're given a leap year, work out the key as taught previously, and then make a mental note that it's a leap year.

January and February both change the days with which they are associated in leap years. January in a leap year moves 1 day back from Thursday to Wednesday. In other words, in a leap year, January, April, and July are all associated with Wednesday, and October is the only month associated with Thursday.

Similarly, February also moves 1 day back from Sunday to Saturday in a leap year. In leap years, then, February and August are both associated with Saturday, and only March and November are associated with Sunday.

These two leap year exceptions may seem difficult to manage at first, but if you just think of January and February moving their day associations back by 1 day, it quickly becomes natural.

Remember that any year which is evenly divisible by 4 is a leap year, except for years ending in 00. Years ending in 00 are only leap years if they're evenly divisible by 400. For example, 2000 was a leap year, but 1900 and 2100 are not.

Practice your association of days with months by going to the Quick Calendar Month Creation Quiz and clicking on the Month and Day Association button.

Once you're comfortable with the month-to-day associations and calculating the key from the year, you're ready to quickly create an entire calendar month, as explained in the next tab.

Credits & Notes

The approach of associating each year and century with a date, and each month with a day of the week is also from “How to find the day of the week on which any particular date falls” by E. Rogent and W.W. Durbin in the August 1927 issue of The Linking Ring.

In the original article, however, Durbin and Rogent's month and day associations were based on dates in the 1800s. In order to make the math work properly with Bob Goddard's First Sunday Doomsday Algorithm, however, the month and date associations have been changed so that they're now based in the 1700s.

Since the month and day association have been changed, the rhymes for remembering each association are original with me.

Creating the Calendar

At this point, many calendar calculation methods would have you do one last calculation to work out the day of the week for the given date.

Since my goal here is to minimize the calculation, this step simply involves creating a calendar for the given month and year. Here's a PDF template for a set of blank calendars, sized for printing on the back of business cards. When you bring out a single one to use, it should appear like this:



This step is where all of your calculation and memory work come together. To show you how, let's say you ask for a year, and you're given 1982. As you write down 1982 in the Date section of the calendar, you run through the calculations in your head: 82 = 80 + 2, which becomes 40 - 2 = 38, cast out 7s so we have 3, add 4 for the 1900s, giving 7, casting out 7s, leaves us with 0. We now know 1982 is a 0 year.

For the month, we'll say that you're given April, which you should immediately associate with Wednesday (Remember? “Wind in April...”), as you write down April in the Date section.

At this point, you know that 1982 is a 0 year and April is associated with a Wednesday, yet the calendar looks like this:



Generally, the rule is to put the day of the week you recalled above the number you were given. If you have 5 and Monday, you'd put Monday directly above the 5 on the calendar. In this case, though, we have 0 and Wednesday, so what do you do?

Simple, think of the “0th” of a month as being 1 week before the 7th. In the case of 0, you always put the day of the week above the 7. Since April is associated with Wednesday, we put the abbreviation Wed above the 7th, like this:



From here, it's easy to see the remaining days of the week should be placed:



Don't forget to give the month the correct number of days. In our example, April is a 30-day month, so we add a 29 and 30 in the bottom row:



At this point, you're done creating the calendar! If this was someone's birthday, and they were born on April 17, 1982, a quick look at the calendar is all you need to see that they were born on a Saturday. You can circle that date and the day of the week, and now they have a souvenir of their birthday, and your amazing calendar skills!

Another Leap Year Reminder

In the previous tab, you learned that when working with leap years, and you're given either January or February as the month, you need to move the day associate back by 1 day. Let's see how this works in practice.

For this example, we'll choose 1992 as the year. Multiples of 4 aren't partitioned, so we start with 92 ÷ 2 = 46, and cast out 7s (46 - 42) to get 4. Adding 4 for the 1900s to that, we get 8, and we cast out 7s to get 1. You've worked out that 1992 is a 1 year, and also realize you need to keep alert if they ask for January and February.

Continuing with this example, we'll imagine that February is chosen as the month. When you hear this, you need to recall 2 concepts. First, February is associated with Saturday in a leap year, and second, that a leap-year February has 29 days. Putting all that together, the resulting calendar should appear like this when finished:



Leap-year Januarys and Februarys are probably the worst-case scenario you'll face in this method. Once you're comfortable with those, you're ready to perform this feat.

Verification by the Spectators

For this feat to really be impressive, they need to know that the calendar you created is correct. In the past, this meant carrying around cumbersome almanacs. Today, though, the handiest and most convenient way is carry an internet-connected mobile device around.

They can verify it with any kind of search, but I usually have them point their browser to Wolfram|Alpha (http://www.wolframalpha.com). Once you have the calendar created, have them enter the year and month followed by the word calendar, and Wolfram|Alpha will generate the calendar for that month!

