When it comes to memorizing various things, numbers are often considered the toughest challenge. How do you turn something so abstract into something easily imagined, and more importantly, something easily recalled?

The most common systems used to remember numbers are the Major/Peg system, which I cover in these links and in these videos. Alternatively, there's also the DOMINIC system.

The major obstacle for most people, however, is that both of these systems require some study on their own before they can be used regularly as a tool. Having said that, those who do put in the work do find them remarkably useful, with the payoff being images for a given number being more readily available.

What if you need to memorize numbers, but not badly enough to study and adopt such as system? There are alternatives that take advantage of what you already know about a given number.

First, there's the Number Shape system, in which you simply remember numbers via objects with a similar appearance:



Not many people are aware that the Wikimedia sites (known mostly for Wikipedia) have an entire section of photos dedicated to the Number Shape system.

In a similar vein, there's the Number Rhyme system:



The system with probably the widest variety of potential for images, at least of the simple systems, would have to be what I call the “conceptual” system. In this approach, you use images that you've related to the number in other parts of your life. In the video below, it initially seems like the lecturer is using the Number Rhyme system, since she uses “sun” as her image for 1 (about 1:30 into the video). However, note that she continues with using “eyes” for 2, and “triangle”. This is, however, a good lesson in feeling free to use other supporting systems, should one not work for you in a particular instance.



Coming up with your own images can be a bit of a challenge for this system, depending on the number and your experience. For example, quick, what image would you use for 32? If you're having trouble coming up with an image, Wikipedia's number pages can be of great help, such as their number page for 32.

Think-a-link, a mnemonic site, also uses this approach, but takes it one step further. In their tips and tricks for numbers section, they take such approaches as splitting up the numbers (using a 4 by 4 for 44), percentages (75 = 75%, possibly a Pac-Man type image), and even Roman numerals (4 = IV, so 4 = ivy).

Looking through this page and other pages on this site with a number you need can be very inspiring. Didn't find an image you could use for 32 at the Wikipedia link? Check out the approaches used in the entries for Franklin D. Roosevelt, Germanium, 4 times 8, and others.

The last simple number system I'll mention in this post has been dubbed by Rememberg.com as the “Count System”, which focuses on the number of letters in common words and phrases. Because of this, it tends to work better for longer numbers. For example, Searching for 32 doesn't bring up any Count System results there, but 320 and 3200 work just fine.

This is not intended as an exhaustive list, but rather as a place to start exploring. Since creativity is so important in creating a mnemonic system, there are many more ways, and many more will be developed. Even at Rememberg, the site I just mentioned, they cover other systems beyond the ones discussed here.

The important thing is to find one that is fun and useful for your needs.

This week's Scam School features yet another creation by Martin Gardner, as submitted by yours truly!

This one is a prediction, but one that is very clean and appears very fair to your audience. It's also a handy impromptu effect to know, since it's adaptable to so many different props.

In this week's episode of Scam School, this routine is performed simply with paper, pens, and coasters:



This routine was created by Martin Gardner, who originally published it in Ibidem #6, back in 1956, as The 1-2-3 Trick. These days, it's more easily found in Martin Gardner Presents under the same name.

There's one important part to this routine that is underemphasized in the video above, and I'd like to focus on that here.

As described in the video, there are two possibilities. The first one is the possibility that occurred in the performance part of the video, where the Ms, the Bs, and the Hs all matched.

The 2nd possibility, revealed with the prediction on the paper, is the possibility I'll be discussing in more detail. Take a close look at the second possibility (about 6:30 into the video) - the left column (from top to bottom) reads MBH, the second column (again, from top to bottom) reads BHM, and the 3rd column (yes, from top to bottom) reads HMB. Notice that each of these groups has no duplicates in it, so that each forms a complete set of 3. This will always be true for 1 of the 2 possibilities in this routine.

In Martin Gardner's original version of the trick, he used Aces, 2s, and 3s of three different suits, and specifically refers to this to the audience as The 1-2-3 Trick.

Imagine it playing out this way: You set down the Ace of Hearts, 2 of Hearts, and 3 of Hearts as the middle row of cards, and after laying down your arrangement, the cards appear in this manner:

2  3  A  
A  2  3

2  3  A  
A  2  3
3  A  2

When peforming the classic Day of the Week For Any Date feat, it often makes a difference how you do it.

