Magic (Squares) with Mirrors

Published on Sunday, February 26, 2012 in , , ,

Chris Wardle's Dual Magic SquareToday, we're going to work a little magic with mirrors!

Instead of using mirrors secretly, however, we're going to use them openly. And instead of stage magic, we're going to play with magic squares.

Numberphile recently posted a great video involving mirrored magic squares. Don't worry if you're not familiar with magic squares, since the video starts with a basic introduction to them:

Inspired by this video, a poster with the screenname rotflmaopmpqxyz developed the following magic square with the same qualities, but using 3 digits and totaling 1,776 in every direction:

UK magician Chris Wardle developed a variation of the magic square in the video, which was posted last year over at Mark Farrar's site. If you mouse over the images, they'll automatically display their rotated versions.

Chris Wardle also created two other interesting magic squares here. Instead of using mirrors, however, these two magic squares rotate. The top one keeps the same total when rotated, and the bottom one changes.

To my mind, however, one of the most impressive mirrored magic square is Werner Miller's Square Bet magic square. This is a 2 by 4 grid that doesn't become a magic square until you hold a mirror up to it. When you do, you get a 4 by 4 square totaling 234 in every direction, even diagonally!

Considering that the creators not only had to take into account the totals, as with all magic squares, but the appearance of the numbers themselves, and how the new number would change in relation to each other, I find all of these to be very impressive!


de Bruijn's Magic

Published on Thursday, February 23, 2012 in , , , , , , , ,

Mathematisches Forschungsinstitut Oberwolfach photograph of Nicolaas Govert de BruijnNicolaas Govert de Bruijn, a noted mathematician, passed away on February 17, 2012, at the age of 93.

He made contributions to many branches of mathematics, but those interested in recreational mathematics will be especially interested in discovery of what is known today as the de Bruijn sequences. What exactly are they and more importantly, how are they used?

If I asked you to list all the possible pairs in which you could combine a 1 and 0, that's simple to do. The possible pairs are 00, 01, 10, 11. If we combine them into one long sequence of numbers, it would look like this: 00011011.

Now, what if I asked you to give me the shortest possible sequence of 0s and 1s that contains all their possible pairs? This is a little trickier, but it turns out that the answer is simply 0011. A quick look and you can find 00, 01, 11 and...wait, what about 10? If you repeat the sequence (00110011...and so on), you'll note that the 10 is found by combining the final 1 of the sequence with the initial 0.

You can't make a shorter cyclic sequence of numbers than 0011 that contains every possible pair of 0s and 1s, so this is a de Bruijn sequence.

In a more general sense, a de Bruijn sequence is the shortest possible cyclic sequence in which given k symbols, you can find all possible permutations of length n. The less technical definition of a de Bruijn sequence is simply “every possible arrangement squeezed into the smallest possible space.”

In our simple example above, we dealt with an “alphabet” of 2 symbols, so k=2, and we were looking for all possible pairs, so n=2. In mathematical notation, this is often simple written as B(k,n), so our example would be B(2,2).

Such sequences are known to have been used as long ago as 200 BCE, but de Bruijn's breakthrough, with help from Tatyana Pavlovna Ehrenfest, was in finding a general approach for determining the sequence for any B(k,n).

If you want to see how challenging this can get as k and n get larger, consider James Grime's combination lock puzzle:

Here, you're using an alphabet of 10 symbols, and looking for every possible sequence of 4 numbers, or B(10,4). Try working out the simpler challenges in the above video, and then watch James Grime's solution video, along with the method used to solve them:

Applying the formula mentioned in the video above, you can quickly determine that B(10,4) is going to be 104, or 10,000 digits long. Fortunately, you only need to find 1 of the many possible de Bruijn sequences.

Besides picking locks, mathematicians have taken advantage of de Bruijn sequences for use in card tricks. Colm Mulcahy explains a basic use in his December 2008 Card Colm column.

Since the study of these sequences is applicable to so many fields, it's amazing where you can find a use for them. Playing around with card tricks based on the de Bruijn sequence led one researcher to discover an improved way to make files smaller.

In the recently-released book Magical Mathematics, authors Persi Diaconis and Ron Graham delve even deeper into de Bruijn sequences and card tricks. You can get a preview of part of that chapter by clicking on the Google Preview button below:

In that book, there are some ingenious uses for de Bruijn sequences with cards, but there's one magician whose work has taken them to an entirely new level. If you're a member of the Magic Café, explore Leo Boudreau's online work, as well as his written work, with de Bruijn sequences. Leo Boudreau work largely involves binary numbers, so if you like his work, learning a good binary number memory system can turn his amazing routines into miracles.

Even with this lengthy post, we've barely delved into only one part of N. G. de Bruijn's body of work. The world was better for having him in it, and he will be truly missed.


Remember All the US Presidents

Published on Sunday, February 19, 2012 in , , , , , , ,

Scott Blake's mosaic of George WashingtonIn the US, tomorrow (Feb. 20th, 2012) is President's Day. Why not use that as an excuse to challenge yourself to remember all of the US presidents?

There seem to be as many ways to do this as there are people who want to do this. In this post, I'll show you a few of my favorites.

A good place to start is right over in the Mental Gym, where I already teach how to memorize the presidents. It features the classic Animaniacs' President Song, “Uncle Dave's” story for memorizing the presidents, Think-a-Link's mnemonics for linking presidents to their numbers, and more. In the More... tab, there are further links for remembering the presidents, including one that teaches you how to memorize the presidents X-Men style!

Another method that's very fun is the approach used in the book Yo, Millard Fillmore!. The simple yet wacky story helps you remember the presidents in order, and even adds mnemonics for every 5th president, so you can keep track of their numbers. Here's how the first 10 presidents are taught:

Besides the standard paperback book, Yo, Millard Fillmore! is also available as an eBook for the iPad, which adds narration, interactive reviews and quizzes.

If you prefer a more physical hands-on approach, you can actually make your own presidents game. Over at BoardGameGeek, you can download files to convert the standard Guess Who? board game into a game of presidents. When you're done, the results should look like this picture, and you simply play with the same rules.

Think you know them all? Try testing yourself with Sporcle's US President quiz!

Did you ever learn to memorize the presidents? If so, what method did you use? I'd love to hear about it in the comments!


Yet Again Still More Quick Snippets

Published on Thursday, February 16, 2012 in , , , , , , , ,

Luc Viator's plasma lamp pictureFebruary's snippets are here, and it's time to have a bit of fun with math!

• Let's start off with the founder of recreational mathematics, Martin Gardner. The January 2012 issue of The College Mathematics Journal is available for free online, and is dedicated to the work and memory of Martin Gardner! Being a math journal, it does get into some heavy math, but even if you don't care for that, there's still plenty of fun math-related experiments and puzzles you can try and enjoy. You can also access each article individually, if you prefer.

• There's a magic tumblr blog called 366 effects. The magic tricks posted there are largely classic effects, many of them math-based. You can even find a nod to Martin Gardner there! The author is very good about giving proper credit, as well.

• Over at Mind Your Decisions, there have been several interesting posts recently. The first one dealt with a puzzle about page numbers: A book has N pages, numbered the usual way, from 1 to N. The total number of digits in the page numbers is 2,808. How many pages does the book have? Back in 2000, this same puzzle was featured on Numericana, too. It's amazing how challenging such a simple problem quickly becomes.

• What's a better value for your money, a 12-inch diameter pizza, or two 8-inch diameter pizzas? One blogger was faced with that decision, did some mental math, and opted for the choice with more surface area, even winding up with a bonus! The Presh at the Mind Your Decisions blog had a similar experience, used a calculator and was beaten by someone using some quick mental math!

The latter version's mental math is especially impressive, as there are several layered mental math tricks used. The first trick is the elimination of constants. Pi, of course, is constant, and we can also assume the thickness is, as well, but those only matter when going for an exact answer, not a comparison. Next, notice that even though the formula for a circular area is Pi × radius2, the mental math genius squared the diameters of the pizza. Again, because we're making a comparison, this is merely a scaled-up version of comparing the same circle's radius. Being able to work out problems such 14/9 in your head was taught here on Grey Matters back in 2009, and figuring out 1.5 squared is just a minor variation of squaring numbers that end in 5. Sometimes, in mental math, it's not just knowing what to do, but knowing what you don't need to do, as well.

• While we're focused on pizza and the Mind Your Decisions blog, here's how to play Nim with an unevenly-divided pizza, and ensure you wind up with the most pizza! If you like this game, make sure to check out my Nim posts. If it's tasty versions of Nim you're after, you'll particularly enjoy Chocolate Nim.

• I'll close with an answer, instead of a question. A poster over at Quora wondered what it was like to have an understanding of very advanced mathematics. An anonymous user provided a wonderfully clear and sincere answer that is a must-read. This is one of those posts that make you want to stand up in front of your monitor and clap, even though you know the author will never hear you.


Remembering Text with iOS

Published on Sunday, February 12, 2012 in , , , , , , , ,

Verbatim 2 screenshotMemorizing a piece such as a speech, poem, or music lyrics is often considered one of the biggest memory challenges, as they need to memorized word-for-word.

With most other memory techniques, you can get away with remembering the general concept. For example, when memorizing playing cards mnemonics, you simply have to remember the mnemonic itself, not an entire sentence structure.

However, for a speeches or poems, the exact wording is imperative. How exactly do you go about this? In this post, we'll look at some iOS apps that help you do just that.

The reason I'm looking at iOS apps in particular is simply because the mobile devices I regularly use use iOS. For those who do use Android or other mobile operating systems, I'll start with two text memorization apps that can be accessed with any mobile device on the web. These first two are also both free to use.

The first is Verbatim 2, my own original entry. I originally released it in 2009, and upgraded it last year. Here's the basic idea behind Verbatim:

The other online tool for remembering text is Memorize Now. The advantage of this tool is that you can keep multiple texts to memorize handy. Here's how Memorize Now works:

Before I continue with more apps, I suggest you watch one or both of the above videos, as they introduce concepts used in many memorization apps. Verbatim's approach of remembering lines in larger and larger groups is known as spiral learning. Note that both apps quiz you by removing information, and challenging you to fill it in. You can find more about spiral learning at the Memory Tools page.

The rest of the apps in this list are all iOS-specific native apps available from the App Store for various prices. They'll use the techniques I just described, and more.

Our first native app is Memorize Anything. With this app, you start by reading the piece out loud, and record it on your iOS device. It's recommended that longer pieces be broken up into separate sound files, each about 3 minutes or so. You then learn the piece simply by listening to it over and over.

When you're ready to test yourself on your chosen text, you have Memorize Anything play it back again, but this time with sound fading in and out at random spots. When you can fill in the muted portions over several plays, you know you've got the piece memorized!

Another app that uses sound, but in a different way, is Learn Anything, formerly released as Loop&Learn. In this one, you also record sound, but in smaller chunks, and you can optionally include graphics, which can be especially helpful if you design them as mnemonics. You then use a spiral learning approach, similar to Verbatim, but now with audio and video to help. Watch the video tutorial to get a better idea of how this app works:

Besides the paid version of the app, there's also a limited free trial version available.

The last three apps I'll discuss all work in a very similar manner. They all work with just the text, and allow you to progressively eliminate more and more of the text, so as to increasingly challenge your recall.

memoRISE is a free app, and will test you with just the first letters of words, or a fill-in-the-blank approach. You can think of this a simplified version of Verbatim 2 above, done as a native app.

Instead of memoRISE's either-or approach, there's Line Memory. This app allows you to use a slider to determine what percentage of the words are blanked out.

Our final app is Memorize Now. Like the Line Memory app, you can progressivly hide more and more words, but it uses a different approach. Instead of a slider, there are distinct buttons, which take you to different levels of missing text. In the earlier levels, the first letters remain, as other letters are removed. As you get to the higher levels, the first letters are replaced with black boxes.

Memorize Now, like the rest of these apps, can be used for memorizing texts like speeches, poems, and lyrics. If you have a particular use in mind, there are also dedicated versions of Memorize Now available that come pre-stocked with appropriate texts. These custom versions include Scripture Mastery Now!, Memorize Bible Verses, and Memorize Famous Poems.

Have you used any of these apps to memorize text? Perhaps you've used another app, iOS, Android, or otherwise, that I neglected to mention here. If so, I'd love to hear about the apps, and your experiences with them, in the comments!


Day One Updated!

Published on Thursday, February 09, 2012 in , , , , , ,

Day One logo4 weeks ago, I announced the release of Day One, my updated and simplified handling of the classic day of the week for any date feat.

Today, I'm announcing an update to Day One, with a new ebook and a new app!

The updates mostly concern additional feats to perform after you've created the calendar. Using the information you worked out while before creating the calendar, these extra feats let you present an incredible range of knowledge.

The first feat is called “Turning 65.” If the calendar you created was an audience member's birthday, this section shows you how to quickly determine the day of the week they'll turn 65. There's much less calculation involved that you might think. As Michael Daniels has pointed out in his review of the update, even if you use another way of determining the day for any date, you can still use this feat!

The second added feat shows you how to work out which months in any given year have a Friday the 13th. Generally, you perform this with the year for which you've just created the calendar. It's especially interesting bit to present if someone was born on a Friday the 13th, but I teach how to present it under all circumstances.

The next feat is tougher than the previous ones, but the astounding results are worth the extra effort. After creating a calendar for someone, you ask for any day of the week, and are able to give all the months of the given year in which that day occurs five times. For example, if you just created a calendar for a month in 1980, and someone asked for Thursdays, you could tell them that the only months with five Thursdays are January, May, July, and October, as seen here. As many people are familiar with the infamous e-mail about 5 Friday, 5 Saturdays, and 5 Sundays happening only every 823 years, this can be a topical feat.

The final feat in the new ebook is different than the others. This feat shows you how to present the Day One feat without creating a calendar card. If you don't have the calendar cards with you, or even if you just want a simpler, less formal presentation, this is a good presentation to know. All the memory work is presented as if you're recalling stories from your past, or stories told to you by loved ones, if the given date was long enough ago.

A new set of apps is included to help train you to perform all these additional feats.

Best of all, the new ebook and app come at no additional charge as part of the Day One package! If you already purchased Day One from Lybrary.com, then the updates are available from there by simple download.

If you haven't bought it yet, you can buy Day One at Lybrary.com for only $9.99!


Around The World in 7.2°

Published on Sunday, February 05, 2012 in , , ,

Erzbischof's diagram of Eratosthenes' experimentBack in the days of Ancient Greece, a mathematician named Eratosthenes managed to determine the approximate circumference of the Earth.

How did he do this? Even better, how could you go about repeating the experiment to teach the idea in modern times?

Let's start with the original experiment. Here's the story of Eratosthenes' experiment, as told by Charles and Ray Eames, whom you may know best from their education film, Powers of 10:

The actual angle measurement Eratosthenes got from the obelisk's shadow was 7°12', or 7.2°. Dividing 360° by 7.2° gives 50, meaning that 7.2° is 1/50th of the full circumference. That's why he multiplied the distance between the two towns by 50.

It's important to note that Eratosthenes was working longitudinally (running north/south). If the Earth were a perfect sphere (it isn't), the circumference would be the same all around. Since this experiment ran longitudinally, you get what is known as the polar circumference, as opposed to the equatorial circumference.

So, how close is the 25,000-mile measurement to the Earth's actual polar circumference? With modern equipment such as satellites, we've been able to determine that the Earth's average polar circumference is 24,818 miles - an accuracy of more than 99.27%!

How could you teach this with modern tools? There's a simple approach taught in this PDF that uses Google Earth. While you can't put a stick in the ground of Google Earth to measure a shadow, Google Earth already gives degrees of longitude. Thanks to that and Google Earth's ruler function for determining distance, this makes the virtual version of this experiment much easier to perform.

Of course, if the real Earth was good enough for Eratosthenes, why not use it today? In Eratosthenes' time, he had to wait an entire year before making the measurement back in Alexandria, so he could know that the Sun and the Earth were in the same relative position.

Modern communication makes it simple to arrange for the same experiments to be done at the same time on the same day in two different places. Rhett Allain of Southeastern Louisiana University and Chad Orzel of Union College in Schenectady, NY got together and did just this, as described in their respective links. They made time lapse videos, coordinating the times of their measurements, and posted their resulting time-lapse videos on YouTube:

Technically, this doesn't really replicate the original conditions of the experiment, as the Schenectady, NY is about a 1,500-mile drive northeast of Hammond, LA.

It might be fun to recreate the experiment more closely, perhaps with live video going between two schools. For example, if Rhett Allain had used a Google Maps Distance Calculator to find out that Bismarck, Missouri is almost exactly 500 miles north of him, he might have arranged to work the experiment with someone there.


Hacking Memory Techniques

Published on Thursday, February 02, 2012 in , , ,

Taro Taylor's Scrabble tiles photoIt's one thing to learn an established memory technique. There are times when you want to remember something, but ready-made systems either can't handle the information itself, or bring up the information in a way you need.

The solution is to develop your own memory system. Not only is it possible, but if you do so, you'll have the satisfaction of not only remembering what you need, but also of knowing that you created the way to handle it.

What if you wanted to memorize all the 2-letter words allowed in Scrabble? That's the example we'll look at in this post.

A post titled “Hacking Scrabble,” provides a wonderful look at the approach one blogger used to meet this challenge.

Although it doesn't specifically say so in this article, I imagine that the person who wrote this article looked at the existing mnemonic systems used by Scrabble players. The most-used memory system here is called anamonics (a portmanteau of anagram and mnemonics), and is a way of recalling all the individual letters that could be added to a set group of letters to form legal Scrabble words.

The problem with using this approach is that anamonic lists generally start with three-letter words, not two.

The first step in working on the system was getting as much of the information needed together as possible. In this case, there's a readily-definable set of all the needed words:

aa ab ad ae ag ah ai al am an ar as at aw ax ay ba be bi bo
by ch da de di do ea ed ee ef eh el em en er es et ex fa fe
fy gi go gu ha he hi hm ho id if in io is it ja jo ka ki ko
ky la li lo ma me mi mm mo mu my na ne no nu ny ob od oe of
oh oi om on oo op or os ou ow ox oy pa pe pi po qi re sh si
so st ta te ti to ug uh um un up ur us ut we wo xi xu ya ye
yo yu za zo

Presented this way, the 124 words seem overwhelming. The next step, then, is to look at exactly how you're going to need the information.

Since it's Scrabble, you're probably going to be looking at a given letter on your tile board, and wondering with what other letters you can use. This suggests that the words be organized by their first letter, followed by a list of possible letters with which that letter could be used. The advantage of this is that now you can start with one of 26 known letters and work from there. Here's the list the original poster developed:
a: abdeghilmnrstwxy   j: ao                 s: hiot
b: aeioy              k: aioy               t: aeio
c: h                  l: aio                u: ghmnprst
d: aeio               m: aeimouy            w: eo
e: adefhlmnrstx       n: aeouy              x: iu
f: aey                o: bdefhimnoprsuwxy   y: aeou
g: iou                p: aeio               z: ao
h: aeimo              q: i                 
i: dfnost             r: e
Now the information is arranged in a more useable way. As so often happens at this point, you realize that there's no simple pattern that will take you from the information you have to the information you need.

So, what we need now is a way to make the information more meaningful. The original post goes into great detail about how this was handled. Basically, the poster realized that words could be made from the letters, and decided to find a list of suitable words. Please read the original post to see exactly how this was done.

The particular mnemonic phrases developed and used by the author are listed in this file. If you scroll down to the chosen mnemonics, you'll note that a new problem developed. Some letters were only associated with consonants, and others were only associated with vowels. As you can see from the above list, some are also associated with just one other letter.

Does this mean that the approach won't work? No. Almost half the alphabet was handled by the anagram approach, which is a good start. From here, you might add on simple additional rules or systems to handle the exceptions.

For example, how do you deal with the lack of consonants that go with the letter u? The only letters that go with it are: shtgnmrpt. What if we turned the disadvantage into an advantage? We could remember that u doesn't go with any other vowels by remembering “u is unique.”

We could then add vowels to make words out of the letters above, as long as we later recall that the vowels are only placeholders, and not to be used with the letter u. The letters shtgnmrpt then become the more memorable phrase: “more shotput gun” (or a similar phrase you prefer). The other letters could be handled in similar ways.

Once you've developed meaningful way to handle all the information, don't forget to put the information together in your mind with the link system! If you don't make the time and effort to remember the information you've organized, then there was no need to organize it in the first place!

Since you know you're always going to start with a single letter in this case, how do you give each letter a memorable image? This article features a great approach to using letters as memory pegs about halfway down, under “Alphabet peg mnemonic system.• You simply remember a as hay, b as bee, and so on.

This works especially well in some cases, as linking a to the letters abdeghilmnrstwxy becomes a matter of linking hay to “my exhaling bedstraw.” Picture a pile of hay exhaling straw perfectly ready to use as bedstraw, and you've got the image locked in!

The short version of creating a custom memory technique for a specific situation is to get as much of the needed information together as possible, consider how you'll need to actually use the information, organize it accordingly with a focus on grouping as much information together as possible, and find a way to make the needed information meaningful. Since there's often no pattern to the information we need, creating wild and bizarre imagery is often the best way to do this.

Have you ever created your own original memory system for a given task? I'd love to hear your story in the comments!