For our April 1982 example, entering the phrase April 1982 calendar into Wolfram|Alpha will display that calendar, and you can see that both start on the same day and end on the same month. You can also verify the February 1992 calendar in the same way.

Practice Creating Calendars

Over in the Quick Calendar Month Creation Quiz, you can practice creating calendars by clicking on the Calendar Quiz button. This quiz is different than the previous quizzes, however, as it opens up full-screen in a new window.

There are two pull-down menus near the top. By default, the one on the top-right reads Gregorian: 1700-2099, which are the years you'll probably want to start with when you're first learning this feat.

Clicking on this menu will bring up other selections, which are: Gregorian: 1582-9999, which tests on years in the current calendar, but in a much wider range (the first 2 items on the tips page can help here), Julian: 1 AD - 1582 AD, and Julian: 45 BC - 1 BC. You won't need to worry about these until you've read the next page.

To use this quiz, select a range from the top-right menu, and then click on the New Date button on the top-left of the screen. Down by the word “Date:”, at the bottom, a random year from the selected range will appear, followed shortly by a random month.

The days of the week towards the top of the screen can be scrolled left or right by using the left and right arrow keys on your keyboard, or by swiping them left or right, if you're using a mobile device. What you want to do is use this ability to scroll the days to align the key you calculated from the given year with the day you associate with the given month.

For example, if you're given 2014, and you quickly work out a key of 2, and then you're given the month of May, which is associated with Friday, you want to scroll the days left or right, so that Friday is right above the 2.

In order to make sure the number of days are right, you use the pull-down menu at the top-left of the screen, which defaults to 28 days. You can also select 29 days, 30 days, and 31 days, and the added days will appear in the bottom row. Continuing with our 2014 May example, you'd want to select 31 days, since that's the length of May, so that 29, 30, and 31 appear in the bottom row.

Once you're satisfied that you've created the month, click the Verify button at the top right to see whether you are correct. It can inform you whether the number of days in the month is wrong, whether the days aren't labeled correctly, or whether you got it correct. If nothing is correct, it will simply say, “Oops. Try again, please.”

To try again, simply click the New Date button, and you're given a brand new random year and month!

Once you're comfortable working with years in the Gregorian calendar, the next tab will show you how to handle years in the Julian calendar, all the way back to 45 B.C.!

Credits & Notes

Creating a calendar month as an alternative to doing calculations involving an individual date is an approach I developed when I created Day One. In fact, the calendar PDF linked in the top of this tab was originally developed for Day One.

The quiz seen here is also a modified version of the calendar quiz I originally developed and included with Day One.

Julian Calendar

The calendar we currently use, known as the Gregorian calendar, was first used by Spain, Portugal, and most of Italy on the Gregorian date of October 15, 1582. The date immediately prior on the Julian calendar was October 4, 1582.

Other countries didn't adopt the Gregorian calendar until later. Britain and its colonies, including America at the time, adopted it in 1752, jumping from the Julian date of September 2 to the Gregorian date of September 14 on the following day.

In the Julian calendar, leap years happened every 4 years, including years ending in 00. With the Gregorian calendar, years ending in 00 are only leap years if they were evenly divisible by 400. For example, 2000 was a leap year, but 1900 was not and 2100 will not be.

With this shifting ahead of days, and different leap year rules, you might think it would be difficult to account for this change in a calendar calculation. I have good news for you. If you want to calculate a Julian date with this method, it's astonishingly simple.

To adjust for a Julian date, the only modification you need to do concerns the century. Instead of using the 0-2-4-5 “No tuna for Friday” approach, simply add the century number itself, casting out 7s if needed.

For example, if someone asked about 1066, you would start with 66 just as before. 66 = 64 + 2, which is modified and solved as 32 - 2 = 30. Casting out 7s from 30 leaves 2. For the Julian calendar, we just add the century number (10): 2 + 10 = 12. Casting out 7s from 12 leaves us with 5.

Next, if someone asks for the month of October, we recall that October is associated with Thursday (“Jan’s and October’s 31...“), so we can create a Julian calendar month for October 1066, by starting with a Thursday the 5th. Yes, Wolfram|Alpha can verify Julian calendar dates, too.

As long as you don't accept dates in 1582 from October 5th to October 14th, you can now handle dates as far back as 1 A.D., and as far into the future as you like!

To practice creating calendar for Julian years and months from 1 A.D. to 1582 A.D., you can go the the Quick Calendar Month Creation Quiz page, and click on the Calendar Quiz button. When the Calendar Quiz opens, select Julian: 1 AD - 1582 AD from the top-right menu to practice these dates.

B.C. Dates

The Julian calendar was first put in place by Julius Caesar in 45 B.C., so wouldn't it be nice to handle those first few dates, as well?