Most performances aim for the instantaneous Rain-Man type of reaction. The required calculations and/or mnemonics can make this challenging, so performers often use various touches than can simultaneously give them time to mentally retrieve the information, while helping, at least in appearance, to speed things up from the audience's point of view.

The year, with its leap year exceptions, is often the hardest part of the date to deal with, so one classic approach is to ask for the year first. Dr. Arthur Benjamin does just this when he performs the calendar feat for audiences:



Recently, another approach was developed that I find interesting. You hand out a book containing perpetual calendars for every year from 1582 to 2399, and as above, you ask for the year first. You then name the page on which they can find the appropriate calendar, and while they look it up, you have time to recall the information about the year. The particular presentation was developed by Hans-Christian Solka, and the prop is available here.

As described, the idea seems to simply be to locate the page as the end of the feat, but I believe it's more effective as a highlight, on the way to giving the correct day of the week. If you click on Preview on that page, you can see the layout of the pages and the table of contents for yourself.

Another angle is to skip a step altogether by eliminating the specific date. Imagine you have a calendar like this printed on the back of your business card. You ask for the month first, then the year, writing them both down on the line to the right of the 31st. After that, you immediately fill out the appropriate days of the week above the dates!

For example, if someone says January for the month, and 2007 for the year, you would fill out the calendar in this way. This approach is especially easy if you're familiar with John Horton Conway's Doomsday approach, and are familiar enough with it to get the doomsday for a given year.

When you're given the month, you simply have to recall the doomsday for that month. In our January 2007 example, as soon as someone says January, you write it down while recalling that the doomsday for January is the 3rd, or the 4th if they name a leap year. Then, when they say 2007, you know that the doomsday for that year is Wednesday (from your calculations or mnemonics), so January 3rd (since 2007 isn't a leap year) is a Wednesday, you simply write “Wednesday” over the 3rd, and write the other days accordingly! This version happens so quickly, that you may want to repeat it, which will also help prove that you're not just guessing at days when checked later.

As a matter of fact, writing down the given date is another tactic often used to make the obtaining of the date appear faster than it is. You can begin your calculations during the writing, especially if someone else is doing the writing, and the audience will often psychologically eliminate the writing time when recalling the feat later. I can't give away too much, but in Paul Brook's The Chrysalis of a Polymath, he has some excellent presentational touches that involve writing it down, and performing it in a way that helps encourage the spectator recall you and your performance long after it is over.

Practice to improve your speed is always essential, of course, but so is staying on top of new developments in calculation. There's an excellent history of development of Doomsday-related techniques here, including the Odd+11 technique I recently discussed. You can find more about techniques at sites like Accelerated Doomsday, An Easier Doomsday Algorithm, First Sunday Doomsday Algorithm and Wikipedia's Doomsday Algorithm page.

November's snippets are here, and we're going to try and jog your memory with them!

• There's a new memory technique teaching site on the web called Memory Dojo. Joining requires a paid membership, but there are several free goodies available on the site, as well. Most notably, there are the video lessons from the site available on their YouTube channel.

• Mental Case for Mac OS X has just upgraded to the long-promised 2.0 version! The new version does require OS X Lion. If you haven't upgraded your Mac to Lion yet, they still offer 1.9 and a serial number for download from the site. Either version will still work with both the iPad and iPhone/iPod Touch apps.

• Speaking of apps, there's another new one out there called Namerick. It's designed to help you remember people's names via two ingenious strategies. First, you enter a name and create a silly-sounding alliterative phrase for the name, such as "Fred floats felines" (there are both randomly-generated choices and the option to enter your own original creations) as a mnemonic. Using spaced repetition, you're reminded of your new acquaintances. through the iOS 5 Notification Center. at intervals designed to maximize your retention. This way, you can focus on the recalling the face and name for later recall.

• OK, I've covered paid products above, so how about some good old fashioned free memory fun? Each week, for the rest of 2011, mental_floss' blog will feature mnemonic posts, encouraging their users to create new ones or post their favorite classics. If you've been practicing your US Geography, mental_floss will also challenge you to name all the state capitals, name the landlocked states, and name the states that begin an end with the same letter.

With almost every memory technique, visualization is the key, but it's usually discusses as visualization in your head. We live in an age of powerful computer graphics, so why not take advantage of this to bring your visualizations out of your head and onto the screen?

There's an amazing array of software and websites out there to help you do this, most of which hasn't even been considered for use in memory training! Let's take a look at a creative new use for some programs with which you may already be familiar.

The one advantage of software specifically designed for memory training is that it knows how to get out of your way. So, when using other software to help, there are two basic rules-of-thumb I've found helpful.

1. Keep everything as a simple as possible. That way, it's easier to keep your focus on memorization as your target.
2. Use computer- or device-based software, as opposed to cloud-based apps. Software residing on your computer generally has a better idea of your computer's capabilities than a similar cloud-based program, so it can offer more options. This isn't a hard and fast rule, however.

This Friday is 11-11-11. It is, of course, Rememberance Day. Many others, are simply taking advantage of the once-in-a-lifetime date as a reason to celebrate, including countless couples who are getting married on that date.

As I did for 9/9/9 here and here, and for 10/10/10 here, I'll be marking the day by looking at the qualities of the number 11, and what kind of fun we can have with it.

Many people's first introduction to the idea that math can be done in your head comes when they learn about multiplying single-digit numbers by 11, as seen in this classic Schoolhouse Rock video:



When you're first learning multiples of 11, you do quickly notice that the nice neat double-the-digit pattern quickly ends after 9. The ease of learning to multiply by 11 in your head doesn't have to end there, however! There's another pattern that, once learned, makes it easy to multiply numbers of ANY size! The best place to learn this technique is via Math World's Math Tricks: Multiply by 11 post, as it includes not only the teaching video below, but some practice exercises, as well.



Demonstrating this ability can be very impressive to an audience, especially since many of them will remember struggling with multiples of 11 beyond 9. One of the best presentations I've seen for demonstrating your ability to quickly multiply by 11 is "The Mental Block" from Paul Brook's book, Chrysalis of a Polymath. With that presentation, the audience member remembers that they brought up 11 (no force, but some very effective psychology), and they time you as you quickly multiply six different two-digit numbers by 11. The whole performance is presented while discussing how people's aversion to math is often a bigger barrier than lack of ability.

Surprisingly, there's also a simple trick for dividing any two-digit number by 11, and giving the answer to three decimal places. It's even simpler, but it's less well-known, and more impressive, due to the seeming mental calculation of multiple decimals.

1/11 is a repeating decimal equal to .090909... and so on. This repetition of 9 makes determining the decimal equivalent of any “11th” easy:

1/11=.090909...
2/11=.181818...
3/11=.272727...
4/11=.363636...
5/11=.454545...
6/11=.545454...
7/11=.636363...
8/11=.727272...
9/11=.818181...
10/11=.909090...

If you know your multiples of 9 up to 10, this is a simple matter. If not, you can get some quick help from my Magic of 9 post.

Knowing this, dividing any two-digit number by 11 in your head becomes a simple matter. Let's divide 63 by 11. First, what's the largest multiple of 11 equal to or lower than the chosen number? In our example, this would be 55. This tells you that the answer will by 5 and some decimal. Next, subtract that multiple of 11 from the given number. We subtract 63 - 55 = 8, so you could give the answer as 5 8/11, but there's a more impressive way to give this answer.

Knowing the pattern of 11th above relies on 9, ask yourself what is 8 * 9? It's 72, so you know the decimal for 8/11 equals .727272..., so you could give the answer for 63/11 as 5.727272... giving the endless decimal pattern. Even better, I suggest limiting the answer to three decimal places, giving the answer in this case as 5.727. Sure, the person verifying your calculation will still see the .727272... pattern, but it will be less apparent and more impressive to the rest of your audience!

If you'd like to be able to divide any two-digit number by any number from 1-11, check out my Mental Division With Decimal Precision post. Alternatively, Blake over at Nerd Paradise turns this feat around, and allows you to give a fraction for any given two- or three-digit repeating decimal.

Going back to the multiplication of 11, there's an even better way to demonstrate your ability to multiply by 11 while hiding that you have to do this with 11. As taught in episode 170 of Scam School, your multiplication ability is hidden behind an addition problem:



The larger the numbers you can multiply by 11, the greater the freedom you'll be able to give to people in choosing their initial numbers. If you can multiply numbers up to and including 1,287 by 11, you can let the other person choose any two numbers from 1-99 as the starting numbers. You can learn more about the working behind this trick in my Scam School Mathemagic post.

We'll wrap this up with James Grime's 11-11-11 video, the first full official video done for the new NumberPhile YouTube channel, talking about all things 11, including an amazing fact about books:



Sadly, I'll only get once more chance to celebrate a day like this in my lifetime, which is next year on 12/12/12.

Persi Diaconis, the Stanford professor who discovered that 7 riffle shuffles are the minimum required to properly mix the deck, and Ron Graham, the UCSD professor who popularized the Erdős number concept (The mathematical world's equivalent of a Kevin Bacon number), have just released a much-talked about book titled Magical Mathematics.

Is it any good? No less than the late, great founder of the field of recreational mathematics himself, Martin Gardner himself wrote the foreword, so it is definitely worth a closer look.

The first thing you notice about this book, especially if you've read previous books on math-based magic effects, is that it doesn't simply stop at explaining the effect and simply stating the mathematical principle. Magical Mathematics' strength is the way in which is delves into the principle to help you develop a better understanding, which in turn can help you spur your own creativity.

Each chapter in the first half of the book is dedicated to one particular type of principle, such as cycles, shuffling, or codes. In the later chapters, the book tends toward discussions of ideas related to those in the earlier chapters. In this latter part of the book, you learn about things like magic routines involving the I Ching, prominent people in the history of mathematical magic, and even the math behind juggling!

Grey Matters readers will recognize many of the names and principles included in this book, such as:

• Martin Gardner (of course)
• The Gilbreath Principle
• Stewart James and his effect Miraskill
• Bob Hummer and his 3-Card Monte
• Robert Neale and his Rock Paper Scissors routine

If you find you like de Bruijn Sequences, you may want to check out Leo Boudreau's work, which you can find online, as well as in his books.

Despite the cover, and even the impression you might get for quickly perusing the book, the routines aren't just limited to card tricks. There are effect here with pencil and paper, origami, chains, and many other items. Even some of the card tricks, once you understand the principle behind them, can be adapted to other objects.

The best thing about Magical Mathematics is that you can take your understanding of the trick as far as you want. Do you want to understand just enough of the effect to perform it without understanding the math? The authors describe each effect and method before explaining the mathematical basis.

Perhaps you prefer to understand the principle better so that you can create your own variations. You can take yourself through the full explanation, or even only part way if you prefer, and find inspiration either way.

If you enjoy my posts on recreational math and magic, I have no doubt you will enjoy and find great value in Magical Mathematics.

Magical Mathematics retails for $29.95, including the PDF and EPUB versions, but you can obtain the hardcover version and Kindle edition for a lower price.

To help give you a better idea of the value of this book, I've posted a Google Books preview of it below. Some of the pictures in the first chapter are removed for copyright reasons, but the publisher, Princeton University Press, has generously made the entire first chapter, including images, available for free (PDF).

Quite a few unusual math videos have come down the pike recently, so I thought I'd round them up here and share them with you.

Most of these include material I've rarely, if ever, come across before.

Over in the Mental Gym, I teach how to square 2-digit numbers. If you've mastered that, you may have wondered about going beyond 100. In the following video, you can learn how to handle squaring the numbers 100 through 199 quite easily:



Part 2, available here, teaches how to square the numbers 200-999. There's also a part 3 that teaches additional techniques, including how to handle some 4 digit numbers!

There are also a few old friends of Grey Matters have recenly-posted videos with math tricks.

This week's Scam School features a feat where you can determine how many cards are cut from a deck of cards. Where's the math? It plays a hidden role, as Brian Brushwood will explain:



The other old friend is Arthur Benjamin, whom you may remember from his popular TED video, or from my recent mention of his free video lecture on memorizing numbers. The newest post is his full hour-long Secrets of Mental Math lecture (More than 4 times longer than the TED video), based on his book of the same name: