February'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.

Memorizing 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!

4 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!

Back 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:

It'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

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

It wasn't that long ago when remembering passwords was a problem most people only ran across in spy movies.

These days, with various assorted internet accounts everywhere, remembering your password, as well as making it difficult to figure out, is becoming more and more of a challenge. Fortunately, there are many memory techniques that can help.

The first rule of internet security is that you can never reach a level where you are absolutely secure. All you can ever do is decrease your risk of a breach.

Last August, XKCD put the problem into an amusing and accurate perspective:



Yes, you've probably often heard that using regular words all in lower case is a bad idea. However, that advice generally refers to using a single regular word. A longer password comprised of multiple words isn't found in any dictionary, and the length alone make it harder to achieve through sheer guessing.

The "bits of entropy" referred to in the above cartoon can be thought of as a way to score the difficulty of uncovering a password. The following Wolfram|Alpha widget accepts a given length of password, and will then generate a password of that length, as well as how long passwords of that type would take to crack:



The XKCD password above, "correcthorsebatterystaple", is 25 characters long. Try putting in 25 and see how long Wolfram|Alpha thinks that would take to crack!

If everyone used that exact phrase, however, it would become well known, and thus easier to discover. Fortunately, the comic inspired this password generator, so you can get your own unique phrase.

While passwords of this type are a good idea, they're unfortunately not always possible to use. Strangely, there are many places that limit your character range and password length. Obviously, maximizing the mixture of digits and upper- and lower-case letters, while staying away from words found in the dictionary.

The trick with this approach becomes memorizing the password. An iOS app called PasswordGear offers an ingenious mnemonic solution, in which each letter and number is transformed into a memorable image as described in the video below. Even if you don't have an iOS device, you can still apply the approach on your own.



For more reading on improving your password security and remembering them, check out Lifehacker's password articles.

25 years ago today, Square One TV debuted on PBS! As a budding math geek, this show was a must-watch for me.

If you're not familiar with Square One TV, it was a show teaching math using comedy skits, music videos, guest stars, and whatever else to teach mathematical concepts. Everything from basic arithmetic to geometry to somewhat-advanaced algebra was covered in a way that was fun and interesting.

My favorite music video they ever did is a great example of this. It was called “Change Your Point of View.” Although largely about solving math problems by looking at the problem from different perspectives, it's also great advice for any type of problem:



To give you an idea of the comedy skits they used, here's a skit called The Adventures of Spade Parade, in which they have to figure out which consultant is which:



Magician Harry Blackstone, Jr. even had his own recurring segment, in which he would perform and teach mathematically-based magic:



Like many PBS shows, Square One TV was 30 minutes long (no commercials meant 30 minutes of content), and broadcast 5 episodes a week. The show itself had a rather unusual format, however. The first 20 minutes would consist of skits, songs, and other segments like the ones above.

The last 10 minutes of the show would always be an episode of Mathnet, a sort of Dragnet parody following the adventures of detectives Kate Monday (later replaced by Pat Tuesday) and George Frankly. A new adventure would start on Monday, and would be continued on each day, winding up on the following Friday.

To get a better idea, you can actually find full episodes online. Here's the very first episode of Square One TV. The very first skit, a song about the concept of infinity, recurs throughout the episode, as if it continued forever. The show's producers even convinced PBS to continue the gag even after that first show was over.

The Mathnet episode, “The Case of the Missing Monkey,” guest stars a young Yeardley Smith, better known today as the voice of Lisa Simpson. This adventure continues in the second episode, and the third episode. I can't find the fourth and fifth episodes online yet, but you can see the rest of the same case in the 39th episode and the 40th episode, when it was re-run.

The show had a good long run, and broadcast its last new show on May 6, 1994, seven years later. Happy 25th birthday, Square One TV!

I've talked about the modular arithmetic before, especially as it related to the day for any date feat.

In this post, we're going to take it out of the calendar feat's shadow, and give it a starring role in its own feat!

If you remember doing division before you learned about fractions, you remember doing problems such as 21 ÷ 4 = 5 remainder 1. Modular arithmetic is simply focusing on the remainder exclusively. 21 modulo 4, for example, just equals 1, because when you divide 21 by 4, 1 is the remainder.

If we're talking at 10 AM, and I agree to call you in 5 hours, then you know to expect a call from at 3 PM. You did 10 + 5 = 15, but you know that hours aren't numbered any higher than 12, so you just subtracted 15 - 12 to get 3. This is modular arithmetic, and is also why it has the nickname “clock arithmetic.”

Let's try comparing the modular arithmetic patterns of two numbers, say, 2 and 4. Since 2 times 4 = 8, we'll compare the remainders as they run from 0 to 8:



What if I were to tell you that I was thinking of a number from 0 to 8. I then gave you a further clue that, when divided by 2, it has a remainder of 1, and when divided by 4, it has a remainder of 3, we run into a problem. Look at the chart. That description fits both 3 and 7, and there's no way to work out which of the two. The problem is that the pattern of remainders, when divided by 2 and 4, repeats 2 times from 0 to 8.

If we try this with, say, 3 and 6, and ran up to 18 (3 × 6) you can see in this chart that there are 3 times where a pattern of 3 remainders repeats.

If we want to identify a number by its remainders alone, is there some way to make sure that no repeating pattern emerges? Notice that, when we used 2 and 4 (and went up to 2 × 4, the remainder patterns repeated 2 times, and that 2 is the largest common factor of 2 and 4. Similarly, when we used 3 and 6 (and went up to 3 × 6), the remainder patterns repeated 3 times, and that 3 is the largest common factor of both 3 and 6.

If we want no repeating patterns, then what we're really saying is that, when performing modulo a and b, running from 0 up to a × b, we would like each number combination to only show up 1 time. For this to be true, we simply have to make sure that the greatest common factor of the numbers involved is 1!

This is the basic idea of the Chinese Remainder Theorem. Martin Gardner discusses this idea in more detail in his book Aha!: Aha! Insight and Aha! Gotcha (Spectrum). You can find the relevant pages online here and here, thanks to Google Books.

When using two numbers, it's pretty easy to make sure their only common factor is 1. If we use, say, 4 and 5, and go up to 20, we can already know that there won't be any repetitions, because the largest factor common to 4 (factors: 1, 2, 4) and 5 (factors: 1, 5) is 1.

The Chinese Remainder Theorem also tells us we can go further, and even use 3 or more numbers, and they won't repeat (up to a × b × c ×...) as long as their largest common factor is 1! The easiest way to do this, of course, is to turn to our old friend, prime numbers.

In the Martin Gardner book linked about, he talks about a version of a trick where someone thinks of a number from 1 to 1,000, and gives you the remainders after dividing by 7, 11, and 13. Since 7 × 11 × 13 = 1,001, you'll get a unique combination of remainders for any number given. But what about the version he mentions from 1 to 100 with 3, 5, and 7? What's the formula for that?

Let's take the approach in his article and apply it. For the remainder after dividing by 3, we need a multiple of 5 × 7 that's 1 greater than a multiple of 3. 35 doesn't work, because 34 isn't a multiple of 3. 70, being 69 + 1, works perfectly, though. OK, we start with 70 × a (or 70a for short).

What about 5? Let's look at the multiples of 3 × 7. There's 21...perfect! It's already 1 more than a multiple of 5. OK, now we've got 70a + 21b. What about 7? 3 × 5 = 15, and 15 is already 1 more than a multiple of 7. For all three numbers, we now have 70a + 21b + 15c. Divide that total by 105 (3 × 5 × 7), and the remainder will be the number you're looking for!

You could do that on a calculator, but if you're familiar at all with Grey Matters, you'll know that I encourage you to do things like that in your head. However, I understand that it can be tricky.

A magician named Tom Harris, back in 1958, proposed a different approach that required no calculation. You memorize the number combinations with help from the Peg/Major system, linking the combined numbers you get to the unique answer for that combination. For example, if someone gives you the numbers 1, 0, and 3, you would recall the phonetic equivalent “twosome”, and remember that you linked that to the word “toes,” which translates to 10.

This is a bit of work, but we have an advantage over someone trying to do this in 1958. Using a spreadsheet program makes an easy grid, and will handle listing the numbers from 1 to 100 for you, and will even handle working out the remainders for you. To find words for each combination, you can use some of the mnemonic generators listed here (My favorite for this would be pinfruit). Those familiar with the Peg System will already have 100 words ready for the answer numbers.

Naturally, I've developed a Wolfram|Alpha widget that can make things easier on your audience members:

I'll leave you with one last related challenge. James Grime needs help counting his juggling balls. Can you use what you've learned to help? When you're ready, here's the answer video.