Grey Matters: Mental Gym: Day of the Week For Any Date (Revised)

Day of the Week For Any Date (Revised)

Published on Tuesday, March 01, 2011 in calendar, feats, memory, mental math

Note: If you're interested in calendar calculations, you also might want to check out my Quick Calendar Month Creation tutorial.

Re-Introduction

In this feat, someone gives you a date, and you quickly state the day of the week on which it fell. This new approach is updated for the 21st century, and employs new tips and tricks that help make this feat simpler to learn and quicker to perform.

Approach:

The Day of the Week For Any Date feat combines both memory and mental math. A relatively simple mastery of both, though, will create a response far out of proportion to the required work.

Before I describe the basics of the approach, I'd like to help you get a good idea of your goal, as well as what's possible, by seeing this feat performed by various people in the following videos:

Here's the basic principles, broken down into simple steps:

1) Day and Months Number Conversion: To work out the days of the week mentally, we need to convert them into numbers. We'll also need to convert the months into numbers, to adjust for their effects. These are taught in an easy-to-remember manner.

2) Addition of 3 Numbers: Without using a calculator, can you tell me what 6 + 6 + 31 is? That's about as difficult as the basic formula gets. If you're comfortable doing that, you won't have a problem working through the formula.

3) Subtracting Multiples of 7: Let's say you're asked about the 27th of a month. Regardless of the month or year, we can state with certainty that the 27th of a month will fall on the 6th (since it's 3 weeks, or 21 days, earlier). Since adding 6 is simpler than adding 27, and will give the same result, why not use 6? If you learn to subtract multiples of 7, this makes the arithmetic so easy that you won't have to worry about addition problems any tougher than 6 + 6 + 6!

4) Year Number Conversion: After becoming comfortable with all of the above when given dates in the years 2000 to 2003, you'll learn how to remember and adjust the leap years in the 21st century to key dates. After learning those, you'll learn a simple way to adjust for any year from 2000 to 2099, and even adjust for other centuries!

Running through all these principles, there will be an emphasis on recognizing and taking advantage of patterns. The quicker you can recognize a pattern, the quicker will be your calculation.

We'll start with the codes for the days of the week, since that is our goal. All the formulas and patterns you'll learn later will result in a number from 0 to 6. This number is turned into a weekday as follows:

Day of Week	Number	Mnemonic
Sunday	0	SUNday=NONEday
Monday	1	MONday=ONEday
Tuesday	2	TWOSday
Wednesday	3	Three fingers look like a W
Thursday	4	FOURSday
Friday	5	FIVEday
Saturday	6	SIXturday

To help get you comfortable with converting numbers into days, take the Weekday Codes quiz here. Once you can get a perfect score in a short time, then continue with this tutorial.

Next, you need to learn the codes for the months. Like the weekdays, they range from 0 to 6:

Month	Number	Mnemonic
January	6	WINTER has 6 letters
February	2	February is 2nd month
March	2	March 2 the beat.
April	5	APRIL has 5 letters (& FOOLS!)
May	0	MAY-0
June	3	June BUG (BUG has 3 letters)
July	5	FIVERworks
August	1	A-1 Steak Sauce at picnic
September	4	FALL has 4 letters
October	6	SIX or treat!
November	2	2 legs on 2rkey
December	4	LAST (or XMAS) has 4 letters

Take the Month Codes quiz here to help reinforce the mnemonics.

Note: It's important to note that, in leap years, January reduces by one to 5, and February reduces by one to 1. The other years don't change in leap years. Leap years will be discussed in more detail later.

Once you're comfortable with both the month and weekday codes, you're ready to start calculating your first dates. In the next tab, you'll learn how to use these codes to work out dates for the years 2000 to 2003.

Formula

Ready for the formula? Here it is: Month Code + Date + Year Code = Day of Week Code. It's a lot simpler than many people think, but there are some fine points to learn.

We haven't covered year codes yet, so I'm just going to teach you four simple ones with which to start out:

2000 = 0
2001 = 1
2002 = 2
2003 = 3

Those shouldn't be too hard to remember, should they? We'll learn more year codes later in this tutorial, but for now, we'll just focus on these years.

Let's start with a simple example. Let's figure out May 1, 2000. The code for May is what? The mnemonic is MAY-0, so May is a 0. The date itself is the first, so we use 1. The year code for 2000 is 0, so our problem is 0 + 1 + 0 = 1.

Which weekday has a code of 1? ONEday is MONday, so Monday is the day of the week on which May 1, 2000 fell. You can verify that at this site. Congratulations, you've just calculated your first date!

Let's try a date that's a little more challenging. Our next date is October 4, 2000. October is 6 (remember “SIX or treat”?), and 2000 is still 0, so that gives us 6 + 4 + 0 = 10. And the weekday that goes with 10 is...

Wait a minute, the weekday codes only range from 0 to 6! What do we do with a 10, or any result higher than 6 for that matter?

If you look on a calendar, the 10th of any month will always fall on the same day of the week as the 3rd, because the 3rd is 7 days earlier. So, with the result, or any number in the formula, we can subtract 7, or any multiples of 7 to reduce the answer. You may want to refresh yourself on the multiples of 7 with the help of Schoolhouse Rock here.

In our October 4, 2000 example, we got 10 as a result, so we can reduce that by 7. 10 - 7 = 3, so our result boils down to 3. Which day of the week is 3? Since 3 fingers looks like the letter W, that's Wednesday. Once again, you can check that here.

This reduction of the multiples of 7 can make the problem itself easier, as well. Let's try figuring out the day of the week for Halloween 2001, or October 31, 2001. October is 6, and 2001 is a 1, so the problem works out to be 6 + 31 + 1 = 38. The closest multiple of 7 to 38 is 35, so we do 38 - 35 = 3, which gives us another Wednesday.

That approach works, but it could be made simpler. When you hear that the date is the 31st, you can reduce that right away by working out that 31 - 28 (the closest multiple of 7 to 31) = 3, and doing 6 + 3 + 1 = 10. True, you would still need to reduce that 10 to 3 again to get Wednesday, but you'd need to subtract multiples of 7 either way.

Note that, by bringing the dates down by multiples of 7, you're making the problem you have to add much simpler. If 6 + 31 + 1 and 6 + 3 + 1 will both give you the same results, wouldn't you prefer to make it easier on yourself?

The scary technical term for subtracting multiples in this manner is modulo arithmetic, which is explained quite clearly at BetterExplained.com.

Let's try this with Valentine's Day in 2003. We start by making sure of the date, February 14, 2003. February is a 2 (remember the mnemonic?), and 2003 is a 3. The problem then becomes 2 + 14 + 3. However, if you spotted that 14 was already a multiple of 7, you should realize that you can drop it out completely! 14, or any multiple of 7, is the same as 0, so you can ignore them. For February 14, 2003, all you really need to add is 2 + 3 = 5. 5 is a FIVEday, or rather a Friday, so that's our answer!

Let's try one last problem before we go. What about February 2, 2000 (Groundhog Day)? February is 2, and 2000 is 0, so the problem we get is 2 + 2 + 0 = 4. 4 is a Thursday (remember FOURSday?), so February 2, 2000 should be a Thursday. Once again, we verify that information here and...OOPS! That site says February 2, 2000 is a Wednesday! What went wrong?

I briefly mentioned this at the end of the previous tab, but it needs to be repeated now. Whenever you're working in January or February dates in a leap year, you need to reduce the month code by 1 to compensate. Effectively the extra day, February 29, hasn't happened yet, so we're subtracting one to adjust for that fact. January becomes 5 (6 - 1) and February becomes 1 (2 - 1).

Since February 2, 2000 is a February date in a leap year, the code for February needs to be 1, not 2. Let's try the equation again, with that in mind. February in a leap year is 1, and 2000 is 0, so the equation is 1 + 2 + 0 = 3, which you should know by now is a Wednesday. As we saw when we originally made the error, Wednesday is indeed the correct day of the week.

Practice the Dates: 2000 to 2003 quiz here, making sure to keep an eye out for January and February dates in 2000, and adjusting your calculations accordingly. Practice these dates until you can calculate them with little trouble, and don't forget to subtract multiples of 7 to make your work easier!

Once you're comfortable with dates from 2000 to 2003, we're going to teach you how to better handle leap years in the next section.

Year Codes: Why?

You'll note that the year codes for 2000-2003 progressed in a nice, simple, 0-1-2-3 order. This is because a normal year consists of exactly 52 weeks (52 × 7 = 364) plus 1 day, to make 365 days. So, from one 365-day year to the next 365-day year, a given date in a given month will fall one day later.

However, a 366-day leap year means that everything must jump ahead 2 days. This also means that the year codes for leap years will jump ahead 2 instead of 1. You might expect 2004 to be a 4, but because it's a leap year, it jumps ahead to 5.

To get you comfortable with the strange nature of leap years, I'm going to start by teaching you the first 7 leap years.

First 7 Leap Years

Here are the year codes for the first 7 leap years, along with handy mnemonics by which to remember them:

Leap Year	Year Code	Mnemonic
2000	0	2000 is mostly 0s
2004	5	Count: 4...5...
2008	3	Right half of 8 looks like 3
2012	1	12 ÷ 12 = 1 (See below)
2016	6	16 ends in 6
2020	4	2 + 0 + 2 + 0 = 4
2024	2	24 ÷ 12 = 2 (See below)

With 2012 and 2024, you'll note that all you have to do is divide their last 2 digits by 12 to get their year code. This pattern keeps working all the way through 2096, which will give you a few extra leap year codes quite easily, assuming you know your 12 multiples:

2012 = 1
2024 = 2
2036 = 3
2048 = 4
2060 = 5
2072 = 6
2084 = 7 = 0 (Remember to drop any multiples of 7!)
2096 = 8 = 1 (Remember to drop any multiples of 7!)

First, practice just recalling the codes with the Leap Year Codes: 2000 to 2024 quiz here.

Once you're comfortable recalling all the codes, practice working out dates for those years with the Leap Year Dates: 2000 to 2024 quiz here. Don't forget that that the month codes for January and February are both reduced by 1 in leap years!

Once you've practiced those years, you're ready to learn how to handle any leap year in the 21st century!

All Leap Years

If you're given a leap year that ends in a multiple of 12, you can already handle those through 2096 quite easily, of course. What about the remaining leap years?

If there were no such thing as leap years, the pattern year codes would simply repeat every 7 years. Because of the effect of leap years every 4 years, however, the pattern of year codes usually repeats every 28 years.

I say “usually” because years ending in 00 are an exception. Years ending in 00 are only leap years if they're divisible by 400. So, 1600, 2000, and 2400 are leap years, while 1800, 1900, and 2100 are not.

Thanks to the 00 exception the calendar only repeats EXACTLY every 400 years. However, when you're dealing with a range of years in which EVERY (without exception) 4th year is a leap year, then you can still rely on the 28-year rule.

This means that you can depend on the 28-year rule for every leap year from 2000-2096! I'll break this down in a simpler manner, so you can see how this is useful.

For the leap years 2028 through 2052, all you have to do is subtract 28 years, and you'll get a year with the same year code! 2028 - 28 is 2000, which you already know has a year code of 0, so 2028's year code is 0. 2032 - 28 = 2004, whose year code is 5, and so on.

While doing 2028 - 28 = 2000 in your head is simple enough, some people find that working out problems like 2040 - 28 or 2052 - 28 during a performance to be a little challenging. There's a way to make it simpler.

If you're worried about subtracting 28 from a number, add 2 and then subtract 30 instead. For example, instead of doing 2040 - 28, work out 2040 + 2 - 30 = 2042 - 30 = 2012. 2012 is a 1 year, so 2040 is a 1 year as well! What year has the same year code as 2052? Add 2 and subtract 30, and you'll get your answer in no time.

Similarly, for the years 2056 to 2080, you subtract 56 to get the year code. The mathematical short cut here, if you feel you need it, is to add 4 and then subtract 60. What's the year code for 2064? 2064 + 4 - 60 = 2068 - 60 = 2008 = a year code of 3! How about 2068? You should get a year code of 1, just like 2012.

How about 2072? Did you start by adding 4? Stop! 72 is a multiple of 12, so we just work out that 2072 is 6 from the pattern of 12s above. Don't forget to take advantage of the 12 pattern when you can! You should ask yourself if a year is divisible by 12 first, before subtracting.

Finally, for the years 2084 to 2096, just subtract 84. For 2084 and 2096, of course, just use the 12 pattern we discussed earlier.

If you need a shortcut for 84 for the remaining years, simply subtract 4, then subtract 80. 2092 - 4 - 80 = 2088 - 80 = 2008 = year code of 3! I'm sure you have the idea by now.

You can practice this process with the Leap Year Codes: 2000 to 2096 quiz here. Again, don't forget to take advantage of the 12 pattern when you can.

Also, don't forget to practice actual dates in these leap years with this Leap Year Dates: 2000 to 2096 quiz.

Once you're comfortable working through both of these quizzes, it's time to learn how to determine the code for every year from 2000-2099 in the next section!

2000-2099

Are you ready to handle any date from 2000 to 2099? You're probably more ready thank you think!

Once you can handle leap years, the remaining years are simple.

When given a non-leap year, you need 2 pieces of information: The year code for the nearest leap year BEFORE the given year, and how far the given year is from that leap year. When you have these two pieces of information, simply add them together (remembering to drop any multiples of 7, as we've discussed before), and you have the year code.

For example, take 2009. The closest leap year is 2008, which has a year code of 3 (remember?), and 2009 is 1 year later. So, we work out 3 + 1 = 4, so 2009's year code is 4!

How about the year code for 2051? The nearest leap year BEFORE that is 2048, and we can use the 12 rule to determine that the year code is 4. Since 2051 is 3 years later, we do 4 + 3 = 7 = 0 (don't forget to drop out multiples of 7!), so 2051 has a year code of 0.

How about a tricky one like 2094? It's 2 years after 2092, which has a year code of 3 (remember how we know that?), so 2 + 3 = 5, so 2094 has a year code of 5.

Since you can't be more than 3 years after a leap year in any date from 2000-2099, this is a relatively simple adjustment.

To get practiced with this approach for determining year codes for any year, use this Year Codes: 2000 to 2099 quiz.

Once you get comfortable with that quiz, move on to the Dates: 2000 to 2099 quiz here.

Being able to determine the day of the week for any date in the 21st century is an impressive feat on its own. Once you're comfortable with doing that, you can move on learning how to handle dates in other centuries in the next section.

Other Centuries

Often you'll get asked about dates in the 20th century, especially if you're discussing someone's birthday. How do you handle those?

For dates from 1900 to 1999, simply work out the similar date for the 2000s, and then add 1. That's it!

For example, Let's say someone tells you they were born on January 20, 1985. Start as if you were working out January 20, 2085. 85 is a 1 year and January is a 6, so 1 + 6 = 7 = 0. Reduce 20 to 6 (cast out multiples of 7!) to get 6, and add 1 for the 20th century to get 7, which drops to 0. That 0 is the code for Sunday, and sure enough, January 20, 1985, was a Sunday.

Once you've worked out the day of the week for a given date in the 21st century, there's a simple pattern to alter the day for other centuries:

2300 to 2399 = add 1
2200 to 2299 = add 3
2100 to 2199 = add 5
2000 to 2099 = add 0
1900 to 1999 = add 1
1800 to 1899 = add 3
1700 to 1799 = add 5
1600 to 1699 = add 0

There is one VERY important note here for January and February dates in the years ending in 00: If a year ends in 00, it's only a leap year if it's divisible by 400. The years 1600, 2000, and 2400, and so on are leap years, while years like 1900, 1800, and 2100 are not. If you're given a January or February date in a year ending in 00, double check whether it's a leap year before you make the leap year adjustment.

Take a few minutes to study these adjustments, and then practice using them in the Dates: 1600 to 2399 quiz here.

Assuming you've put in the practice, you should be ready to give the day of the week for any date. In the next section, I'll provide a few tips and some background that can help improve your performance.

Calendar Background

The current calendar system we use is known as the Gregorian calendar, since it was introduced by Pope Gregory XIII. It was first put into use in 1582 by the Catholic countries, so the calculations you've learned aren't really effective for dates before 1600.

In addition, many non-Catholic countries didn't adopt the Gregorian calendar until much later. Britain and its colonies didn't adapt the calendar until 1752. The use of the Gregorian calendar as a worldwide standard, however, didn't happen until the 1920s!

Tips

• I can't emphasize enough the speed advantages of dropping multiples of 7, and becoming comfortable with that process. After you get use to doing this for dates from the 7th to the 31st several times, it almost becomes automatic.

• Carry a perpetual calendar! It's one thing to do this feat and know you're right. When you're doing it for an audience, they'll need some way to verify that you're correct. Originally, this meant carrying around a bulky book of calendars, but many mobile devices today make this much easier.

You'll generally want an app that mainly generates calendars for a wide variety of year, without appointment features, such as QuickCal for the iPhone and iPod Touch. iPad users can use YearViewer, and Android users can use Two Hundred Year Calendar or Day of Week.

• Want to practice on the go? Download these free mp3 files that give a date, then pause, then give the day of the week. The pauses range from 30 seconds down to 3 seconds, so you can challenge yourself as you get better. They're available in both DATE/MONTH/YEAR order (common in the UK, Australia, and Europe) and MONTH/DATE/YEAR (common in the US).

• As you've seen, working out year codes can take longer than just remembering the month, date, and week codes. When performing, the smart thing to do is ask for the year first, work out the year code as needed (including whether a leap adjustment will be needed), and only then ask for the specific date.

That way, not only do you get the year calculation out of the way, but you'll be able to determine the weekday more quickly and it will appear more impressive to your audience.

• When you're comfortable performing the feat this way, but you find you desire to be quicker, there is a more advanced step you can take. You can completely eliminate the calculations of the year code by memorizing the 100 codes needed for the years 2000-2099.

To do this, you'll need to be familiar with the Link System, the Shape Peg System and the Phonetic Peg System (AKA the Major System) (with images for 0 to 99).

Once you've practiced those systems and are comfortable with them, you use the Phonetic Peg System for images to represent the last two digits of the year (0 to 99), and the Shape Peg system for the year code (from 0 to 6). You then use the Link System to mentally link those two images together.

If you decide to go this method, here's a complete chart of the years from 2000 to 2099 with their corresponding year codes. Because the images people use with the above systems are so widely varied, I've avoided suggesting any mnemonics.

Year	Year Code
2000	0
2001	1
2002	2
2003	3
2004	5
2005	6
2006	0
2007	1
2008	3
2009	4
2010	5
2011	6
2012	1
2013	2
2014	3
2015	4
2016	6
2017	0
2018	1
2019	2
2020	4
2021	5
2022	6
2023	0
2024	2
2025	3
2026	4
2027	5
2028	0
2029	1
2030	2
2031	3
2032	5
2033	6
2034	0
2035	1
2036	3
2037	4
2038	5
2039	6
2040	1
2041	2
2042	3
2043	4
2044	6
2045	0
2046	1
2047	2
2048	4
2049	5
2050	6
2051	0
2052	2
2053	3
2054	4
2055	5
2056	0
2057	1
2058	2
2059	3
2060	5
2061	6
2062	0
2063	1
2064	3
2065	4
2066	5
2067	6
2068	1
2069	2
2070	3
2071	4
2072	6
2073	0
2074	1
2075	2
2076	4
2077	5
2078	6
2079	0
2080	2
2081	3
2082	4
2083	5
2084	0
2085	1
2086	2
2087	3
2088	5
2089	6
2090	0
2091	1
2092	3
2093	4
2094	5
2095	6
2096	1
2097	2
2098	3
2099	4

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289 Response to Day of the Week For Any Date (Revised)

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Bram

June 19, 2011 at 1:47 PM

Hello,

Great explanation (as always). Small typo on the first tab: "The other years don't change in leap years" --> the other months.

Bram

Sapphire Ace

October 11, 2011 at 3:12 PM

Thank you so much! It was so worth it! And couldn't have found a better teacher! XD

CaptainNegatory

October 11, 2011 at 4:34 PM

The difficult part is finding the nearest leap year. Or is it just me that's missing something?

Gabriel

October 11, 2011 at 6:14 PM

This comment has been removed by the author.

Gabriel

October 11, 2011 at 6:19 PM

In Portuguese, the days of the week are ordinal numbers, so it's even easier for me to apply the rules. This is great, thank you!

Anonymous

October 11, 2011 at 9:57 PM

Maybe I'm doing it wrong, but November 18, 1969 doesn't seem to work. 2+4+6+1=13, which makes it a 6, so it should be Saturday? But, instead it is Tuesday?

CaptainNegatory

October 11, 2011 at 11:30 PM

This comment has been removed by the author.

CaptainNegatory

October 11, 2011 at 11:31 PM

@radicalpi

November = 2
Day=18
Year=2
1900's= +1

Explanation:

68 is the nearest leap year, minus 56 (68-56=12) since 12 goes to 2012, or the multiple of 12 rule it actually equals 1. Plus the distance from your year of 1969 (1+1) makes the yearcode 2.

So November + 18 + 1969= X
Or 2 + 18 + 2 = 22
22/7(Remember to remove the 7 multiples) = 21 remainder ONE.
Now, because it is in the 1900's we add 1 to the day code giving us TWO for TWOsday.
So, November 18th, 1969 is a Tuesday.

I choose not to worry about the 7 multiples in the daycode. It makes no difference if you do it with the day code, or add them all together and then do the multiple of 7, so I do multiples last and just add everything straight up. It makes easier memorization of all the number jumbo in my head.

Hope that helped you, it took me a few tries to learn this. Please let know if it helped and if not ill try to re-explain!

Best of Luck,
CaptainNegatory

Anonymous

October 12, 2011 at 12:11 AM

My question is what's the method behind the month code? Rather than memorize them (and forget), I'd prefer to be able to figure it out (and forget the method ;).

After looking at the first 7 leap-year codes, I found it easier to divide the year by 4, then multiply that number by 5 to get the year code. For example, 2068 is a leap year. 68/4 = 17, 17 * 5 = 85. If you remove the multiples of 7 you get 1, which is the year code. Or you could just add the 85 in and do the multiples at the end.

Anonymous

October 12, 2011 at 1:11 AM

Also, the '00 leap-year exception doesn't seem to apply when finding year codes. For instance, to find the code for 1703, you would use the code for 1700 (0) and add 3, even though 1700 isn't divisible by 400 and thus isn't actually a leap year.

Day of the Week For Any Date (Revised)

Re-Introduction

Approach:

Formula

Year Codes: Why?

First 7 Leap Years

All Leap Years

2000-2099

Other Centuries

Calendar Background

Tips

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