The first snippets of 2015 are ready!

This time around, I have some clever and fun approaches to math to share. I think you'll be surprised by them, even (or especially) if you don't usually like math.

• This January marks the 28th anniversary of Square One TV, an educational program that taught math with the use of skits, songs, and other fun approaches. While it's not on TV anymore, YouTube user Anton Spivack has been making full episodes available. I've been gathering them together in playlists by season if you want to experience this show for yourself:

Square One TV: Season 1
Square One TV: Season 2
Square One TV: Season 3
Square One TV: Season 4
Square One TV: Season 5
Square One TV: Mathnet

• While I'm thinking about YouTube channels, check out Funza Academy's site, as well as their YouTube channel. Being interested in math shortcuts, I especially enjoy their Math Concepts and Tricks playlist, as it teaches some impressive math shortcuts, including rapidly multiplying any 2-digit numbers together!

• Magic Cafe user RedDevil, author of the RedDevil Mentalism blog, recently shared a great tip for my Day One routine. Day One is my approach to minimizing the work required for the classic Day of the Week For Any Date feat.

RedDevil took this one step further by pointing out that you don't need to remember all the year information I teach in there. Instead, you can only memorize just the leap years, and move 1, 2, or 3 days forward as you go 1, 2, or 3 years ahead respectively.

If you have Day One, you'll understand this. If you don't have Day One, it's still available for only $9.99! If you're a member of the Magic Cafe with at least 50 qualifying posts, you can read his tip in more detail in RedDevil's original thread.

Yes, the snippets are short and sweet this month, but there's still plenty to explore in these links if you take the time to learn and enjoy them!

Happy New Year!

With a new calendar year, you deserve a couple of new calendar feats to go with it. In this post, you'll learn how to quickly give the day of the week AND the moon phase for any date in 2015.

Even better, both of these feats are much easier than they sound!

DAY OF THE WEEK FOR ANY DATE IN 2015: The method to do this is quite simple, and is known as the Doomsday method, originally developed by John Horton Conway.

Start by going to last week's post, Calendar Calculation Made Simple, and learning the simple calendar calculation techniques taught there.

To work out dates in only 2015, all you have to remember is that 2015's "Doomsday" is Saturday. If you think about it, you can already work out any date in February using just this knowledge.

For example, Valentine's Day, Feb. 14th, must also be a Saturday, because it's exactly 2 weeks before Feb. 28th. How about Feb. 2nd (Groundhog Day)? Well, Feb. 7th is a Saturday, and Feb. 2nd is 5 days before that. What's 5 days before a Saturday? The answer is Monday! Therefore, Groundhog Day will be on Monday in 2015.

On which day will Christmas fall in 2015? We know from the technique taught in last week's videos that December 12th is a Saturday, so 2 weeks later, December 26th, is also a Saturday. Since Christmas is one day before that, it must be on a Friday!

When is July 4th this year? It's exactly 1 week before July 11th, so it must be a Saturday, as well.

St. Patrick's Day, March 17th, is 3 days after March 14th (Pi Day, mentioned in the videos from last week), so it's 3 days after a Saturday, making it a Tuesday in 2015.

January 15th is Martin Luther King, Jr.'s birthday, but what day does it fall on in 2015? January 3rd is a Saturday this year, and so is January 17th (2 weeks later). Take back 2 days, and we get January 15th being a Thursday this year!

With the knowledge from last week's videos, and a little practice, you can quickly and easily determine the day of the week for any 2015 date. You could get practice at the Day For Any Date (Mentalist Challenge) page, changing the year to 2015, and then trying to determine the date before you click the Show button.

When you're demonstrating this ability for someone, it's nice to be able to prove that you're right about the date. I use Wolfram|Alpha and/or timeanddate.com's calendars.

MOON PHASE FOR ANY DATE IN 2015: 2 years ago, I posted a new tutorial about determining the moon phase for any date. Similar to the year calculations, focusing on a particular year, such as 2015, greatly simplifies the required calculations. Like the doomsday algorithm above, this formula was also developed by John Conway.

In fact, working out the moon phase for any date in 2015 is even simpler than working out the date! How simple is it?

(Month key number + date + 8) mod 30

It's probably best if I explain each part:

Month key number: January's key number is 3, February's key number is 4, and all other months' keys are their traditional numbers; March is 3, April is 4, May is 5, and so on up to December, which is 12.

Date: This is simply the number represented by the particular date in the month. For the 1st, add 1. For the 2nd, add 2. For the 3rd, add 3, and so on.

+ 8: The addition of 8 takes the starting point of 2015 into account, which is why this particular formula works ONLY for 2015.

mod 30: If you get a total of 30 or more, simply subtract 30. Otherwise, just leave the number as is. Betterexplained.com has an intuitive explanation of modular arithmetic.

The resulting number will be the approximate age of the moon in days, from 0 to 29. This formula only gives an approximation, so there's a margin of error of ±1 day.

As an example, let's figure the phase of the moon on July 4, 2015. July is the 7th month, and the 4th is the date, so we work out (7 + 4 + 8) mod 30 = (11 + 8) mod 30 = 19 mod 30, which is just 19.

In that example, we estimate the age of the moon to be 19 days old.

What exactly does the age of the moon in days mean in practical terms? Here's a quick guide:

• 0 days = New moon (the moon is as dark as it's going to get)
• 0 to 7.5 days = Waxing crescent (Less than half th moon is lit, and it's getting brighter each night)
• 7.5 days = 1st quarter moon (Half the moon is lit, and gets brighter each night)
• 7.5 to 15 days = Waxing gibbous (More than half the moon is lit, and getting brighter each night)
• 15 days = Full moon (The moon is as bright as it's going to get, and will start getting darker each night)
• 15 to 22.5 days = Waning gibbous (More than half the moon is lit, and it's getting darker each night)
• 22.5 days = 3rd quarter moon (Half the moon is lit, and gets darker each night)
• 22.5 to 29 days = Waning crescent (Less than half the moon is lit, and it's getting darker each night)

As we wrap up the year, it's natural for thoughts to turn to the calendar.

Yes, I've talked about calendar calculation many times before, but I'm always on the lookout for better methods and better teaching. MindYourDecision.com's Presh Talwalkar, whom you may remember from last week's post, is back this week with some great calendar calculation lessons!

Let's face it, calendar calculation can sometimes seem daunting. One of the best approaches to learning such a skill for the first time is to ease yourself into it. Presh's starting approach is to teach you how to work out New Year's Day for any date from 2000 to 2099:



Try practicing this skill for yourself. First, use Wolfram|Alpha to get a random year in the 2000s, workout the day of the week for New Year's Day as above, and then verify your answer with Wolfram|Alpha.

It's not difficult, and once you get the hang of this skill, you're ready to move on to the next step.

Using the New Year's Day skill as a starting point, Presh then introduces you to John Horton Conway's well-known Doomsday approach to calendar calculation:



Once again, practice is the key here. You can use Wolfram|Alpha to generate a random date in the 2000s, work it out using the above method, and then verify the correct date using Wolfram|Alpha again.

If you're interested in calendar calculation in general, I not only have several posts about it here on Grey Matters, but numerous lessons, including quizzes, about calendar calculation over in the Mental Gym. Once you get the knack, it's amazing where you can take this skill!

A little over a year ago, I teased Grey Matters readers with a mystery skill. First, they had to learn to easily multiply by 63, then learn how to easily multiply by 72. The skill itself was revealed to be how to roughly convert any whole number of years into seconds!

In this post, you'll learn a similar skill: how to convert weeks into minutes instantly!

Back in the days before computers and calculators, this was a popular feat among entertainers who performed as human calculators. It was quick and direct to perform, yet was highly impressive to audiences.

One week has 7 days, and each day has 24 hours. Every hour, of course, has 60 minutes, so if we multiply out 1 week × 7 days/week × 24 hours/day × 60 minutes/hour, we get 10,080 minutes in a week. The number 10,080, as it happens is very easy to multiply by almost any number of weeks. If you keep the number of weeks at or below 124 (about 2.37 years), the numbers are even easier to work out.

STEP 1: Ask for any number of weeks less than 2 years (104 weeks). As an initial example, we'll say an audience member gave the number 36.

STEP 2: Write down the number they just gave you. In our example, you'd write 36.

STEP 3: Multiply this number by 8 in your head, and write this result to the immediate right of the first number you wrote. This is simpler than it sounds; all you have to do is double the number 3 times. For 36, doubling once gives you 72, doubling a second time gives you 144, and doubling a third times gives you 288. Writing 288 next to the 36 you wrote earlier gives you 36288.

NOTE: In step 3, it's very important to always treat the answer as a 3-digit number. For weeks from 13 to 104, it will be, but for weeks from 2 to 12, it will be a 2-digit number. You can change this into a 3-digit number simply by adding a 0 to the left of it. If you're given 7 weeks in step 1, you write down the 7 as in step 2, then multiply 7 × 8 to get 56, which becomes 056. You would write 056 as your step 3 answer, giving 7056, and then continue with step 4.

STEP 4: Write a zero to the immediate right of the other numbers, add commas where appropriate, and you're done! In our example, we add the zero to the right, giving us 362880. With commas, that result is 362,880. This means that there are 362,880 minutes in 36 weeks!

With a little practice, you'll be astounded as to how quickly you can pick this impressive skill up. You can quiz yourself by having Wolfram|Alpha give you a random number of weeks from 2 to 104, and then use it to verify whether you've worked out the correct answer.

HANDY BONUS TIP: You can make this more impressive for an audience by having someone with a calculator verify this in a long, drawn-out manner. Tell them to put in the number of weeks given, then multiply by 7 for the number of days in a week, then multiply by 24 hours in a day, and then multiply by 60 minutes in a week. Multiplying it out the long way makes this feat seem more difficult, as you're hiding the simple 10,080 conversion factor.

Try it out and amaze your friends!

As you can probably tell from this recent post and this recent post, I've spent quite a bit of time thinking about the Knight's Tour lately.

These thoughts have reminded of a different type of Knight's Tour puzzle. This unusual variation involves moving the knight around a calendar.

It was 4 years ago, during September or October, that I was looking for blog post inspirations and ran across a thread on the XKCD forums, titled “Knight's Tour revamped”, which suggested playing the Knight's Tour on a calendar.

There was an added challenge, however: With your starting square being considered as move #1, how many dates could you land on that were the same as the move number? For example, if move #1 started on the 1st of the month, both the move number and the date would be 1.

As you can see in the original thread, the original poster used a July 2010 calendar and managed to find a complete Knight's Tour on which the 2nd, 6th, 11th, and 23rd moves landed on the dates of the 2nd, 6th, 11th, and 23rd respectively. Not surprisingly, it was Jaap of Jaap's Puzzle Page who found an 8-match solution.

I filed this in the back of my mind, but never really did anything until I ran across the Solving the Knight’s Tour on and off the Chess Board post which I mentioned last week. I liked the basic idea of being able to input a shape, and have the computer work out the tour, and especially the idea of using it to work out the XKCD forum's calendar challenge.

With a little knowledge of Java and graph theory under my belt, I managed to work out a program to solve it. For my fellow Java programmers, here's the main portion of my program, and here's the KnightsTour class I wrote to support it. Most of the hard work is done by lines 590 to 749. Those and lines 20 to 23 can removed if you're interested more in the general Knight's Tour than the particulars of the calendar challenge.

One of the first things I did, not surprisingly, was to find out how many day-to-move matches I could find in this month's calendar. I also found 8, all of which are highlighted below in red:



Yes, I've gone through every possible calendar, starting on every possible date, and learned quite a few interesting things in the process:

• Due to the fact that the number of days in a week (7) is odd, and the fact that the knight always moves an odd number of spaces (3), this means that a Knight on a calendar will always move from an odd date to an even date, and vice versa (just like what your teacher taught you about adding even and odd numbers). This, in turn, means that it's impossible to get ANY date matches if move #1 begins on an even-numbered date, as all the odd moves will land on even dates, and vice-versa.

• The above fact also means that if you start on an even date in a month with an odd number of days (29 or 31), you won't be able to complete a Knight's Tour.

• Yes, Jaap's 8-match path is the best one possible for July 2010 in particular. It also happens to be the only way to get 8 date-to-move matches in a Knight's Tour of a 31-day month beginning on a Thursday.

• Given any random month and year, you can always find a complete Knight's Tour and at least 6 date-to-move matches. Surprisingly, these minimum matches aren't found in the shortest months, as you may expect. With 30- and 31-day months starting on a Saturday, as well as 31-day months beginning on a Friday, 6 is the highest number of date-to-move matches you'll be able to find.

• There are months with 9 date-to-move matches, but none with more than that. 9 date-to-move matches can be found in a 29-, 30-, or 31-day month starting on a Tuesday or a Wednesday. In a 29-day month starting on a Thursday, or a 31-day month starting on a Monday, you can also find 9 date-to-move matches. You can often find more than 1 way to get to these matches, as well.

As it happens, next month (October 2014) is a 31-day month starting on a Wednesday, and here's one of the 3 possible ways to get 9 date-to-move matches:



I chose this one simply for the elegance of the column containing 16-23-30 and the diagonal containing 12-20-28. I also find it interesting that so many powers of 2 have date-to-move matches (2-4-8-16).

For the more math-inclined geeks, I'll wind this post up with all the maximum number of matches I've found, including the dates on which they start:

28-day months, starting on:
Sunday:    7 matches, beginning from the 1st or 23rd
Monday:    7 matches, beginning from the 1st or 21st
Tuesday:   8 matches, beginning from the 25th
Wedneday:  8 matches, beginning from the 1st
Thursday:  8 matches, beginning from the 1st or 5th
Friday:    8 matches, beginning from the 1st or 15th
Saturday:  7 matches, beginning from the 23rd or 25th

29-day months, starting on:
Sunday:    7 matches, beginning from the 1st
Monday:    7 matches, beginning from the 1st or 27th
Tuesday:   9 matches, beginning from the 25th
Wedneday:  9 matches, beginning from the 1st or 11th
Thursday:  9 matches, beginning from the 1st
Friday:    7 matches, beginning from the 1st, 5th, 27th, or 29th
Saturday:  7 matches, beginning from the 1st

30-day months, starting on:
Sunday:    7 matches, beginning from the 7th or 23rd
Monday:    8 matches, beginning from the 27th
Tuesday:   9 matches, beginning from the 25th
Wedneday:  9 matches, beginning from the 11th
Thursday:  8 matches, beginning from the 1st
Friday:    7 matches, beginning from the 1st or 7th
Saturday:  6 matches, beginning from the 1st or 25th

31-day months, starting on:
Sunday:    8 matches, beginning from the 23rd
Monday:    9 matches, beginning from the 7th or 31st
Tuesday:   9 matches, beginning from the 1st, 23rd, or 25th
Wedneday:  9 matches, beginning from the 7th
Thursday:  8 matches, beginning from the 5th
Friday:    6 matches, beginning from the 1st, 5th, 7th, or 31st
Saturday:  6 matches, beginning from the 1st, 23rd, 29th, or 31st

Since I've changed my posting schedule, I seem to have neglected my monthly snippet posts!

Not to worry, however, as we're kicking off September with a good round-up of different takes on some of my favorite mental feats.

• One of the longest-standing tutorials on Grey Matters is the classic Knight's Tour. The traditional version usually happens on an 8 by 8 chessboard. What about other irregular, non-rectangular shapes?

Over at the Wolfram Blog, Jon McLoone explored that question using Mathematica in his post Solving the Knight’s Tour on and off the Chess Board. If you're interested in the programming and the math, there's plenty in this article. Even if you don't care for all the math and programming, the variety of boards with successful Knight's Tours is amazing and amusing. Who knew Pac-Man could play the Knight's Tour so well?

• Over in the Mental Gym, I have a full tutorial on squaring 2-digit numbers in your head. I've often wanted to move on to squaring 3-digit numbers, but never really found a method that suited me. However, I recently ran across a video tutorial from Mind Math called Mental Math Trick to Square 3-digit Numbers for Faster Calculation. It breaks the problem up into 2 steps, working with the hundreds digit followed by the remaining 2 digits as a group. If you're used to squaring 2-digit numbers, this method isn't difficult to learn and adapt:



• Back in March, I wrote a post about calculating powers of e in your head. At the time, I was unaware of Colin Beveridge's post, Secrets of the Mathematical Ninja: Estimating Powers of e, which featured a quicker, yet less accurate estimate.

After seeing my post, Colin took it upon himself to develop an improved method, which he posted as Powers of e Revisited: Secrets of the Mathematical Ninja. When you're done exploring those posts, check out the rest of Colin's Blog!

• Another favorite blog topic of mine is calendars. Beyond the standard day of the week for any date feat, there's plenty of interesting mathematical patterns and shortcuts waiting to be discovered in the calendar. One of the best round-ups I've found on the internet is P.K. Srinivasan's Number Fun with A Calendar (PDF version). Besides the PDF version, there's a zipped .DOC version and even a video demonstration of some of the topics from the book:



That's all for this month. I hope you found these enjoyable and useful!

People are often confused as to why the dates of Easter moves around so much from year to year. It moves so much much because Easter is the first Sunday after the first full moon after the first day of spring.

If this sounds confusing on its own, consider that the Roman Catholic and Eastern Orthodox churches use different calendars, which can yield different dates as a result!

Thanks to the work of John Conway, though, it is possible to work out the date of both Roman Catholic and Orthodox Easters in your head!

Some basic understanding and practice are all that's really needed to be able to calculate the Easter date in your head for any year from 1900 to 2099. In order to help make everything clearer, I've posted my new Easter Date For A Given Year tutorial over in the Mental Gym. To make it easier to learn, the tutorial is broken up into several steps:

• The introduction explains the rules for Easter calculation in detail, as well as what you need to know to get started.

• The next section explains how to calculate the date of the traditional Roman Catholic Easter. After learning how to work out the date of the Paschal full moon (the first full moon after the first day of spring in a given year), you then learn how to work out the date of Easter for that same year.

• If you want to impress others by performing this feat, there's an entire section of presentation tips that can help make this feat entertaining.

• The method for calculating the date of Orthodox Easter is covered another section, as well. Assuming you can work out Roman Catholic Easter, there are surprisingly few changes involved in working out the Orthodox Easter date.

• Finally, there's another section for those adventuresome souls who want to venture on and work out Easter dates in other centuries. Here you can find out what changes need to be made to the original calculations.

Since practice is important, I've also developed a set of interactive Easter date quizzes. Since you work through each section verbally in a step-by-step manner, the quizzes work the same way. In the first quiz, you simply work out the paschal full moon date for Roman Catholic Easter. In the next quiz, you're asked about the paschal full moon and Easter dates. The Orthodox quizzes are similar, and start with the Roman Catholic dates first, since you need that information as a starting point.

If you put in a little understanding, a little practice, and a little time, you may surprise yourself (and others) with an impressive new skill!

It's been over a year since I posted a tutorial over at the Mental Gym, so I figured it was about time for a new one!

This one is a new spin on working out the calendar for a given month and year. Yes, I have an existing Day of the Week For Any Date tutorial, and even a commercially-available version, but this new one is remarkably simple!

The new tutorial is dubbed the Quick Calendar Month Creation. It's a combination of a little-known, yet surprisingly simple calendar calculation method published by W. W. Durbin and E. Rogent in 1927, Robert Goddard's First Sunday Doomsday Algorithm, and my own approach of creating a full-month calendar to was calculations.

For those familiar with Doomsday algorithm, this isn't yet another variation of John Conway's fine work. Instead, I started with Durbin and Rogent's unusual and simple approach to dealing with the year, and adjusted the math so it meshed with Goddard's powerful work. The exact details and credits are given at the bottom of each section of the tutorial.

The result is a calendar creation routine that's quick and simple to learn, yet powerful enough to let you create calendar for any month and year back to 45 B.C., when the Julian calendar was first used!

To help you practice it more effectively, I've also developed a quiz page. The initial quizzes are simple, in order to help you master the calculation and recall required by each step, and then there's a more complete quiz, which simply has you create a calendar month for random dates. Some of you may recognize this as a modified version of my calendar quiz from Day One.

If you've never tried to do a calendar calculation before, try this out, and you might surprise yourself. If you've tried to do calendar calculations before and given up because of the difficulty of other methods, try the Quick Calendar Month Creation tutorial, and see the ease and power of this approach. Either way, I'd love to hear what you think about it in the comments!

Last week, I gave you some free magazines to peruse.

This week, I have some books I'd like to share with you, and they're filled with a great selection of classic mathematical feats and magic!

The first book I'll share with you is a 1952 book titled Mental Prodigies, by Fred Barlow (embedded below). The first section is concerned purely with arithmetical prodigies, or what we might call lightning calculators or human calculators today. The next section discusses related types of prodigies, such as chess or music. Next, is a section on memory, including chapters on famous memorizers, as well as mnemonics and memory techniques for actors.

The chapter I think will be the most interesting to Grey Matters readers is the section on Mental Magic, starting on page 183. It includes many of the standard feats covered here on Grey Matters, such as day for any date, squaring, cubing, and finding roots.

However, there are some more unusual ones, such as calculating the number of farthings in a given number of guineas, or how many barleycorns there are in a given number of yards. Granted, these might not play too well today, but the same technique, as the author explains, can also be used to give the number of minutes in a given number of weeks!

After the embedded book below, scroll down further for another math magic book.



If you'd prefer more than just a few small chapters about mathematical magic, check out the 1950 classic, Math Miracles, by Wallace Lee.

Among the more unusual tricks here are the Ne Plus Ultra Lightning Multiplier and its variations. The Miscellanies chapter is filled with numerous quick and unusual tricks, including the 100 version of our old friend Nim!

I don't want to rob you of the joy of discovery however. Take a chance to page through these e-books, and you may find some unexpected treasures!

Many of you are spending today with your mother in honor of Mother's Day, so I won't strain your brain too much today.

In fact, I'll just leave a few free magazines on the table for your perusal when you have some time later.

I'll start with the brand new Recreational Mathematics Magazine. This magazine is available as a whole PDF, or as PDFs of individual articles. The first article that caught my attention here was “The Secrets of Notakto: Winning at X-only Tic-Tac-Toe”. It caught my attention because I'd written about Notakto strategy 2 years ago, including how to win playing on 1 or 2 boards, and then how to win when playing on 3 or more boards.

Don't let me rob you of the joy of discovery, however. The other articles, including the one about Lewis Carroll's mathematical side, the one about vanishing area puzzles, and others are all waiting to be discovered.

The next math magazine I'd like to draw your attention to is Eureka, published by the Archimedeans, the Mathematical Society of the University of Cambridge, since 1939. New issues are being made available online for free by mathigon.org. This is no minor mathematical publication, either. It was the Archimedeans' Eureka magazine that, back in October 1973, had the honor of being the first to publish John Conway's Doomsday Algorithm for calculating the day of the week for any date.

Generally, The College Mathematics Journal isn't available online for free, but they have generously posted the full contents of their January 2012 Martin Gardner issue online for free! It's full of the kind of recreational mathematics which Martin Gardner loved and Grey Matters readers are sure to appreciate and enjoy. There are too many articles to single any one out for special attention, so I suggest jumping in and seeing what catches your eye first!

The final magazine I'll set out for your perusal isn't a mathematical magazine, but rather a magic magazine called Vanish, which is free to download, or read online as a page-flipping e-magazine. The reason I'm including it here with math magazines is because of Diamond Jim Tyler's article on “The Game of 31”. This is variation of our old friend Nim. For a Nim variation, 31 has a surprising amount of its own variations, including a dice version, a finger dart version, and a version which you can still scam someone after teaching them the secret!

That's all for now, so I'll wish you happy reading!

Happy New Year!

With a new calendar year, you deserve a couple of new calendar feats to go with it. In this post, you'll learn how to quickly give the day of the week AND the moon phase for any date in 2014.

Even better, both of these feats are much easier than they sound!

DAY OF THE WEEK FOR ANY DATE IN 2014: The method to do this is quite simple, and is known as the Doomsday method, originally developed by John Horton Conway. Don't worry, learning this method for one particular year is very simple.

The "Doomsday" from which the method gets its name always refers to the last day of February, whether it's the 28th or 29th. For 2014, the "Doomsday" is Friday (Feb. 28th, since it's not a leap year). If you think about it, you can already work out any date in February using just this knowledge.

For example, Valentine's Day, Feb. 14th, must also be a Friday, because it's exactly 2 weeks before Feb. 28th. How about Feb. 2nd (Groundhog Day)? Well, Feb. 7th is a Friday, and Feb. 2nd is 5 days before that. What's 5 days before a Friday? The answer is Sunday! Therefore, Groundhog Day will be on Sunday in 2014.

It's also fairly simple to learn the even-numbered months. There's a very simple pattern to remember them: 4/4 (April 4th), 6/6 (June 6th), 8/8 (August 8th), 10/10 (October 10th), and 12/12 (December 12th) will always fall on the same day of the week as the "Doomsday" (the last day of February, remember?).

On which day will Christmas fall in 2014? We know December 12th is a Friday, so 2 weeks later, December 26th, is also a Friday. Since Christmas is one day before that, it must be on a Thursday this year!

The odd months aren't much harder, but the patter is not the same. 5/9 (May 9th) and 9/5 (September 5th) will also always fall on the Doomsday, as will 7/11 (July 11th) and 11/7 (November 7th). This is easy to remember with the following simple mnemonic: "I'm working 9 to 5 at the 7-11". It helps you remember that 9 and 5 always go together, as do 7 and 11.

When is July 4th this year? It's exactly 1 week before July 11th, so it must be a Friday, as well. If you've got all the previous dates down, you've already got the mental capability to determine the date for 10 out of the 12 months!

The easiest way to handle March is to think of Feb. 28th as also being "March 0th". Working forward from March 0th, it's easy to see that March 7th, 14th, 21st and 28th will all be Fridays. St. Patrick's Day, March 17th, is 3 days after March 14th, so it's 3 days after a Friday, making it a Monday in 2014.

In January, it's usually the 3rd day of the month that falls on the Doomsday. In a leap year, however, January 4th falls on the Doomsday. Remember it this way: "3 times out of 4, it's January 3rd. On the 4th year, it's January 4th." In 2014, since it's not a leap year, you only have to recall that January 3rd is on the Doomsday (Friday, for 2014).

January 15th is Martin Luther King, Jr.'s birthday, but what day does it fall on in 2014? January 3rd is a Friday this year, and so is January 17th (2 weeks later). Take back 2 days, and we get January 15th being a Wednesday this year!

With the above knowledge, and a little practice, you can quickly and easily determine the day of the week for any 2014 date. You could get practice at the Day For Any Date (Mentalist Challenge) page, changing the year to 2014, and then trying to determine the date before you click the Show button.

When you're demonstrating this ability for someone, it's nice to be able to prove that you're right about the date. I use QuickCal on my iPod Touch (similar calendar are available for many portable devices).

MOON PHASE FOR ANY DATE IN 2014: 1 year ago, I posted a new tutorial about determining the moon phase for any date. Similar to the year calculations, focusing on a particular year like 2014 greatly simplifies the required calculations. Like the doomsday algorithm above, this formula was also developed by John Conway.

In fact, working out the moon phase for any date in 2014 is even simpler than working out the date! How simple is it?

(Month key number + date - 3) mod 30

It's probably best if I explain each part:

Month key number: January's key number is 3, February's key number is 4, and all other months' keys are their traditional numbers; March is 3, April is 4, May is 5, and so on up to December, which is 12.

Date: This is simply the number represented by the particular date in the month. For the 1st, add 1. For the 2nd, add 2. For the 3rd, add 3, and so on.

- 3: The subtracting of 3 takes the starting point of 2014 into account, which is why this particular formula works ONLY for 2014.

mod 30: If you get a total of 30 or more, simply subtract 30. Otherwise, just leave the number as is. Betterexplained.com has an intuitive explanation of modular arithmetic.

The resulting number will be the approximate age of the moon in days, from 0 to 29. This formula only gives an approximation, so there's a margin of error of ±1 day.

As an example, let's figure the phase of the moon on July 4, 2014. July is the 7th month, and the 4th is the date, so we work out (7 + 4 - 3) mod 30 = (11 - 3) mod 30 = 8 mod 30, which is just 8.

In that example, we estimate the age of the moon to be 8 days old.

What exactly does the age of the moon in days mean in practical terms? Here's a quick guide:

• 0 days = New moon (the moon is as dark as it's going to get)
• 0 to 7.5 days = Waxing crescent (Less than half th moon is lit, and it's getting brighter each night)
• 7.5 days = 1st quarter moon (Half the moon is lit, and gets brighter each night)
• 7.5 to 15 days = Waxing gibbous (More than half the moon is lit, and getting brighter each night)
• 15 days = Full moon (The moon is as bright as it's going to get, and will start getting darker each night)
• 15 to 22.5 days = Waning gibbous (More than half the moon is lit, and it's getting darker each night)
• 22.5 days = 3rd quarter moon (Half the moon is lit, and gets darker each night)
• 22.5 to 29 days = Waning crescent (Less than half the moon is lit, and it's getting darker each night)

It's time for December's snippets.

I've noticed the simpler, more direct skills prove popular, so this month will feature more skills you can learn, use, and demonstrate quickly.

• We'll start off with a simple skill: figuring out your longitude by looking at the night sky, assuming you're in the northern hemisphere. First, you need to find the star Polaris, which is why you need to be in the northern hemisphere for this to work. If you don't know how to do that, my post from September about learning to find various stars will be of help here.

The next step is to determine how many degrees above the horizon Polaris is located. This post from One Minute Astronomer shows how to measure the approximate angle using only your hands! This is a fun skill to demonstrate and teach, as well.

• From arrangements of stars, we come down to earth to arrangements of numbers. Michael Daniels, over at mindmagician.org, has posted a new magic square generator which can handle any integer from 34 through 9999. If you're curious about the method used to create these, you can learn more about it in his ebook, Mostly Perfect. You can even download free excerpts from the book for free!

• One of my favorite feats, the calendar feat, is taught in a very simple and direct version in the following video from Mister Numbers:



If you're not already familiar with Mister Numbers' work on YouTube, check out his channel, and see some of his other work in number patterns. He details more about this calendar procedure in his Kindle ebook, Amazing Calendar Math Magic.

This method has it roots in John Conway's Doomsday Method, and I show how to build on this basis in a simple way to handle almost any year in my ebook, Day One.

• Also from Mister Numbers, here's an impressive video that quickly teaches kids, or anyone really, to be able to handle multiplying the numbers from 1 to 40, and beyond, by themselves in a simple way:



I take advantage of this same basic pattern in my lessons on extracting the roots of perfect squares over in the Mental Gym, so this is a very useful pattern to know!

I hope you've found something quick an interesting. Have any quick and interesting math tips or patterns of your own? I'd love to hear about them in the comments!

Many in the US are enjoying a lazy 4-day weekend as I write this. That being the case, I'll keep the feats relatively simple.

In this post, you'll find out how to easily give others a new perspective by looking at time and space in new ways!

EARTH PHASE FROM THE MOON: If you've practiced working out the moon phase for any date in your head, whether you do the full version, or just memorized how to do it for 1 particular year, this feat is surprisingly simple.

Once you've determined the phase of the moon on a given date, the phase of the Earth as seen from the moon will be exactly the opposite phase! If the moon, as seen from the Earth, is in a full moon, then the Earth, as viewed from the moon, will be a “New Earth” (the Earth will be unlit). If the moon is in a waxing gibbous phase (more than 50% lit, and getting brighter each night), then the Earth, as seen from the moon, will be in a waning crescent phase (the Earth will be less than 50% lit, and getting darker each night).

Why does it work out this way? Take a look at the moon phase diagram below. Pick a phase, and follow that phase's line from the Earth to the moon, and imagine extending it through the moon. Imagine yourself out in space, along that line, looking at the opposite side of the moon that everyone on Earth sees. It's not hard to understand that the moon on this side must be in the opposite phase. If one side is getting brighter, the other side must be getting darker, and vice-versa.

Now, imagine yourself on that same line, but now you're between the Earth and the moon, facing the Earth. The sun is far enough away (90+ million miles!) that it's going to be lighting the opposite side of the moon and the Earth in the same way.



Just as in the original feat, you can verify this with Wolfram Alpha. If someone asks for the moon phase for, say, December 10, 2014, you would use the standard feat to estimate that the moon would be 19 days old (18-20 days old, including the margin of error), so you'd know it's in a waning gibbous phase, which means the moon is more than 50% lit, and getting darker each night.

Conversely, the Earth, as viewed from the moon, must be in a waxing crescent phase, so the Earth is less than 50% lit, and getting brighter each night. Wolfram Alpha can verify this for you.

1 MILLION SECONDS AGO: When you hear large numbers tossed around, it's really hard to get a sense of scale. How big is something like 1 million? To put it into perspective, imagine we're talking about 1 million seconds. When was it 1 million seconds ago?

Determining this isn't hard, especially if you just want to give the correct date. 1 million seconds is roughly 11.5 days. You can work out in your head what day 12 days ago was, or just cheat and use Wolfram Alpha to find out. If your local time is 1:46 PM or before, 1 million seconds ago was 12 days ago. If your local time is 1:47 PM or after, 1 million seconds ago was 11 days ago.

I'm writing this paragraph on December 1st, 2013, at about 11:45 AM local time, so 1 million seconds ago was November 19th, 2013. If I'm asked this afternoon at, say, 3:30 PM when 1 million seconds ago was, I'd say it was November 20, 2013, instead, because that is after 1:47 PM.

If you're interested in giving the exact minute, take the current time, add 13 minutes, then add 10 hours. 1 million seconds before December 1st at 11:45 AM would be November 19th of the same year at 9:58 PM, because 11:45 AM plus 13 minutes is 11:58 AM, and 10 hours after that is 9:58 PM.

If you're challenged to work out the exact second it was 1 million seconds ago, add 13 minutes and 20 seconds before adding the 10 hours. On 1:46:40 PM local time on any given day, 1 million seconds ago was exactly midnight, heading into 11 days ago.

As always, people can verify your answer using Wolfram Alpha.

1 BILLION SECONDS AGO: Since we're talking about large numbers, many people don't realize the difference in scale between 1 million and 1 billion, so when was 1 billion seconds ago?

1 billion seconds is over 31 years ago, so don't try and work out the exact date in your head. For this one, just look it up in Wolfram Alpha. As I write this on December 1, 2013, 1 billion seconds ago was March 25, 1982.

Working out the exact time is even simpler for 1 billion seconds ago, as it happens. First, add 13 minutes (and 20 seconds, if desired), just as before, but subtract 2 hours instead of adding 10 hours. December 1, 2013 at 11:45 AM minus 1 billion seconds is March 25, 1982 at 9:58 AM. Yes, your calculations can be verified with Wolfram Alpha.

You can make older dates like this more vivid by looking up those days on Wolfram Alpha or Wikipedia's year pages. For example, just a quick scan of those pages, I can remember that Danica Patrick was born, the first computer virus was only 2 months old, and the Vietnam Veteran's Memorial in Washington, D.C. would be opened the next day for the very first time.

1 TRILLION SECONDS AGO: 1 trillion seconds ago is the easiest, because that was 31,689 years ago, before modern clocks or calendars existed. This is roughly around 30,000 BCE, so ideas like the bow and arrow were still new, and not a single person was living in Japan yet. Obviously, if you include this, it's more for the sense of scale as compared to 1 million and 1 billion seconds ago.

If you like more mind-blowing changes in perspective, check out my Astronomical Scale post, and be ready for even more surprises!

While going through Mental Floss' Be More Interesting columns mentioned in my previous post, their post on how to navigate with stars caught my attention.

I've posted on how to calculate the moon phase for any date in your head, so why not learn more about the rest of the night sky?

The advice in Mental Floss' star navigation post is good as far as it goes. Yes, there are really only a few constellations you need to know to find your way around the sky, but the column stops short of practical teaching.

A website called quietbay.net used to feature a great tutorial on finding the important constellations, but that site has vanished from the internet. Fortunately, the Internet Wayback Machine has come to the rescue!

Here is the archived version of quietbay's clear, visual, and interactive constellation tutorial. It only takes about 15-20 minutes for the full tutorial. Being an archived version, there are a few images missing here and there, and only once or twice are those missing images are essential to finding the stars in the tutorial, but overall, it's still quite workable, and will quickly teach you how to located Polaris, Betelgeuse, Orion, the Big Dipper, Cassiopeia, and even Jupiter, if it's in the sky.

You should also note that it's a northern hemisphere-based tutorial, so the constellation Crux isn't included. Unless you're viewing from the southern hemisphere or the northern tropics, you won't be able to see Crux. If you can see it, Crux is one of the easier constellations to locate.

Try out the tutorial, read the Wikipedia article on Crux, and practice with the real night sky, and you'll be amazed how quickly you can get a good, basic knowledge of the night sky!

UPDATE: This site goes as far back as 2003. This approach was turned into a book in 2010, titled Stikky Night Skies. It teaches 6 constellations, 4 stars, a planet, and a galaxy, and only takes about an hour to read. There is a sample tutorial on the book's website, teaching only about Orion and Betelguese.

If you'd like to learn more in this same way, I highly recommend Laurence Holt's Stikky Night Skies!

Instead of posting on Thursday, I thought I'd wait until today, so we can enjoy some Friday the 13th fun!

Do I believe in the superstitions about Friday the 13th? No. Without them, however, Friday the 13th is just another day, so they do serve at least one good purpose!

Let's start off with a nod to those superstitions, then, including why even the non-superstitious may need to take them seriously:



How long until the next year without a Friday the 13th? Well, if you look at a calendar, the only time a Friday the 13th can occur is when the 1st of the month falls on a Sunday. Let's take a close look at when each month begins.

January 1st can, of course, fall on any day of the week. We'll refer to this uknown day as d. Now, regardless of which day d is, the first of February happens 31 days later. More accurately, it happens 4 weeks and 3 days later. So, whichever day of the week d is, February will begin 3 days of the week later. Working through the individual months in this way, here's what we find for a non-leap year:

January begins on day d.
February begins on day d + 3.
March begins on day d + 3.
April begins on day d + 6.
May begins on day d + 1.
June begins on day d + 4.
July begins on day d + 6.
August begins on day d + 2.
September begins on day d + 5.

Stop at this point and notice an important feature here. We have months than begin on d, d + 1, d + 2, d + 3, d + 4, d + 5, and d + 6. So, by September of a non-leap year, we must have a month that began on a Sunday. That, in turn means at least one of those months must have a Friday the 13th!

Is this also true for leap years? Let's work through a month of leap years in the same way and find out:

January begins on day d.
February begins on day d + 3.
March begins on day d + 4.
April begins on day d.
May begins on day d + 2.
June begins on day d + 5.
July begins on day d.
August begins on day d + 3.
September begins on day d + 6.
October begins on day d + 1.

In a leap year, it takes until October to guarantee that a month will begin with a Sunday, and therefore to guarantee that a month will have a Friday the 13th.

In other words, if you're waiting for a year without a Friday the 13th, it's not going to happen without another major calendar reform.

Now, as many Grey Matters regulars already know, the Gregorian calendar (the one most people currently use) repeats itself exactly every 400 years. So, September 13, 2413 will also be a Friday the 13th!

Out of curiosity, what is the most common day of the week on which the 13th falls? Over at QED Cat, they took the time to examine all 4800 months, and found that the 13th is more likely to fall on a Friday than any other day! Thursday and Saturday are the least common days, occuring only 684 times each out of 4800 months, while Friday occurs 688 times every 4800 months! Granted, this is only a difference of about 14.25% versus about 14.3%, so it's not like Friday occurs wildly out of proportion to the other days.

I wrap this post up with one final thought. 2013 has 2 + 0 13ths on a Friday. The next one will be in December, exactly 13 weeks from today!

There's a book out called The Mental Calculator's Handbook by Jan van Koningsveld and Robert Fountain, which naturally piqued my interest just by the title.

How does this book compare to existing books on mental math? Check out this review and find out!

First, you'll probably want to know what kind of mental math expertise the authors have. Robert Fountain is a British calculating prodigy who was the first of only 3 current International Grandmasters of Mental Calculation.

German mental calculation champion Jan van Koningsveld has held several world records relating to mental calculation, including taking only 3 minutes and 6 seconds to solve 10 problems, each of which involved multiplying two 5-digit numbers. You can find several videos online of his performances, and even if you don't speak German, they're easy to follow due to the numbers and his results being displayed.

At first glance of the contents, The Mental Calculator's Handbook doesn't seem to be much different than, say Arthur Benjamin's The Secrets of Mental Math. The first few chapters cover addition, subtraction, multiplication, division, and fractions.

Once you delve into the chapters themselves, they do begin with basic techniques similar to other books. I was pleased to discover, however, that they do take even these basic techniques farther than most books. The exercises at the end of each section, and the detail given about the techniques is written very clearly, so it's easy to understand.

This early attention to detail and emphasizing the finer points really begins to pay off when you begin learning the techniques in the later chapters, which include working out classic feats such as finding roots, our old friend calendar calculation, and the rarely-discussed factoring of numbers into their prime components.

The section on prime factorization was an especially interesting eye-opener. I was familiar with the basic techniques from my own work on primes in mental math, but the techniques here went much farther. Testing for divisibility by 2, 3, 5, and 9 are simple enough, but when many primes provide a challenge for divisibility tests, such as 7, 11, 13, and 37. The authors turn these into almost trivial challenges by showing how working with much larger numbers, such as 999 and 1,001.

Regular Grey Matters readers won't be surprised to know that I enjoy reading about and working out calendar-related challenges, and even here I was surprised! Besides just the basics of working out the day of the week for any date, you learn how to handle questions such in which years between 2000 and 2099 will Halloween fall on a weekend, and in which months of 1961 the 29th fell on a Sunday.

The Mental Calculator's Handbook winds up with brief biographies of various past mental calculators and their performances. This section especially was a very enjoyable read, and gives you an idea of just what can happen when such feats are demonstrated, and learn the sometimes sad and often amazing ways in which these performer's lives were affected.

If you're not sure of your own interest in mental calculation, I suggest starting with a more basic book, such as The Secrets of Mental Math and see if it's something you'll enjoy. Once you're ready to pursue it further, then you're ready for The Mental Calculator's Handbook, and it's greater attention to detail and mastery of the field. As a matter of fact, this book is a great bridge between the simpler mental math books, and the far more advanced ones, such as Ronald W. Doerfler's Dead Reckoning: Calculating Without Instruments.

Overall, if you're interested in mental math, and want to go beyond the basics, The Mental Calculator's Handbook is an excellent resource to take you to those next steps.

Have you practiced your multiplication by 63 skills? Good! Have you practiced your multiplication by 72 skills? Great!

Today, you'll learn to put these skills to use in a surprising and amazing way!

How many seconds are there in a 365-day year? To find out, we'd multiply 365 days per year by 24 hours per day by 60 minutes per hour by 60 seconds per minute, for a grand total of 31,536,000 seconds per year.

How about the number of seconds in two consecutive 365-day years? We just multiply the previous total times 2, and we get the rather interesting total of 63,072,000 seconds in 2 years!

Since we've learned to multiply by 63 and 72 quickly, we can now use this knowledge to quickly estimate how old someone was in seconds on their last birthday!

As you've seen in the previous posts, writing down the answers as you develop them is very helpful, and is an important part of working through the feat of estimating someone's age in seconds. You should write down the answer as you go, using a writing instrument and surface which allows you to erase, such as a pencil and paper, or a dry-erase board and marker. This is because you may need to make minor corrections as you proceed.

When you start, it's important that you ask for the person's age only, and have no idea of their birthday. Since the above calculations involve multiplying by 365, leap years are not taken into consideration. As you'll see later, the emphasis should be on the speed of doing an estimate in your head, not the exactness of the answer with every detail taken into consideration.

Once you have someone's age, the first step is to divide it by 2. This is important because 63,072,000 seconds is the amount of seconds in two 365-day years. You're effectively figuring out the number of seconds in how many pairs of years there are in their age.

The person's age divided by 2 will be treated as the given number, as used in the 2 previous posts. There are 3 different kinds of challenges you'll face in this feat, and I'll explain them below from easier to more difficult.

EVEN AGES UP TO 26: If the person's age is an even number of 26 or less, this is the simplest calculation. When their age is divided by 2, the given number will also be an integer (whole number), and the skills you've developed so far will be all you need.

As an example, let's assume you ask someone their age, and the reply that their age is 26. As you write their age down at the top of the page (or board), you work out that 26 ÷d; 2 = 13, so your given number is 13.

Now, you'll be effectively multiplying 13 by 63 million instead of 63, so I start by putting down a comma. To the immediate left of this comma will be the millions, and to the immediate right will be the hundred-thousands place. So far, the writing surface would look like this (not including the age at the top):

,

Now, we work through our process for multiplying by 63: “13 tripled is 39...*write down the 9 to the immediate left of the comma*...39 doubled is 78...78 plus 3 is 81...*write down the 81 to the immediate left of the 9*...” Before we move on to the multiplication by 72, let's take a look at the writing surface as it would look now:

819,

Remember, this represents 756 million at this point. The places are important, of course. The next step is to multiply by 72,000, which means putting down another comma to the right of the above answers, and putting dow the comma between the thousands and hundreds place (dashes used to represent empty spaces):

819,---,

From here, you continue your calculations to multiply by 72, picking up from where you left off: “78 doubled is 156...*write down the 6 to the immediate left of the comma*...78 plus 15 is 93...*write down the 93 to the immediate left of the 6*...” At this point, you should have an answer which appears like this:

819,936,

The final step is always to write 3 zeroes to the right of the rightmost comma:

819,936,000

Sure enough, if you have a person multiply their age, 26 in this example, by 365 by 24 by 60 by 60, they'll find that 819,936,000 is the right answer!

Even better, that's the process they'll assume you went through in your head to get your answer, so you're getting credit for doing the full work, when you're really only using a short cut.

EVEN AGES FROM 28 to 66: Ages ranging from 28 to 66 will result in given numbers being integers ranging from 14 to 33. When multiplied by 72, any of these numbers are over 1,000. This means, of course, that when multiplied by 72,000, these numbers will result in answers of 1 million or more, so you'll have to deal with carrying that 1 over into the answer you received when multiplying by 63.

For this example, let's say the person's age is 42, which makes the given number 21.

Start running through the process as before, thinking and writing as you go: “21 tripled is 63...*write down the 3 to the immediate left of the comma*...63 doubled is 126...126 plus 6 is 132...*write down 132 to the immediate left of the 3, include comma to denote 1 billion, add in comma before thousands place*...” So far, you would have this:

1,323,---,

Continuing on to multiply by 72, you'd think and write, “126 doubled is 252...*write down the 2 to the immediate left of the comma*...126 plus 25 is 151...*write down 151 to the left of the 2*...” Here's where you run into the challenge. As you start to write the 151 to the left of the 2, you realize there's going to be overlap. Write the tens and ones digit of 151 (the 5 and rightmost 1) as you would normally, like this:

1,323,512,

Next, erase the 3 and mentally add 1 to it, giving 4. So, you replace the 3 with a 4, and the answer will look like this:

1,324,512,

Don't forget to add the 3 zeroes to the right of the rightmost comma, so you have the full answer:

1,324,512,000

Dealing with carrying seems tricky at first, but once you get the hang of it, it's not much of an obstacle.

ODD AGES UP TO 65: Odd ages will result in a given number ending in .5, as you've seen in the previous tutorials. When dealing with these numbers, you may have to deal with carrying, as above, but you'll have to make 2 more simple adjustments.

The first, of course, is writing “&,5#148; instead of “.5”. When you're only multiplying by 63, the number just represents .5, but when multiplying by 63 million, that .5 starts representing 500,000! I'll discuss the other adjustment when we come to it.

For this example, we'll say that the person is 35, which, when divided by 2, makes your given number 17.5.

You would start just as before: “17.5 tripled is 51...52.5...*write down the 2,5*...52.5 doubled is 105...105 plus 5 is 110...*write down 110 to the immediate left of the 2,5, add comma for thousands*...” So far, you have:

1,102,5--,

At this point, you can probably see the other problem coming. Not only will you have to carry as before, but you now have to deal with that 500,000. Don't worry, this is easy to solve. When you're working out the final addition for the multiplication by 72, mentally add 50 to it. Once you have this number, erase the 5, and write the total as you would before. If needed, use the carrying technique taught earlier.

Continuing with our example from where we left off: “105 doubled is 210...*write down the 0 to the immediate left of the comma*...105 plus 21 is 126...126 plus 50 (to deal with the 500,000) is 176...*erase the 5, write down the 76, erase the 2, add 1 for the carry, write 3 and final zeroes*” Your final answer, with the 3 zeroes should look like this:

1,103,760,000

Yes, this is the correct answer for 35 × 365 × 24 × 60 × 60.

Once you become familiar with both carrying and dealing with ages of an odd number, you're ready to handle any age up to 66 in this feat!

TIPS:• You can use this link to practice with even ages only, and then use this link to practice with any age from 1 to 66. Don't forget to check your answer with this link, setting x to the age you were given (default age at the link is 42).

• After going through the calculation, I like to explain, “I only asked for the age, so I have no idea if the birthday they're talking about was last year, this year, or even next year. So, instead of trying to take every last detail such as leap years or seconds since their birthday into account, this is just a quick mental estimate.” They'll be startled to find out that this estimate is so exact, when you explain that you're multiplying their age by 365 days in a year, by 24 hours in a day, by 60 minutes in an hour, and by 60 seconds in a minute!

• You can grab interest at the beginning by talking about how people hear the terms million and billion, but never really think of the difference. Explain that 1 million seconds is only about 11¹⁄₂ days, while 1 billion seconds is roughly 31²⁄₃ years!

If you know you're going to be performing this feat ahead of time, you can use Wolfram|Alpha's calculations and Wikipedia's year and month pages to find historical events you could mention. It's one thing to say that 1 billion seconds ago was December 1981, but more vivid to say something like 1 billion seconds ago, Britney Spears was born.

Practice this feat and have fun using it to amaze people!

Doomsday Improvements - I bet those are two words you never expected to see together!

Doomsday, in this context, refers to the Doomsday algorithm developed by John Conway, which helps determine the day of the week for any given date. Since it was originally published in 1973, there have been many variations and improvements, and we'll look at some clever ones in this post.

The classic Doomsday Algorithm starts by determining the day of the week in a given year on which the last day of February falls. From there, you're able to work out any date in the rest of the same year.

There are other approaches, of course. Another popular approach for calendar calculation returns results from 0 to 6, where 0 represents a Sunday, 1 represents a Monday, and so on all the way up to 6, which represents a Saturday. Effectively, these methods are based on working out how far a date in a given month and year are from the first Sunday. We can call this a “First Sunday” approach, to differentiate it from the Doomsday approach.

The great thing about the Doomsday approach, of course, is how easy it makes some of the calculations. The great thing about the First Sunday approach is the ease of interpreting the results. Back in 2009, Bob Goddard combined the best of both worlds with his First Sunday Doomsday (FSD) algorithm!

If you've learned the Doomsday approach, you can quickly adapt to this new approach. Let's take a close look at it, and maybe even improve on it.

YEAR CALCULATION: Step 1 determines a key for the year, based on the popular Odd + 11 approach, described on Grey Matters here, and on Wikipedia here. If you read carefully, however, there is an important difference.

The last step of the standard Odd + 11 approach requires you to subtract the result of the previous steps from 7. In the FSD doomsday approach, this step is skipped entirely! Naturally, the fewer the calculations, the quicker you can get to the result.

In 2008, Mike Walters published a quicker way to get the same result as Odd + 11 that would work well here. You can read it in detail at that link, but I'll describe the practical steps here.

Note that Mike Walters' approach also involved subtracting from 7 as the last step. Since it's not used in Bob Goddard's version of Odd + 11, we also won't be using that subtraction from 7 here, either. That way, the results remain consistent.

When given a year, ask yourself whether the last two digits are evenly divisibly by 4. If they aren't, keep adding 11 years at a time, until you get to a year that is evenly divisible by 4. You'll only ever have to add 11, 22, or 33 years, and you don't need to worry if the total goes over 100. Once you're dealing with a year which is evenly divisible by 4, divide that year by 2, and then subtract the nearest multiple of 7 which is equal to or greater than that number. You're done!

Let's use Mike Walters' example of 1953, so we'll only focus on the 53. 53 itself isn't evenly divisible by 4, so we add 11 and get 64, which works! 64 divided by 2 is 32, and 32 - 28 (the nearest multiple of 7 equal to or less than 32) = 4, we now know our key number for the year is 4!

Notice that in Mike Walters' write-up, he gets 3 instead of 4 for 1953? Again, that's because Bob Goddard's FSD approach differs from the standard Doomsday approach by removing the final subtraction from 7. Removing this final step from Mike Walters' approach ensures that we get the correct results required for the FSD approach.

In practice, any even number which is NOT evenly divisible by 4, such as 1962, just needs to have 22 added (62 + 22 = 84). Also, any odd numbered years, such as 1933 or 1947, will require either 11 or 33 to be added to get to a multiple of 4 (33 + 11 = 44, and 47 + 33 = 80).

CENTURY CALCULATION: This is where Bob Goddard's FSD approach really shines! As you may know, the Gregorian calendar repeats exactly every 400 years. Bob Goddard's approach uses the 1700s as a basic century, and the century keys adapt from there, using his ingenious “No tune for Friday” mnemonic.

The 1700s are used as the basic century for 2 reasons. First, because the Gregorian calendar wasn't used in Britain and its colonies (including America) until 1752. Second, it's easy to think of the 1700s as the century in which America got its start, so you can think of the Gregorian cycle as an American history cycle, too.

The astounding thing about the FSD algorithm, though, is that it handles Julian dates easily! Instead of using the 4 century pattern for Gregorian dates, you simply use the century number for Julian dates. For the 1400s, you add 14, and for the 900s, you add 9. It's surprisingly simple.

If you're familiar with how the traditional Doomsday approach handles months, you'll note the same mnemonics are used in the FSD approach, so that's an easy adaption.

