I teach a version of Day of the Week For Any Date feat, also known as the human calendar feat, here on Grey Matters, but I thought it was time to look into this feat in more detail.

To help you get a more complete picture, I've put together a Day for any Date Online Toolbox. Best of all, each one of these tools can be accessed for free!

Gregorian Calendar: The calendar we use today wasn't developed until 1582, so most days of the week before 1583 can't be calculated using the formulas we'll be examining. In Britain and its American colonies, it wasn't adopted until 1752. Some countries, including Greece and Russia, began using it less than 100 years ago. For a fascinating look at the history of humanity's search for an accurate calendar, check out James Burke's Infinitely Reasonable.

Leap Years: Most people think of leap year as being every 4 years. It's important to realize that years ending in 00 are only leap years if they're evenly divisible by 400. For example, 2000 was a leap year, while 1900 and 1800 were not. The next year ending in 00 that will be a leap year will be 2400. Thanks to this rule, the calendar repeats exactly every 400 years.

Modular Arithmetic: If there's one thing you're going to have to know when doing any version of the day for any date feat, it's modular arithmetic, also known as “Calendar and Clock Arithmetic”. Specifically, you're going to need to know how to take a number and work out modulo 7. If you remember doing problems like 23 ÷ 7 = 3 with a remainder of 2, then working out any number modulo 7 is easy. Modular arithmetic focuses only on the remainder, so 23 modulo 7 would just be 2 (the remainder of 23 ÷ 7)! Obviously knowing your multiples of 7 helps greatly.

Key Numbers: Almost every system for calculating days of the week include turning the year, month, and day into adjustment numbers commonly referred to as key numbers, or simply keys. Often, people who have learned one way of calculating run into others who do the same feat, and they're shocked to find that their key numbers are different. The key number can be changed, as long as they all compensate for the change in the same way. I've run across one method that used 1 for a Sunday and 7 for a Saturday (most day keys, due to the mod 7 calculation, run 0 through 6), yet it always gave the correct date, because the rest of their system compensated for this.

Day of the Week Keys: For most version of the calendar feat, you'll find that 0=Sunday, 1=Monday, 2=Tuesday (very easy to remember!), 3=Wednesday, 4=Thursday, 5=Friday, and 6=Saturday. Even people who are used to calendars that have Monday as the first day of the week use this, since 1=Monday. Again, variations of this system will still work out as long as the other keys properly compensate.

Mnemonics: Mnemonics aren't required to perform calendar calculations, but they do allow you to speed up the process by helping you to recalling the results of other calculations, rather than actually working them out during the feat itself. If you choose this route, and I highly recommend you do, take the time to learn and practice the Link System, the Number Shape or Number Rhyme systems, and the Major/Peg System (all accessible through the link above).

Perpetual Calendars: One of the original drawbacks of doing this feat was the need to carry a perpetual calendar around with you, so the audience could verify the day. They ranged from impressive monstrosities to ingenious mechanical devices. Today, a free app on your mobile device, or even someone else's, as they're likely to trust it more, are easily available, making this feat more commercial than ever before!

Year Key Number

In calendar calculations, a mention the year specifically refers to the rightmost two digits (such as the 11 in 2011), as the other part of the full year isn't considered unless adjustments for the century are required.



(year + ⌊year ÷ 4⌋) mod 7: Straight out of Zeller's congruence (denoted as K in the formula at the link), this is the classical formula for determining a key code for the year from 0 to 6. The odd-looking L-shaped bars (⌊ and ⌋) refer to the floor function, which simply means to drop any fractional parts remaining after the divison. For 1987, you'd do (87 + ⌊87 ÷ 4⌋) mod 7 = (87 + ⌊21.75⌋) mod 7 = (87 + 21) mod 7 = 108 mod 7 = 3, so the code for 87 would be 3. This many mathematical steps is not something that is easily done under pressure of performance, so alternatives have been developed.

Decade and Leap Year Offset: In this method you memorize keys for each decade then add the last digit of the year, then add 1 more for each leap year inbetween. Since you're recalling 1 of only 10 numbers, and only adding and doing a mod 7 calculation, this is obviously quicker. If you work through the system as described at the link, you'll see that 1987 becomes a 4, not a 3. As mentioned earlier, it still gives the correct date, since the rest of the system compensates for that difference.

