Two years ago, about this time, I reviewed Presh Talwalkar's previous Infinite Tower book.

Since then, not only has Presh not only been hard at work on his Mind Your Decisions blog, but also another book guaranteed to interest Grey Matters readers! This newest book is titled The Best Mental Math Tricks. Presh was kind enough to send me an advance copy, so I'll share my review in this post.

Probably the first thing to stand out about this book, when reading the table of contents, is that it's organized almost exactly backwards to most arithmetic, and even most mental math books. It starts out with a variety of mental math shortcuts for specific situations, then moves on to squaring shortcuts, followed by multiplication shortcuts, then division shortcuts, and it closes with another variety of shortcuts.

There's nothing bad about this approach. As a matter of fact, since the subject is mental math, this actually allows the shortcuts to be described in a rough order of simpler to more complex. It's also a nice change from the standard order of adding/subtracting to multiplication/division to roots/powers.

When you learn this book is put out by the author of a blog, you might be concerned that this is just a collection of previous mental math blog posts that you could access online for free. While there is some overlap, there's plenty of material in the book that has never been posted on the author's blog. Conversely, there are also several mental math shortcuts on his blog which don't appear in the book, so Presh's book and site wind up complementing each other quite nicely.

Even when there is crossover, the entry isn't simply copied straight from the blog to the book. For example, Presh wrote a post titled Understanding the rule of 72: a popular rule that has little practical value that was highly critical of this standard shortcut. In the book, however, the rule of 72 is taught with a less critical review, while still giving the reader an understanding of when the rule is and isn't appropriate to use.

The structure of each shortcut is also well thought-out. Each one starts with a description of the shortcut itself, followed immediately by practice problems which help you internalize it. Just before providing the answers to the practice problems, however, Presh explains the proof behind each shortcut, so you can get a better understanding of why it works. This is probably one of the most useful and important aspects of the book. It's one thing to learn a rule, but another thing to understand the reasoning behind it.

If you're already familiar with mental math shortcuts, you're still likely to find enough new shortcuts to make this book worthwhile. If you're new to mental math, this book is a definite treat for the mind!

At this writing, The Best Mental Math Tricks isn't available yet, but Presh Talwalkar assures me that it will be released in the near future. When it is released, The Best Mental Math Tricks is now available at Amazon.com. I recommend to anyone interested in improving their mental math skills!

The first snippets of 2015 are ready!

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

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

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

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

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

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

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

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

One of my favorite mental challenges, as many regular Grey Matters readers know, is the Knight's Tour. The challenge is, using only the chess knight's L-shaped move, to land on each of the 64 squares once.

Mentalist Richard Paddon recently released a download resource titled The Knight's Tour: A Scenic Journey. In this post, I'll take a close look at this new take on a classic feat.

We'll start with a quick peek at Richard Paddon himself performing the Knight's Tour, via the teaser ad:



The Knight's Tour: A Scenic Journey comes as a set of 3 files: A 45-page PDF of the same title, a 16-minute MPEG file of Paddon's performance, and the Knight's Tour Windows application used by Paddon, and programmed by Dave Everett.

In the PDF, right away, the author emphasizes the importance of developing drama in the Knight's Tour presentation. The first parts of the actual instruction, however, focus on developing the path through the board. Much of this part of the book may be familiar to readers of the “Knight's Tour” section of Paul Brook's Chrysalis Of A Polymath. However, Richard Paddon does add some new and helpful notes, such as the section on what he has dubbed “delta values”, which are familiar to those who have programmed a Knight's Tour, but little discussed in the use of performances.

In the next half of the book, Paddon discusses the presentational details. He starts with the benefits of the Knight's Tour, including its uncommon nature, and its huge potential on an emotional and theatrical scale. The thoughts behind the presentations are well laid out. Even if you disagree with any aspect of the presentation as written, you at least have a good starting point of why particular choices were made.

One of the more interesting choices is ending on a selected square, as seen in the above video. As the board empties, the chosen square becomes a more and more important focus, and becomes a natural point of building tension. The PDF winds up with a detailed description of how to use the program.

There are very few weaknesses in this product overall. One of the one that stands out to me as both a programmer and a blogger of mental feats was the choice of the Comic Sans font for the numbering of the board. If you're taking as much care as this author does to make an impact on the audience, there's probably better ways to label your board than a font designed specifically to have a comic-book appearance. On an equally minor note, the lightning in many of the shots of the performance video could be better. The importance of the video is for a more complete understanding of the presentation, so this isn't a huge drawback.

Overall, this is an excellent value for anyone seriously interested in performing the Knight's Tour. The basics of working through the path may be easily accessed in multiple sources, but the depth of knowledge that is presented, as well as the use of multiple media to demonstrate this make this the most complete lessons about all aspects of the Knight's Tour and its proper performance.

It's available for only $9.95 over at Lybrary.com and is a remarkably great value for that money. If the Knight's Tour interests you, Richard Paddon's The Knight's Tour: A Scenic Journey is a must-read.

It's time for December's snippets.

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

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

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

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

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



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

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

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



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

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

Even if you're not into mathematical magic and mental math, you're probably familiar with Dr. Arthur Benjamin from one or more of his TED talks.

Another video of his mathemagical feats has surfaced on the web, but this one includes the methods of each routine!

This video lecture is titled The Magic and Math of Mental Calculation, and was held at the 2013 Martin Gardner Celebration of Mind in Washington DC, courtesy of the Mathematical Association of America and Math For America-DC.

The Magic and Math of Mental Calculation is done in full lecture style, and runs about 78 minutes. It is introduced by MAA's Ivars Peterson and Thinkfun (Amazon.com link) CEO Bill Ritchie:



Granted, the single unmoving camera angle could make things hard to follow, but I've gathered numerous links which I hope will make everything clearer.

First, Ivars Peterson mentions that April is Mathematics Awareness Month, with the 2014 theme being Mathematics, Magic and Mystery, after the Martin Gardner book of the same name (Amazon.com link).

Dr. Benjamin starts out by squaring 2-digit numbers in his head. This feat is relatively easy to learn, and the Mental Gym even features a 2-digit squaring tutorial and quiz. The later explanation features some excellent advice on working up to squaring 3- and 4-digit numbers.

This is followed by the missing digit feat, which is explained much later in the video, so I'll come back to it.

Next up is a magic square feat. The explanation can be tricky to follow. Fortunately, Dr. Benjamin has posted the instructions for his Double Birthday Magic Square online for free. There are several essential tips in the video that make the performance of this far better than if you'd just learned from the PDF alone.

When he talks about how he developed the magic square routine in the first place, he mentions a 2003 magic square article in a magic magazine. This seems to be Harry Lorayne's article, 4×4 Magic Square Breakthrough??. The original magazine article isn't easy to find, but the entire article was reprinted in Harry Lorayne's book, Mathematical Wizardry (Amazon.com link), which I reviewed here back in 2006.

The calendar feat, as many Grey Matters readers already know, is a favorite of mine. You can follow along Dr. Benjamin's somewhat brief explanation of the feat with the help of the Day of the Week For Any Date tutorial and quiz here. I have done my own work simplifying the calendar feat in my Day One ebook.

Impressively, Dr. Benjamin even fields a question about mentally determining whether a 3-, 4-, or 5-digit number is prime or not, despite not performing any feats related to this. If you're wondering why he's using this particular approach, my prime number testing post from earlier this year may make things clearer.

Coming back to the discussion of the missing digit feat, it's hard to make this much clearer than it is on the video. There is the amusing question of whether zero is an even number, which Numberphile tackled in one of their videos.

Dr. Benjamin also discusses here what to do when you're not sure whether the missing digit is a 0 or a 9. My preferred approach here would be to say, “I'm not getting anything. It wasn't a zero, was it?” Note that by making this a negative question, you can follow up their answer with “I thought so” or “I didn't think so”, which makes you sound like you knew all along, even though you're just asking a question.

The lecture is wrapped up with the mental multiplication of 2 five-digit numbers. This isn't done as quickly as the other squaring feats. Instead, this is done with lots of verbal calculation and what seems to be some nonsensical words thrown in. First, as he explains after getting the number 37,947 to square, he points out that he's going multiply 37,000 by 947, double that number, square 37,000, square 947, and add all those results together.

Why is he doubling that first calculation? Effectively, he's breaking the problem down into (37,000 + 947)(37,000 + 947). As with any problem of the form (a + b)(a + b), Wolfram Alpha shows that the result must be a² + 2ab + b².

The mysterious words he's uttering are actually ways of remembering numbers. Arthur Benjamin has another free lecture available online that details how to memorize numbers like this.

As with many live lectures, this one winds up with several mentions, including that of Harvey Mudd College, where Dr. Benjamin teaches.

Several of Dr. Benjamin's books and DVDs are promoted in the lecture. Since Grey Matters is an Amazon.com affiliate, you can help support this blog by buying Dr. Benjamin's books through our affiliate link, his Secrets of Mental Math DVD (from which the above free number memorization lecture is taken), his Joy of Mathematics DVD, and/or any of the Amazon.com links listed above.

It's time for some magic!

Don't worry, there's no complicated sleight-of-hand in these tricks. Not only does math make them easy, but you don't even have to do any math during the routines, since all the math involved has been worked out ahead of time.

I'll start with the simpler of the tricks. In this first one, you have someone think of any hour of the day, and you tap numbers on an analog watch while they silently count up to 20. When they reach 20, they say “Stop!”, and your finger is on the hour they secretly chose!

The method behind this simple trick is described in Futility Closet's On Time post.

At first, the workings may confuse you, but a little experimentation with different numbers will help you understand it. Obviously, this is also true for anyone for whom you perform it, so don't treat this as a big mystery, but rather as a simple and interesting experience.

The basic tapping presentation has a long history in magic. In Martin Gardner's book, Mathematics, Magic and Mystery, there's an entire section on tapping tricks. Thank to Google Books, you can read the entire section online for free, running from page 101 to page 107.

The next trick, courtesy of Card Colm, is a little more involved. You have someone name any card suit, have a regular deck of cards shuffled, and then the number cards (Ace through 9) are removed in the order in which they're found in the deck. You then make an unusual bet based on divisibility of various numbers formed by those cards.

This trick is called the $36 Gamble, and the method is found in Card Colm's post, The Sequence I Desire. Magic: When Divided, No Remainder. Beyond just the mathematical method, there's plenty to explore under the hood of this routine, including Arthur Benjamin's method for determining divisibility by 7, and a very deceptive shuffling method, which appears fair.

If you enjoy the deceptive shuffles discussed in the above post and its links, you also might enjoy Lew Brooks' book Stack Attack, which features the False False Shuffle. The false shuffle and the routines in Stack Attack mix well with the principles behind the $36 Gamble. In my 2006 review of the DVD of the same name by the same author, you can get a better idea of the contents.

Even though I've only linked to 2 tricks here, practicing them, understanding them, and digging in to the variations I've mentioned is more than enough to get your mental gears turning, so have fun exploring them!

If you've read even a little of this blog, you know I love playing with math.

Naturally, when mobile apps are released that let me play with math on the go, I'm even more enthused! Earlier this month, an app called Geogebra was released, and it's a great way to explore math visually!

Geogebra is what is known as a computer algebra system, or CAS, for short. As you input various mathematical equations, you get a visual representation. The video below gives you a better idea of how Geogebra is used. While it focuses on the original online version, much of the operation is the same in the new versions:



The tablet versions of Geogebra were announced on Sept. 1st, and are available for Android, iOS, and Windows 8 tablets. Take the time to explore the other links on that page, as there's great support for the program in the form of a community, examples, videos, and much more!

If you already use the Wolfram|Alpha app on your tablet, Geogebra makes a nice compliment to it, for those who enjoy exploring math.