The first thing to remember is that the calendar we use today went from 1 B.C. directly to 1 A.D., so there was no year 0.

Handling dates such as these only requires one additional adjustment. When given a B.C. date, subtract the year from 57, and then treat it like any other Julian year.

If you're asked to create a calendar in, say, 25 B.C., the first step is to subtract it from 57: 57 - 25 = 32. Now you can handle that as a Julian year. 32 doesn't need to be partitioned, so we just do 32 ÷ 2 = 16, and casting out 7s from 16 leaves 2. Since the century is effectively 0, we just leave the key at 2.

Now, if someone asks for June, we recall that June is associated with a Monday (“June day? Moon-day!”), so June in 25 B.C. must've had a Monday the 2nd, which Wolfram|Alpha verifies for us!

When dealing with Julian and/or B.C. dates, don't forget the leap year adjustment, if needed. For example, since 25 B.C. is a leap year, you'd still work out that the key number for it is 2, but if someone asked for January, you need to recall that it's associated with Wednesday in a leap year. That way, you know January in 25 B.C. had a Wednesday the 2nd.

You can also practice BC dates by going to the Quick Calendar Month Creation Quiz page, and clicking on the Calendar Quiz button. After the Calendar Quiz opens, select Julian: 45 BC - 1 BC from the top-right menu to practice these dates.

Once you're comfortable working with Julian calendars and calendars back to 45 B.C., check out these tips to help your performance along.

Credits & Notes

The Julian and B.C. date modifications both come directly from the approach used in Bob Goddard's First Sunday Doomsday Algorithm. Basing the Gregorian century modifications on the 1700s is what makes this possible.

Tips

• Only the last 4 digits of a year matter. Working with the year 15739562 is no different than working with the year 9562.

• Another way to think about the centuries, especially when dealing with years far away from the 1700s, 1800s, 1900s, and 2000s:

• If it's divisible by 400, like the 2000s, add 5.
• If it's even, but not evenly divisible by 400, like the 1800s, add 2.
• If it's just after a multiple of 400, like the 1700s, don't add anything.
• If it's just before a multiple of 400, like the 1900s, add 4.

• If you're comfortable with using the Mnemonic Major System, you can speed up this feat even further. Since dates within a century repeat in a pattern every 28 years, all you have to do is memorize the 2-digit year keys for the years 00-27. You not only have quick recall of those years, but of all the other years as well:

• Years ending in 28 through 55: Subtract 28, then recall the corresponding key.
• Years ending in 56 through 83: Subtract 56, then recall the corresponding key.
• Years ending in 84 through 99: Subtract 84, then recall the corresponding key.

• You don't have to use the business card calendar PDF. Any calendar that allows you to display an entire month AND adjust which days go with which months will work. Some examples:

• Calendars you can make:
  
  • Casio Craft Calendar 1
  • Casio Craft Calendar 2
  • The Perpetual Strip
• Commercially available calendars:
  
  • Calendreco
  • Magnetic perpetual calendars
  • MoMA Acrylic perpetual calendar
  • MoMA Spindle perpetual calendar
  • Monthly Calendar Pocket Chart
  • Perpetual Calendar Clipboard

• There's a simple formula for working out how many days needed to be added to the Julian calendar to adjust it to the Gregorian calendar. You can learn it from my Changing Calendars Mentally post. When working with the Julian calendar, this can make an impressive additional detail.

• Which months have a Friday the 13th? Once you've taken the year and century into account and you have your key number, ask yourself which day of the week is that many days after Saturday, and then ask yourself which months are associated with that day of the week. The months will be the only ones with a Friday the 13th!

For example, if you're given a year whose key you calculate as 3, ask yourself what day is 3 days after Saturday? The answer, of course, is Tuesday. Which months are associated with Tuesday? September and December are associated with that day, so you can state that September and December are the only months that year with a Friday the 13th!

This makes an impressive extra feat to add in to your performance. Note: As always, don't forget that January's and February's associations move 1 day back in leap years!

• What do you do if someone asks for a year before 45 B.C.? Explain that the pre-Julian Roman calendars were just too different. For example, one version had only 304 days each year, and another had 355 days each year, with leap years making some years as long as 378 days. The last pre-Julian year, 46 B.C., was adjusted to 445 days long!

• Hans-Christian Solka, author of the Day of Week Calculation Blog, also suggests the following related online resources for further reading:

• An Improved Doomsday Algorithm, by Michael K. Walters
• A Perpetual Calendar in Your Head, by Dr. YingKing Yu
• Methods for Accelerating Conway's Doomsday Algorithm (PDF: Part 1, Part 2), by Chamberlain Fong and Michael K. Walters