“Dozens, remainder, and fours in the latter...”: This approach, and the particular phrasing, comes from John Horton Conway's Doomsday approach. You start with how many full sets of 12 years (“Dozens”) have passed, which, in our 87 example, would be 7 (7 × 12 = 84), to which we add the remaining years of 3, which gives 7 + 3 = 10. No leap years happened between 1984 and 1987, so we stay with 10. 10 mod 7 gives us 3 once again! This is a slightly easier variation of the decade approach above.

28 Year Pattern: If two years are a multiple of 28 years apart, and they both begin with the same 2 digits (For example, 1902 and 1958 are 56, or 28 × 2, years apart, and both begin with 19), then you can always count on their calendars to be exactly the same. You can take advantage of this by memorizing the key for only the first 7 years that are multiples of 4 (00 -regardless of leap year status-, 04, 08, 12, 16, 20, and 24), and then realizing that every other year is covered by adding 28, 56 or 84 years (again, as long as the first two digits remain the same). The non-leap years can be covered by just adding the difference in years. For 1987, we realize that 1900 (even though it wasn't a leap year) has a key of 0, thus so does 1984. 1987 is 3 years later, so 0 plus gives us 3 once again!

Mnemonics: As I mentioned earlier, this is usually the fastest approach for getting the year key. You can consult a chart or do the calculation earlier, and simply use mnemonics to recall those results during performance. For 1987, I'd use one of the above methods to work out that 87 means a 3, and then link FoG (87 in the Major/Peg system becomes FoG) to lips (3 in the Number Shape System looks like the side view of a pair of lips), perhaps by thinking of a day that's so foggy I can only see a pair of lips approaching me. During performance, all you do is think, “87 → FoG → lips → 3” and you've got the year key!

No Year Key: Some people do a simple version of the Day for any Date feat by working out dates that are only in the current year. As long as the rest of the keys take that into account, you can get away with not using a year key!

Month Key Number

January, of course, has 31 days. That's four full 7 day weeks, plus another 3 days. So, when you go to figure a key for the month of February, it's always going to have a key that's 3 numbers greater than January. Basically, whatever number is used for January, the other months will always have the same consistent relation.

Those who learned the feat in the 20th century usually used the following system:

• January = 0
• February = 3
• March = 3
• April = 6
• May = 1
• June = 4
• July = 6
• August = 2
• September = 5
• October = 0
• November = 3
• December = 5

Groups: It can help just to remember groups of 3 or 4 numbers together, as in a telephone number or ID card. For example, at that link, the month keys as grouped as 622-503-514-624. It's still easy to determine which number goes with which month.

Similar Months: If often helps to categorize the months by their number. This can help in determining which months in a given year begin on the same days. For example, September and December will always start on the same day of the week, and May and June will never be like any of the other years.

Mnemonics: At this link, I give some very simple mnemonics for the month keys (scroll down to the bottom of the post), with the assumption that they're being used to calculate dates from 2000 to 2099 (compare them to the above numbers, they're all 1 less).

Doomsday: This is where John Horton Conway's Doomsday approach really shines. For a given year, you only need to know 1 simple key! As long as you know the date on which the last day of February will fall in a given year, you can work out the day of the week for ANY date in that same year! Bonus Tip: If you've used mnemonics to remember your year keys, you can always find one number to add that will always give you the last day of February (this will change for different centuries, of course), thus allowing you to quickly adapt to working with the Doomsday approach.

Practice, Practice, PRACTICE!

Day of the Week For Any Date Quiz (Revised): This is a step-by-step quiz, used in guiding you along the particular step-by-step approach I teach. However, the Dates: 2000-2099 button and the Dates: 1600-2399 button can be used to test yourself regardless of the system you use.

Mind Magician's Day for any Date System: At the bottom of this tutorial is an interactive quiz that not only gives you feedback on whether you were correct, but also your average time.