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

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

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

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

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

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

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

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

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

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

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

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

Perhaps out of necessity, science fiction authors tend to have a better grasp of math than your average fiction author.

I thought it would be fun to take a close look at some of the mathemtical understanding of sci-fi authors, so we can see what happens when these worlds collide.

I'll start with a rather amusing anecdote about John Taine, as posted over on futility closet:

Eric Temple Bell led two lives. By day he was a mathematician at Caltech; by night he wrote science fiction as John Taine.

By a happy chance the two personalities met in 1951, when the Pasadena Star-News asked Taine to review Bell’s book Mathematics, Queen and Servant of Science.

Not one to lose an opportunity, he accepted. “The last flap of the jacket says Bell ‘is perhaps mathematics’ greatest interpreter,’” Taine wrote. “Knowing the author well, the reviewer agrees.”

July's snippets are ready!

Instead of trying to pick a theme this month, I decided to return to the original purpose of snippets, and just grab random items of interest to Grey Matters readers:

• For those who enjoyed my post on temperature conversion, and the added trick to make it easier, there are simple tricks for other metric conversions, as well.

Nurse Debra Mallory teaches this simple trick for converting pounds to kilograms. Need to work out a metric distance? Here's a quick trick for converting kilometers to miles.

• Last month's snippets focused on hexadecimal numbers. If you want to learn more about those, check out this introduction to hexadecimal numbers. The approach taught on that page is taught quite clearly, so it's easy to grasp.

• Many of the feats I teach on this site require mental arithmetic. If you want to take your mental math to the next level, check out Aaron Maxwell's online book Inner Algebra: How To Do Algebra In Your Head. The whole book is available at that link, and you can also buy a paperback copy on Amazon.

• Just for fun, and completely unrelated to just about anything else I've ever posted on this site, check out this fun and amazing toy, known as Tumble Rings. You can build them yourself, using several sets of key rings. If you're familiar with the toy known as a Jacob's Ladder, you'll recognize the same principle at work here:



That's it for July's snippets. If you have anything you'd like to add, let me hear about it in the comments below!

It seems like Wener Miller just never stops creating!

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

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

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

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

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

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

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

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

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

Canadian Mel Stover (1912-1999) was a well-respected magician and puzzle creator whose works all seemed to have both impish and amazing qualities.

His works have been ignored far too long on the pages of Grey Matters, and I'm going to correct that today.

To give you an fun first impression of his work, read Bob Friedhoffer's remembrance of Mel Stover (PDF version). The described puzzle itself is simple enough, but when you're first confronted with it, it seems quite challenging.

Even if you're not familiar with the man himself, you've probably run across his puzzles. Puzzles.com features some of his better known puzzles, and many instantly recognize either the vanishing pencils or the matchstick giraffe right away.

As a matter of fact, Mel Stover developed numerous vanishing puzzles similar to the pencil one above. On this page, the pencils, the beer glasses, and the vanishing leprechaun were all his creations. Watch the vanishing leprechaun in the video below:



My favorite Mel Stover creation has to be what he dubbed the Beadless Abacus. As you can see below, it's an array of hexagonal cells, each containing a simple arithmetic problem. If you were challenged to sum up the answers to, say, any 3 cells in a straight line, could you do it? How about summing a line of 4 cells, or better yet, summing an array of 7 cells (1 in the center, and the 6 cells surrounding it)? Read the article below, and see if you can figure out an easy way to do these.



If you want to verify your answer, or even if you're just stumped, the answers can be found here. The feat above is an excerpt from Games Magazine's The Giant Book of Games, which is available at amazon.com, if you'd like a physical copy.

I hope you've enjoyed this look at some of Mel Stover's creations, and perhaps even found a new favorite magic trick or puzzle to share with others.

(Note: This is a repost, with some link updating and minor rewriting, from about this same time 5 years ago. I repost it because it has become relevant over this Memorial Day weekend.)

If you do math at all at the gas pump, it's probably either related to how many gallons you can get for a given amount of money, or how much money will be required to get a needed amount of gas. If you're willing to do a bit of math and planning before you go get your gas, you can actually work a surprising amount of real savings into the equation, as well.

How do you save on gas? The obvious first answer is to find the cheapest gas you can. My grandfather's method for this was to drive around looking station by station, but that only works well when you're sure you can find gas lower than 35 cents/gallon. Unsurprisingly, the internet is here to help! Sites such as fueleconomy.gov, FuelMeUp, and GasBuddy make short work of finding the lowest gas prices in your area.

Unless you find the cheapest gas in your immediate area, another question begins to raise its head at this point. Sure, if you go a little farther to that station with the cheap gas you can save some money, but if you factor in the gas you'll burn going the extra distance, and the added gas you'll require, are you really saving money? With the current level of gas prices, this isn't a trivial question.

Fortunately, Kimberly Crandell, better known as Science Mom, tackled the question of whether nearby expensive gas or cheaper gas across town was cheaper in July 2007.

As I've explained, there is some math involved, but there are only five different factors involved: The number of gallons needed, the gas mileage of the car, the cost of the closer (more expensive) gas, the cost of the farther (cheaper) gas, and the miles out of the way for the cheaper gas (Google Maps, Yahoo! Maps, or MapQuest will come in handy here). In the article, you learn the formulas to process this, and how to solve for the savings you'll get, as well as the break even points for cost per gallon, total gas gallons, and distance.

Understanding and working through the formulas is one thing, but how about if you would just like to get your answer and go? Once again, the internet is here to help. My favorite tool for this step is Instacalc, which I first mentioned in August 2007.

I've created an instacalc version of Kimberly Crandell's equations where all you have to do is plug in the five factors (remembering that the two prices requested are both price per gallon).

If you prefer, I've also created a metric version of this calculator, for readers in other countries. Whichever version you use, I hope this helps save you some money and that you find it useful!

Update: If you enjoy William Spaniel's Game Theory 101 videos, you'll enjoy his method of finding cheap gas without perfect information. This method, based on game theory, is equally mathematical, but requires fewer calculations.

There's an old magic trick out there that's been in the public domain for so long, its origins seem to have been lost.

In the classic version of the trick, a card is chosen, and a mysterious person is called. Somehow, this person is able to name the correct card, despite not even being in the same room, or even the same state or country!

Magicians know this trick as “The Wizard”, as most of them learned the version by that name from the book, Scarne on Card Tricks. You can read that particular trick for free online (page 42, page 43).

As with many tricks, the presentations grow and change over the years. Some magicians also know this same trick as “The Phantom” or some other equally mystic name. When Scam School taught this routine (YouTube link), their figure of choice was a secret member of a government conspiracy:



If you think about it, any bit of data which can be identified by two simple pieces of information, in a manner similar to grid coordinates, can be coded in a similar fashion. It's quite obvious that playing cards can be broken down into 2 bits of information, their value (Ace through King) and their suit (clubs, hearts, spades, diamonds). What if the data to be coded didn't have 2 such obvious factors? If we could manage that, this routine could be even more deceptive!

Max Maven developed a version called “Remote Pager” in which a word is chosen from the following old letter

Impossible, but true! A demonstration of intuition, custom tailored for you by Mister Zulu. Cnoose any word in the paragraph of at least four letters. After you choose a word, contact me by phone. Believe it or not, I'll announce the word you are thinking of! Imagine tne surprise ~ but be on guard: I presume my demonstration is going to haunt you...

Mister Zulu

Werner Miller has certainly been keeping busy!

Not long after the release of sub rosa 3 and 4 comes his newest book, EZ-Sqaure 5!

E-Z Square 5 is available as an ebook from Lybrary.com, available in English and in German.

As with previous books in the series, this one features a particular routine concerning magic squares. The major difference here being that these magic squares are created using playing cards, similar to Richard Wiseman's The Grid and Chris Wasshuber's Ultimate Magic Square, both of which are acknowledged in E-Z Square 5.

Werner Miller explores the possibilities through 3 main routines, and a bonus routine. The first routine is the simplest, in which the spectator generates a total by selecting 4 cards out of 16, and you quickly deal a 4 by 4 square with 16 different cards whose rows columns and diagonal give the same total. The second routine, which is my personal favorite, has the spectator cut off about half the deck, and you as the performer are able to create a 4 by 4 grid whose rows, columns, and diagonals are equal to the number of cut-off cards.

In the 3rd routine, the spectator cuts off a group of cards, and deals them into 2 piles, while the performer uses the remainder of the deck to create a 5 by 5 grid of cards. When the magic total is revealed, it proves to be the same as a number created from the top 2 values on the spectator's piles!

The bonus routine may be familiar if you've purchased Werner Miller's da capo 3, as it is Squaring the Cards. In this 4 by 4 magic square routine, the magic square's total is equal to the total of the remaining cards not used in the routine!

If you're nervous about handling the various arrangements and calculations required in normal magic square routines, EZ-Square 5 is an excellent choice, as the routining and use of playing cards takes care of much of the work automatically. As any Werner Miller fan already knows, not much more than basic card knowledge is required in his routines. I recommend E-Z Square 5 highly!

Is it time for April's snippets so soon? It only seems soon because March's were so late.

This month, the focus is on resources which help you remember more effectively!

• Just today, Forbes.com posted a wonderful article titled 6 Easy Ways to Remember Someone's Name. In addition to the standard advice, I especially like the tip of asking them a question, so you can take some time to mentally link their name with their face.

If you want to examine this in more detail, I've prepared a YouTube playlist focusing on memorizing names and faces. There's also an excellent book titled How to Remember Names and Faces: How to Develop a Good Memory (originally published in 1943, but the advice is still very sound!). I've also covered various mobile apps that help you practice these techniques.

• Speaking of apps, there's a new free iOS app called Brain Athlete (iTunes link). This focuses on memory-competition feats, including memorizing numbers, word lists, and playing cards. If you've read Joshua Foer's Moonwalking With Einstein and/or read my PAO system post, you should have a good understanding of the basics.

If you get stuck finding a certain person for your PAO system, here are links to lists totaling 10,000 famous people to help. No actions are objects are included, as these need to be developed based on how you imagine each of these famous people.

• Every so often, I run across free memory web apps that I find useful, such as these. The newest one I've found is the Major System Database. It's very simple and direct. You can find words for a given number, the numeric equivalent of a given word, or even break up numbers into small groups and give you mnemonics for each group!

• For Windows users, there's a new free program available, simply titled Memorization Software. It's designed to help you remember various types of texts, such as lyrics, poems, and speeches word for word. The tutorial video below (no audio) gives you an idea of the various approaches used here.



If you like this approach, but don't have a Windows machine (or even if you do!), my web app Verbatim 2 (Video Tutorial link) is also free, works in a similar manner, and runs in any modern browser.

If you're familiar with Werner Miller's work, you're in for another treat from him!

If you're not familiar with Werner Miller's work, you're in for several treats!

For those not familiar with him, he's a creator of mathematical magic from Germany. What makes his work so special is his knack for ingeniously and effectively disguising the mathematical principles used in his routines. If you haven't already done so, go back through this site to explore some of his other effects. You'll quickly get an appreciation for his style of thinking.

Last October, he released the first two volumes in a new series called Sub Rosa (Latin for in strictest confidence). Earlier this month, Werner Miller released two new books in the series, Sub Rosa 3 and Sub Rosa 4.

Thanks to the math involved, none of the routines require any difficult sleight-of-hand. For example, one of the routines from Sub Rosa 4 is titled, “ESPecial Countdown,” and performed with ESP cards. After the ESP cards are mixed, an audience member chooses a card. This card is set aside face down, and no one, not even the audience member, knows which ESP symbol was chosen.

The performer then removes 8 cards, and asks for a number (8 or less) from which to countdown. The performer then runs through a procedure in which the cards are counted down. When the countdown reaches 0, the card at that position is removed. Interestingly, it proves an exact match for the card previously chosen by the spectator.

Generously, Werner Miller has allowed me to reveal the method to “ESPecial Countdown” here on Grey Matters. The method is explained below, and the PDF can be downloaded via this link.



I'd like to thank Werner Miller for his generosity in letting me share this routine with you. If you'd like to show him your thanks as well, you can buy Sub Rosa 3, Sub Rosa 4, and his other works over at Lybrary.com.

Robert Neale is a magician who is an incredible thinker in terms of methods, as well as a very spirited artist when it comes to presentation.

In this post, you'll get a glimpse of the mind of this amazing performer and inventor.

Back in 1983, Robert Neale released what has become one of his best-known routines, called the Trapdoor Card. This week's episode of Scam School teaches this classic, but sadly neglects to credit the inventor:



Robert Neale himself has given Scam School permission to perform and teach his classic Rock/Paper/Scissors routine back in episode #71, published as “My First Trick” in Neale's book, Tricks of the Imagination:



Perhaps part of Neale's genius is in creating things that are so simple and yet so deceptive that those who aren't directly familiar with his work just think they're things that “have always been around.” Earlier this year, bicyclecards.com presented his Hypercard, first as a puzzle, then in a post explaining how it's made. They correctly credit Martin Gardner with popularizing it, but never mentions Robert Neale himself. Here's a video created by a reader of that tutorial:



If you really want to see the man himself at work, as well as learn some of his amazing thinking, the easiest place to begin is his Celebration Of Sides video. In this video, you see not only his amazing and ingenious approach to creating routines, but his warm and charming ways of presenting even the most abstract concepts. Here's a quick glimpse of some of the things shared on this video:



Hopefully, you've enjoyed this look at some of the work of one of the most creative thinkers in magic, and maybe even learned something new.

For Day One, my calendar calculation routine, I recently released a custom-designed calendar clipboard which helps make the feat visible to a small crowd.

In today's post, I'll show you how to put together your own calendar prop inexpensively, and even some other directions you can take the basic idea.

To start, you'll need a magnetic dry-erase board with a blank calendar pre-printed on it. I used an 11 by 14 dry erase calendar from Expo, which comes with two magnets. The two important features in the board are that it be magnetic, as well as small enough to use and carry for a performance, while still being visible for your audience. You'll also need to make sure your chosen board has 5 weeks to mark.

You'll also need a dry erase marker (usually included with dry erase calendars), a dry erase eraser, and a permanent marker, such as a Sharpie. Optionally, you may want a ruler for making your marks consistent. If you choose a ruler, I suggest a cork-backed ruler to minimize damage to the board.

What you're going to do is use the permanent marker to write the dates in the squares from 1 through 28, similar to the way the calendar clipboard is laid out. Most dry erase calendars have a small space in a corner for the date, such as the corner notches seen on this dry erase calendar, but for better audience visibility, you'll want to use as much of each date's square to write the date.

If you prefer, you can use the ruler and a dry erase marker to create even guidelines for your dates first. Personally, I didn't do this. In performance, I need to write and/or erase 29, 30, and 31 on the board, and those will usually be written freehand, so they tend to stand out if the other dates aren't written freehand, as well.

What happens if you make a mistake when writing with a permanent marker on your dry erase calendar? Don't worry, there are numerous ways to remove permanent marker off of your dry erase board. The simplest and most surprising of them uses only a dry erase marker and eraser to remove permanent marker.

Once you've written the dates from 1, in the upper-leftmost square, through 28, in the rightmost square of the 4th week, using large numbers as discussed above, you can put the permanent marker away.

Not surprisingly, most dry erase calendars have the days of the week permanently marked at the top. Yet, you need to be able to change the days of the week in routines like Day One. To solve this problem, I simply use magnets printed with days of the week. This is why I emphasized the importance of a magnetic dry erase calendar earlier. Even on larger boards, these days of the week magnets cover the pre-printed days, and highlight the days printed on the magnets.

Before each performance, you'll use a dry erase marker to write 29, 30, and 31 on the first 3 squares of the 5th week on the calendar, and have the days of the week magnets arranged in the remaining 4 days of the 5th week in the calendar.

When you're given the month and year, write them in the space for the month at the top, and use the Day One technique to determine where to place each day of the week magnet. After placing the magnets, erase any of the last 3 dates as needed (For example: If you're given a February in a leap year, erase the 30 and 31, leaving the 29) and your calendar should be arranged correctly!

After each performance, erase all the dry erase markings, and put the magnets back down in the 4 empty squares of the final week. If you're about to do another performance, write the 29, 30, and 31 back in. If you're not, you can simply put the board and magnets away until you're ready to perform again.

The basic idea of using permanent markers to create a custom design (and knowing how to remove it in case of mistakes), should start the gears turning for other ideas. Starting from a blank dry erase board, you could create things like a grid for a magic square or a chessboard for the Knight's Tour.

The cork-backed ruler I mentioned earlier is an essential when designing grid-based layouts, of course. For the chessboard, I recommend creating the board in a blue that's dark enough to be distinguished easily from the white squares, yet still light enough to contrast with dry erase numbers written in black.

Don't forget that using a magnetic dry erase board can also be a great way to display magnetic playing cards, either in full-size or in miniature.

For one last idea to inspire you, how about a Sudoku grid? You could use it to display your apparent Sudoku genius as taught in Werner Miller's Swindle Sudoku routine!

That should be enough to be enough to inspire you and get you thinking about different ways to customize and present the mental feats you've learned here on Grey Matters.

For my next puzzle, I'll try figure out why custom-printed dry erase Sudoku boards are so much more readily available than custom-printed dry erase chess boards.

On October 21, 1914, Martin Gardner was born. After having an enormous and positive effect on the fields of magic, mathematics, philosophy, literature, science, and many other fields, he died on May 22, 2010.

On the 98th anniversary of his birth, I thought it would be fun to look back at Martin Gardner's influence.

Back in 1956, Martin Gardner submitted an article on flexagons to Scientific American. They not only ran the article, but offered Gardner his own monthly Mathematical Games column. The first column ran in January of 1957, and he kept it going regularly for the next 25 years! Martin continued to write the occasional column until the May 1986 issue, retiring both himself and the column.

In honor of the flexagon column that started it all, and Martin Gardner's birthday, Vi Hart has dubbed October 2012 the month of the flexagon. Below is the first video she made to honor this month:



Vi Hart also made a second video on their surprising mathematical qualities, and even a third on flexagon safety.

All of Martin Gardner's columns were assembled into a series of 15 books. The books themselves have since been assembled into the Martin Gardner's Mathematical Games CD-ROM. This series of PDF eBooks is searchable and portable, making the joy of surprise and discovery even more accessible.

Even though it's Martin Gardner's birthday, you can get a few free goodies to get an idea of his influence. The January 2012 issue of the College Mathematics Journal was entirely dedicated to his work, and remains free to download as either the full issue or individual articles. The Gathering 4 Gardner Foundation has several Martin Gardner posters available for free download, and even a full free PDF copy of a tribute book called The Mathematician and the Pied Puzzler.

Probably one of the most famous Gathering 4 Gardner giveaways, thanks largely to this YouTube video, would be the paper dragons whose heads seem to turn and follow you. The PDFs that will let you build your own green dragon, red dragon, and/or blue dragon are still available as free downloads.

Many of the puzzles he created and/or popularized over the years can be found in Puzzles.com's Martin Gardner section. It's surprising how many now-classic puzzles can be traced back to him. He even gets a mention in Bill and Ted's Excellent Adventure:

[Bill and Ted are working on their history report]
Bill: Okay, Ted, George Washington. One: the father of our country.
Ted: Two: born on President's Day.
Bill: Three: the dollar-bill guy.
Ted: Bill, you ever made a mushroom out of his head?