If previous approaches have been giving you trouble, try Bob Goddard's FSD algorithm. You just may be surprised how quickly you can pick it up.

It seems like Wener Miller just never stops creating!

He's just released E-Z Square 6, the latest in his series of magic square books!

E-Z Square 6 is a bit different from the previous works. Vols. 1-5 each focused on magic squares with a particular theme, such as birthdays, playing cards, and so on. What makes E-Z Square 6 different is that it goes back and updates and improves the methods and routines from past books.

The first routine is an update on the birthday magic square from E-Z Square 1. You start by putting the spectator's age in the center square of a 5 by 5 grid, and then you fill the remaining squares in a seemingly random way. When you're done, the magic total of every row, column, diangonal, and even several cross patterns, total the year the spectator was born! While the effect is the same, the method is greatly improved. Once you have the first few numbers, which is easy enough, the rest isn't much harder than counting.

The next routine is also an update on a bonus, this time on the magic square routine involving a measuring tape from E-Z Square 2. This one is a little sneakier than most of the routines, so it manages to pack an extra punch.

In E-Z Square 5, Werner Miller focused on magic squares with playing cards. The main problem with one of the feature routines, however, is that the resulting 4 by 4 squares usually featured duplicate numbers. In this volume, Werner Miller shows how to solve that problem once and for all, with a little inspiration from Richard Wiseman's The Grid, which also feature playing card magic squares.

Just when you think you've seen everything, the author goes on to teach other playing card magic square ideas with 3 by 3, 4 by 4, and 5 by 5 grids!

This ebook then rounds out with some fun magic square puzzles. One set of puzzles challenges you to cut an existing magic square into 2 smaller magic squares. The other set of puzzles require you to complete magic squares with only a few numbers with which to start. These very same puzzles, I'm proud to say, were first shared by Werner Miller to Grey Matters readers back in 2010 (puzzle 1, puzzle 2, puzzle 3, puzzle 4, answer to puzzle 4).

Technbically, you don't need the previous volumes to get use of E-Z Square 6, but reading this volume will certainly attract your curiosity about all the other routines.

If you're looking for a different take on magic squares, E-Z Square 6, which is also available in German, provides plenty of great routines and food for thought.

If you've ever practiced determining the day of the week for any date in your head, especially if you've used one of my methods such as the version I teach in the Mental Gym or Day One, you've probably run into the fact that these methods really only work back to seemingly random dates, such as Sept. 14, 1752 or Oct. 15, 1582.

This is due to the switch from the Julian calendar to the Gregorian calendar, as well as the fact that the calendar calculation formulas are designed only to work with the latter. What if you'd like to calculate dates in the Julian calendar?

In today's post, you'll learn how to handle Julian dates all the way back to the 200s!

The first official switch from the Julian to the Gregorian calendar happened in 1582, as noted in the video below. October 4, 1582 was not followed by October 5, but by October 15, 1582, effectively skipping 10 days.



Great Britain made the switch to the Gregorian calendar by jumping from Sept. 2, 1752 to Sept. 14, 1752, effectively skipping 11 days. Since Great Britain and its colonies were the majority of the English-speaking world at the time, native English speakers use this date for the conversion. Here are the years various countries made the switch, and here are the specific switching dates by country.

Since the switch was made due to the inaccuracy of the Julian calendar, the question becomes one of how to compensate for the change? As mentioned above, the change in the 1500s was only 10 days, yet in the 1700s it was an 11-day change.

To start, take the given year, and drop the last (rightmost) 2 digits. For example, 1491 becomes 14, an 943 becomes 9. This century number, which we'll call x, can be used to find the number of days that need to be added to the Julian date, in order to get the corresponding Gregorian date.

UPDATE (June 3, 2013): I've run across a much easier formula for determining the days needed to go from the Julian to the Gregorian calendar. I'll post the updated method here, and keep the method from the original post below the line.

Once you've got the century digits as x, multiply it by 3. If 3x isn't already a multiple of 4, round it up to the next multiple of 4. Once you've done that, divide the number by 4, subtract 2, and you've got your answer!

For example, let's find the adjustment needed for the 1200s. In this case x = 12, and 12 × 3 = 36. 36 is already a multiple of 4, so there's no adjustment at this point. Finally, we divide by 4 and then subtract 2, so 36 ÷ 4 = 9, and 9 - 2 = 7. This means we add 7 days to Julian dates in the 1200s to get the corresponding Gregorian dates.

What about the 900s? 9 × 3 = 27, which isn't evenly divisible by 4, so we round it up to the next multiple of 4, which is 28. 28 ÷ 4 = 7, and 7 - 2 = 5, so we would add 5 days to any Julian date in the 900s to get the corrresponding Gregorian date.

Naturally, once you have the correct Gregorian date, you can use standard calendar formulas to get the day of the week for the given date.

Since I posted this on May 30, 2013, let's try the Julian date of May 30, 1013. What would the corresponding date be in the Gregorian calendar? 10 × 3 = 30, and we round 30 up to the next multiple of 4, which is 32. 32 ÷ 4 = 8, and 8 - 2 = 6. Now we add 6 days to May 30 to get May 36, or more accurately, June 5th (36 - 31 days in May = 5). If we double check with Wolfram|Alpha, we see that May 30th, 1013 in the Julian calendar is indeed June 5th, 1013 in the Gregorian calendar.

Once you have a Gregorian date, you can treat it as you would any other date in your system. With practice, you can now go back as far as the 200s, since the adjustment for the 200s is 0. You can actually go farther back, but the calculations have additional things you have to deal with, such as negative adjustments, and compensation for date with BC or BCE before them.

In performance, and with a little practice, you can buy yourself the extra time needed for this extra calculation by performing the calculation while explaining briefly about the Julian and Gregorian calendars. Practice with Julian dates, and you'll surprise yourself with a wider range on your calendar calculation abilities!

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ORINGAL POST:

It's done with the following formula:


Those funny-looking brackets with an upside-down L-shape simply mean to round any fractional answer you get upwards. Mathematicians and programmers know this as the ceiling function.

As a simple first example, let's assume we're working with a date in the 1200s. For any such date, we drop the last 2 digits, leaving us with x = 12. First, subtract x from 16: 16 - 12 = 4. Next, divide that result by 4: 4 ÷ 4 = 1. Finally add that result to x again, and subtract 6: 1 + 12 - 6 = 13 - 6 = 7.

That result of 7 means that, for any date in the 1200s, we would need to add 7 days to the Julian calendar to get the corresponding Gregorian date. October 10, 1252 in the Julian calendar becomes October 17, 1252 in the Gregorian calendar.

Naturally, once you have the correct Gregorian date, you can use standard calendar formulas to get the day of the week for the given date.

That 1200s example worked out nicely because 4 divided by 4 comes out even. What about when we're dealing with a century that doesn't work out so neatly? Let's try a date in the 900s.

For the 900s, of course, x = 9, so let's start going through the formula again. The first step is 16 - 9 = 7. Next, we have to work out 7 ÷ 4. The exact mathematical answer is 1.75, but since the next step is to round up any fractional answers (remember the upside-down L-shaped brackets?) up, all you really need to know is that 7 ÷ 4 = “1 and some extra”. When you round this up, you get 2.

Now, you can work through the rest of the formula just as before: 2 + 9 - 6 = 11 - 6 = 5. So, for any date in the 900s, you would need to add 5 days to find the corresponding Gregorian date. For the 800s to the 1700s, here's the adjustment required for each century.

Since I posted this on May 30, 2013, let's try the Julian date of May 30, 1013. What would the corresponding date be in the Gregorian calendar? We know x = 10, so 16 - 10 = 6. Next, 6 ÷ 4 = 1 and some more, which rounds up to 2. 2 + 10 - 6 = 12 - 6 = 6, so we need to add 6 days to get the Gregorian date. May 30th + 6 days = May 36, or more accurately, June 5th.