Random.org's Calendar Dates: Random.org will give you random calendar dates, but it won't tell you on which day they fell. If you use this page as a quiz, you'll need to provide your own perpetual calendar.

Free Mobile Apps: Regardless of your chosen mobile operating system, you can almost always find free apps to quiz you on day of the week for any date. Many that aren't intended to quiz you can often be adapted by using a random setting and covering the place that displays the day of the week.

I talk quite a bit about standard memory techniques on Grey Matters, but sometimes you have to be creative not only in your mnemonics, but also in how you approach what you're memorizing.

Today, we're going to look at some varying memory challenges, and learn how different people approached these tasks.

One of my favorite memory feats on this site is the Day of the Week For Any Date feat. That lesson is geared towards being a memory demonstration, however. If you're looking for more a practical immediate use, such as what day of the week a date will fall on this year, here's one way to remember this year's calendar.

The method involves remembering on what day of the week the 7th of each month falls in the current year. It's also a good lesson in doing research to see what others have done when creating an original approach for the task. All this person really had to do was remember the last day of February for a given year, and that would've given them the whole year!

Another popular feature here is the Knight's Tour, where you can build up to learning how to do a Knight's Tour when someone give the beginning AND ending squares. If that's not impressive enough, imagine taking it to the next level while doing it blindfolded!

This Frezco games page discusses the basics of doing a Knight's Tour blindfolded. The first challenge you run into when doing this version blindfolded is that the named squares need to be of different colors, so one of the first things you learn on this page is how to quickly identify a square's color by it's name. Frezco also has an interesting PDF on similar puzzles, such as Bishop's, Rook's and Queen's Tours, and how to make them interesting and challenging.

Lybrary.com's Chris Wasshuber offers more detailed blindfolded Knight's Tour instructions for sale on his site, including performance tips.

One memory feat I haven't covered on Grey Matters is the challenge of memorizing the world map to the point where you can not only identify every country in the world, but even draw a map locating all of the countries. There's an excellent introduction to a very creative approach in this PDF.

That PDF itself is a pitch for a full book, which no longer seems to be available, unfortunately. However, the PDF does describe the basics so well, that you may be able to sit down with a world atlas and develop your own mnemonics and designs. Alternatively, you can find another structured approach in the book Mapping the World By Heart.

Just as with many memory challenges, there are often patterns you can learn about to help you approach any sort of memory task. Classically, the multiplication tables are a source of dread for anyone just learning about them. However, when taught about the underlying patterns, they can become fun and easy! Take a look at Mister Numbers' Times Tables videos, and you'll see a very interesting approach to the times tables. Here's a great and simple way to learn about the 3s, 6s, and 9s all at once:



What creative approaches have you found that helped you memorize a subject?

I'm seeing that it's time to start updating my original memory-training program, Train Your Brain and Entertain (TYBE)!

The original is still available at Lybrary.com and from various magic dealers. However, it was written in 2005, which doesn't seem that long ago. Quite a bit has changed since then, especially considering that this was 2 years before the word “iPhone” would enter our vocabulary.

I'm already getting mail from OS X Lion users about the incompatibility of the original version of TYBE. This isn't surprising, since Lion no longer offers support for the old PowerPC chip systems, for which the original TYBE was written.

The solution is not only to update it, but to bring it to mobile devices! That's my current project, so TYBE 2 is not available yet. However, I am making some great progress, and wanted to give my readers a peek at what it's going to be like.

Like the recently-released Verbatim, I'm writing TYBE 2 for mobile devices, as well as any browser with HTML5 and CSS3 support.

In the screenshot below, you'll see the index, where you can select all the original memory challenges:



If you're wondering about the difference between the light and dark bands, the dark bands are sections I haven't completed programming yet. In the final version, the entire menu will be the same color.

Another thing you'll notice absent from the screenshots are the preferences and help sections. Those will be included, I just simply haven't programmed them yet.

The biggest change so far is that the sequential noun/verb list challenge and the sequential abstract word challenge, which were originally two separate listings, have been merged into one.

Here's some screenshots from that section. On the left, you'll see the main settings for the list, and on the right, a sample list generated with the timer running. Even while you scroll through that list, the timer remains at the bottoms of the screen.



Just as with the original TYBE, I will be selling this program. However, if you've purchased the original TYBE in the past (either the hardcopy in stores, or the download from Lybrary.com), I will be making arrangements for you to get it at no extra charge.

If I'm going to get this ready, I'd better get back to programming it, so that's all for now. If you have any questions about the upcoming release, or any ideas for requested features, let me know in the comments!

Do you have paper, scissors, markers, and glue handy? Then you're not only ready to go back to school, you're ready for today's post!

Today, you'll learn how to make some fascinating paper toys. They all have a mathematical basis, but we're not going to delve into the math behind them. Today's mission is simply the fun of creating and playing.

Our first toy is called a flexagon. It's a hexagonally-shaped paper toy with 3 sides, even though it only appears to have 2.



You can find patterns and templates for designing your own patterns on the author's site.

A 3-sided shape like that is neat, but what about one with more sides? Shortly after Martin Gardner passed away, James Grime filmed this tribute to him, teaching how to make a square 6-sided flexagon:



If you're wondering just how far you can take these, check out flexagon.net to get an idea. They feature wild variations there, include magic square flexagons, and even dodecaflexagons!

Another wild paper toy is called a kaleiodocycle, which you can think of as a 3-dimensional equivalent to a flexagon. Here's a simple way to create a hexagonal kaleiodocycle from three squares of paper:



If you're up for more of a challenge after that, try making the same model from a single piece of paper:

In August's snippets, we take another look at past links on Grey Matters that lead us to all new links!

• If you're old enough to remember the 1990s, you're probably old enough to remember Kevin Trudeau's infomercials for his Mega Memory audio tapes. Thanks to the Internet Archive, the full audio of Mega Memory is now available as a free download! Also available are the full audio of Advanced Mega Memory, and an hour-long lecture video titled Never Forget Another Name.

They original Mega Memory course part of an FTC fraud case against Kevin Trudeau, as he made the claim that these tapes could give you a photographic memory, even if you suffered from learning disabilities. As discussed back in August, a true photographic memory has quite different implications from a trained memory. These audio and video downloads, however, do provide a good basic introduction to mnemonic techniques, so they're still worth checking out.

• I first brought up the Monty Hall dilemma on this blog in 2006, and discussed it again with the help of Scam School last year. If you have Flash on your system, you can try the game yourself repeatedly on StayOrSwitch.com. Besides playing the game itself online, there are several good explanation videos on the site, as well. If you're on an iOS device and don't have Flash, you can still play the game for free with the Monty Hall Paradox app.

• Test Your Vocab is a fun internet quiz as well as a good self-improvement benchmark. It quizzes you by asking about sample words, and uses that to determine the estimated size of your vocabulary. If you desire to expand your vocabulary, there's the Mnemonic Dictionary, as listed on the Memory Tools page, and Thinkalink's vocabulary section, a mnemonic site I covered last year.

• Doug Canning has recently made his book Canning's Card Capers, available as an eBook. This is the book where Mental Shopper (and its practice quiz) originated. If you appreciate think kind of ingenious thinking and generosity, purchasing his eBook would be an excellent way to show it.

Scam School has just released an episode featuring yet another submission from Grey Matters!

Besides showing you the trick, originally published by Martin Gardner in Mathematics Magic and Mystery, I'll teach you an extra tip that wasn't included in the episode!

This trick is called The Purloined Objects:



As Brian says, yes, it was I who came up with the sentence "ABsolutely, BriAn's ACtive shows BeCkon CheAting ContriButions" as a way to help promote his show. In Mathematics Magic and Mystery, Martin Gardner includes other sentences you can use to remember the needed information. Some are for specific objects, such as coins and bills, but he does include some for assorted objects.

How exactly does the trick work? As you might recognize from the use of 1, 2, and 4 as the multiples, the binary system is involved, but we don't even need to delve into the basics of binary in order to explain the workings of the trick.

You might think that 3 people and 3 objects mean that there are 9 possibilities. Actually, since any object chosen means that the others can't choose that object, there are actually only 3 × 2 × 1, or 6 possibilities. Let's take a look at each possibility. As in the video, 1, 2, and 3 will represent the people, and A, B, and C will represent the objects:

1-A  2-B  3-C: 1 takes 1, 2 takes 4, 3 takes 12 = 17 Tic-Tacs taken
1-A  2-C  3-B: 1 takes 1, 2 takes 8, 3 takes 6  = 15 Tic-Tacs taken
1-B  2-A  3-C: 1 takes 2, 2 takes 2, 3 takes 12 = 16 Tic-Tacs taken
1-B  2-C  3-A: 1 takes 2, 2 takes 8, 3 takes 3  = 13 Tic-Tacs taken
1-C  2-A  3-B: 1 takes 4, 2 takes 2, 3 takes 6  = 12 Tic-Tacs taken
1-C  2-B  3-A: 1 takes 4, 2 takes 4, 3 takes 3  = 11 Tic-Tacs taken

After creating Verbatim 2, I started thinking about apps for other types of memory feats.

With most memory techniques and memory feats, it helps to have some sort of random data with which to practice. Having this as a mobile app means you could practice anywhere!

With that in mind, I've begun work on Randomizer, another jQuery Mobile-based app. It's still in its early stages, but it's already usable. Currently, it has 3 functioning sections.

The first section concerns word lists. You can select what type of contents you'd like for the word lists, including nouns, verbs, nouns AND verbs, and more abstract words. The abstract words are picked randomly from lists of last names of US presidents and vice presidents, country names, Kentucky Derby winners, or chemical elements. Using the slider, you can set how many words will be on the list, and the toggle switch will let you determine whether the words on the list are numbered.

Once you click the Generate Word List button, the list appears below a bar that will let you hide and show it by clicking on it. This way, you can quiz yourself on lists in order, or in and out of order quite easily. If you play around with the controls, it doesn't take long to get used to using them.

Although the other sections have different controls, they all share essentially the same style. They all have controls of various sorts to let you control the settings, a Generate button, and a place where the data is shown which includes the ability to hide or show the important data.

The Numbers section is, at this writing, the simplest section. You use the slider to determine how long the generated number will be, and the toggle switch to determine whether commas will be displayed. You click generate, and you get a long number (up to 30 digits) to challenge your memory.

Since one of my favorite feats is the calendar memory feat, I've also included a Date section. The checkboxes let you select which centuries from which to choose (1600-2199). If you don't select at least 1 century, the Generate Random Date button is disabled. A toggle switch lets you determine whether to exclude leap years, since figuring out those can be tricky when you're first learning. At the bottom, a set of radio buttons let you set the date format, so you can read the date in the way with which you're familiar.

In the case of dates, once you click the Generate Random Date button, the date will always be displayed, but the day of the week on which that date falls will be hidden. Once you think you know on what day of the week that day falls, click the bar to reveal the actual day of the week to see if you're right.

As I mentioned earlier, this app is still a work in progress. I have plans to add things such as random cards sequences and more. Right now, other than hiding and showing the generated data, there's no true quiz function. In future upgrades, I might add more interactive quiz features, too. A manual is also in the works.

Similar to the Verbatim 2 app, this app is completely self-contained, and can be run offline in a similar manner.

What do you think of the app so far? What suggestions do you have for improvements? Let me know in the comments!

(Note: I originally published this back in April of 2009. With the school year about to begin, it seemed like a good time to repost it.)

Did you ever wonder if there was a way for the average student to improve their learning abilities? Sure, there are simple tools, such as SQ3R, but what about a more comprehensive, detailed approach to learning as a whole?

Memory trainer Graham Best, who once demonstrated his memory skills on the Alan Thicke Show, spent more than 3 decades as a teacher in the Vancouver school system, and spent much of that time figuring out exactly what students could do to improve.

The result was a course you could purchase at his now defunct memory-learning.com site (still viewable via the Internet Archive Wayback Machine).

While the paid course itself is no longer available, the free overview that came with the course still exists. This overview was called 12 Things I Wish My Students Knew, and was available as a series of 12 videos. The first one, not surprisingly, was titled First Things First, and is shown below: