Many regular Grey Matters readers will be familiar with Michael Daniels' Mind Magician site, where he teaches numerous math and memory feats, such as calculating cube roots in your head instantly. He's recently written a new ebook on the 4-by-4 magic square, titled Perfectly Possible. I found it to be well worth the time and money invested, and wanted to share my thoughts with Grey Matters readers.

This is going to be a difficult review, as I can't give too much away, but I also want to share with you the quality of this method. I'll start with the qualities promoted by Michael Daniels himself:

• Completely impromptu. No set-up, gimmicks, or cribs.
  
• New, improved method - minimal memory and the simplest of calculations.
  
• Suitable for close-up or stage performances.
  
• Produces elegant magic squares.
  
• Can be immediately repeated for different totals.
  
• Includes a browser application that helps you to learn and practice (Internet connection not required).
  

It's now been a few months since Grey Matters was back, so now it's time to bring Quick Snippets back!

This time around, we have plenty of mathy goodness, so it's best to just jump right in!

• Besides the Clay Institute's famous selection of Millennium Problems, which will make you a millionaire if you prove or disprove any one of them, there's a lesser known set, known as John Conway's $1,000 problems. Not long ago, the 5th conjecture, which claims that working through a certain procedure (described in the link an video below) will always end in a prime, was disproven by physicist James Davis. The Numberphile video below details the problem and James Davis' counterexample:



For more about the million-dollar Millennium Problems, watch the BBC's Horizon documentary, "A Mathematical Mystery Tour of Unsolved Mathematical Problems."

• Speaking of fun discoveries in recreational mathematics, check out Allan William Johnson's "Magic Square of Squares", discovered back in 1990, and just recently posted over at Futility Closet.

• James Grimes introduces us to a different sort of "Square of Squares", in his latest Numberphile video, "Squared Squares". The challenge here is to make a perfect square shape from a set of smaller square shapes:



• Presh Talwalkar, of Mind Your Decisions, posted an interesting puzzle recently. It's titled, "The Race To 32,768. Game Theory Puzzle". Read the article up to the point where you're challenged to work it out yourself, or watch the video below up to the 2:12 mark, and try and figure it out for yourself. If you get stuck, try going over my Scam School Teaches the Game of 15 post for inspiration.



• Late last month, mathematical video maker 3Blue1Brown posted a must-watch video on the visualization of all possible Pythagorean triples. Even if you remember everything from you math classes about Pythagorean triples, this video is both eye-popping and an eye opener:



• We'll wrap this set of Quick Snippets up with help from Mathologer. His videos are always interesting, but his latest one is one of those unusual approaches to math that makes you appreciate its beauty. This video is titled, "Gauss's magic shoelace area formula and its calculus companion", and it teaches an simple but unusual method for working out the area of any polygon that doesn't intersect itself. The host even goes on to show how this approach can be adapted in calculus to work out the area contained by curves!

Ever since I started this blog, I've been waiting for this day. I started Grey Matters on 3/14/05, specifically with the goal of having its 10th blogiversary on the ultimate Pi Day: 3/14/15!

Yes, it's also Einstein's birthday, but since it's a special blogiversary for me, this post will be all about my favorite posts from over the past 10 years. Quick side note: This also happens to be my 1,000th published post on the Grey Matters blog!

Keep in mind that the web is always changing, so if you go back and find a link that no longer works, you might be able to find it by either searching for a new place, or at least copying the link and finding whether it's archived over at The Wayback Machine.

2005

I love it when two old friends visit and get along!

Numberphile recently took a look at one of my favorite challenges: The Knight's Tour! This is a chess-based puzzle that challenged some of the greatest minds in mathematics.

In a rare video appearance, Brady himself describes some of the fascinating aspects of the Knight's Tour in the following video:



If you'd like to try the Knight's Tour out for yourself online for free, I've created 2 versions you can play. This version is in the Mental Gym (select level 1 for the classic challenge), and here's a more modern version hosted on Dropbox (select “New Game” > “All 64 Squares” for the classic challenge).

When it's a new challenge, it can seem quite difficult. Often, you get past about 50 squares, and then start having difficulty. If you want to be able to tackle this challenge, I have provided a complete Knight's Tour tutorial over in the Mental Gym. If you can understand and remember a few simple patterns, you can not only solve the Knight's Tour starting from anywhere, you can even have someone select a starting AND ending position, and still be able to solve it!

My dropbox version of the Knight's Tour offers various settings, including the ability to show a numbered path, as in the video. This version also auto-detects whether the numbered path is a semi-magic square, as discussed in the video, starting at about the 2:23 mark.

If you can learn to solve it, as in the Mental Gym tutorial, is it possible to learn to start anywhere and create a semi-magic knight's tour square? The answer is almost. Magician Harold Cataquet has done some incredible work on working out just how to do this, and it's written up in the ebook Mind Blasters, by Peter Duffie. If you're really interested in being able to the Knight's Tour AND finishing with a semi-magic square, the article is worth the price of this one book alone.

As Brady mentions in the video, there's an amazing amount of mathematical research done on the Knight's Tour. You can see many of the directions in which this challenge was taken over at Knight's Tour Notes, for a start.

Play around and enjoy the Knight's Tour. If you have any interesting discoveries you'd like to share, I'd love to hear about them in the comments!

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.

If you've ever done decimal division, you've no doubt run acros numbers that endlessly repeat, such as 0.333 0.666.

If you've ever dealt with decimal division by 7, as taught in my Mental Division With Decimal Accuracy post, you've noticed an unusual phenomenon that goes just beyond the repeating of the digits. It's that odd quality we're going to discuss in today's post.

When you divide 1 by 7, you get the number 0.142857142857... and you notice that the digits repeat endlessly. Even weirder, thought, is the fact that dividing 2 by 7 gives you the same digits in a different order: 0.2857142857142857.... and so on. In fact, as discussed in the mental division tutorial, dividing any number from 1 to 6 by 7 will always return the same digits with only the starting point changing.

Numberphile recently turned its attention to the number 142857, and explains many of the unusual qualities behind this and similar numbers:



Let's take things a step further than in the video above. If we arrange the multiples of 142857 from 1 to 6 in order, we get the following square:

1 4 2 8 5 7
2 8 5 7 1 4
4 2 8 5 7 1
5 7 1 4 2 8
7 1 4 2 8 5
8 5 7 1 4 2

01/19 = .0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
02/19 = .1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2
03/19 = .1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3
04/19 = .2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4
05/19 = .2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5
06/19 = .3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6
07/19 = .3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7
08/19 = .4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8
09/19 = .4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9
10/19 = .5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0
11/19 = .5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1
12/19 = .6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2
13/19 = .6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3
14/19 = .7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4
15/19 = .7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5
16/19 = .8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6
17/19 = .8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7
18/19 = .9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8

NOTE: This post originally appeared on Grey Matters back in July 2011. I'm reprinting it today because Werner Miller's mathematical magic really is worth a second look.

Grey Matters favorite Werner Miller is back, and he's brought more of his amazing mathematical wizardry with him!

If you're not already familiar with him, he's a retired mathematics teacher in Germany who has created some of the most original and compelling magic routines I've ever come across. He is the author of several magic books, including Ear-Marked, which is available in the Grey Matters store.

Starting off, we have a couple of good routines that are perfect as promotional tools, since you can print them on your business cards or brochures, and have people perform them for themselves or others without understanding how they work. There's Vive Le Roi!, which includes several variations of routines where you move your finger from card to card, eventually winding up on a predicted card. You can have two people do this together, as they'll be on different cards until the last card.

The other trick along this line is Magic Patchwork, a similar trick with a magic square. He mentions that it was inspired by Pedro Alegria’s El cuadro de colores, but the link given is no longer functioning. Fortunately, it was captured by the web archive. The original is here, with a translation to English via Google Translate available here.

Werner Miller also created a very sneaky calculator trick, called You Push the Button... that seems to be a mathematical trick, but isn't. The use of the calculator helps conceal the outright sneaky method.

Getting back to his specialty of mathematical magic, he offers a great routine with dice. It's called Lined Up, and has two different phases, both of which begin with different-colored dice arranged with the numbers 1 through 6 in numerical order. In Phase 1, you have someone choose a die and turn that number face down. After getting the new total of these dice, you announce which color die has been turned over. Phase 2 is similar, except that you have someone choose a die and turn over every die EXCEPT for the chosen one!

I've saved my favorite for last! It's called Ghost Rider, and uses a chess knight and some file cards. One of the file cards is signed, then mixed into the pile and dealt out into a 3 by 3 square. The spectator then uses the knight and their own free choices to find their own signed card! Part of the principle is taught here on Grey Matters in my Knight Shift post, as mentioned generously in Werner Miller's article. His added touches, however, make this a very impressive trick.

If you like Werner Miller's style and would like to see more, check out the rest of Werner Miller's work here on Grey Matters!

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.

Today, you'll learn an interesting new mathematical game in which the object is to obtain 3 numbers that total 15.

Well, it's not exactly new. As a matter of fact, it's something with which you are probably very familiar!

Let's jump right into the game. Watch the 274th episode of Scam School (YouTube link) below, and once they've played the game and the ad starts, stop the video and ask yourself if you can come up with any simple way to play and win the game. Once you've either figured it out or given up, go ahead and watch the remainder of the video.



Were you surprised? Yes, it's your old friend Tic-Tac-Toe (or Naughts and Crosses)! This is just one of numerous ways that have been developed to disguise the true nature this classic game. It's almost embarrassing how effective such a simple disguise can be.

Last August, I delved into strategy for the game of 15, with Part 1 teaching you the basics and how to win when you go first, and Part 2 teaching you how to win, or at least avoid losing, when you go second.

I created those posts so you can ideally play the game without ever referring to a Tic-Tac-Toe board. There's still one hitch with the game, however, and you can see it in the above video. When the game is introduced, it's explained as a mathematical game, and people immediately get apprehensive. The game is already unfamiliar, and the mathematical aspect often just adds stress.

Since you generally want to put people at ease, perhaps it's best to make the game seem more familiar. Instead of using 15 as the magic total, use 21! How would you do this? Simply increase the numbers in each part of the magic square by 2. Instead of the top row being 8, 1, and 6, you change it to 10, 3, and 8. The whole square should look like this:

10   3   8
 5   7   9
 6  11   4

Yes, the methods of mental feats are very important step in mastering them, which is why I spend so much time on them.

However, once you have the method down, how you bring that ability to an audience in an entertaining way? In today's post, we'll show you a few performers who take these feats to that all-important next level.

Our first performer is Gerry McCambridge, doing the Knight's Tour in his TV show, The Mentalist. Note the importance he pays not only to the feat itself, but with making sure that everyone understands the challenge and the difficulty.



With chess, people already have a preconceived notion of intelligence and difficulty being involved. What about if you're doing a magic square, a feat which boils down to putting down numbers, then repeatedly adding them up? If you can make that entertaining, that's impressive. If you can bring an audience to a standing ovation with it, you know you've really got something!



We'll wrap this post up with a rare US TV appearance late Shakuntala Devi. She was a woman from India known for her mental calculation skills. In Ricky Jay's TV special, Learned Pigs And Fireproof Women, she does root extractions in an extraordinarily fast and impressive manner. For you poker fans, that young man on the computer is probably better known to you as Chris “Jesus” Ferguson.



The whole point of this post, of course, is not to get you to copy these performances directly, but to inspire you to think of what unique and different qualities you can bring to your own performances.

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!

It's time to introduce 2013 to our tradition of snippets!

This month, we'll explore a wide variety of unusual mathematical treats,

• My post about making math visual with Wolfram|Alpha may have been slightly ahead of its time. Wolfram|Alpha's latest blog posts cover the use of equations to draw pictures and the visualization of basic arithmetic problems.

• James Grime is back, and he's covering a topic that is near and dear to his heart. In the video below, he talks about the Enigma machines that the Germans used to code messages during World War II, and the race to break this allegedly unbreakable code:



If you enjoyed this, there is another video detailing the flaw that allowed the Enigma codes to be broken, as well as some tidbits and outtakes.

• I've always encouraged people to learn at least a few amazing feats. Cracked.com is now doing the same thing, albeit with a harsher title, "5 So-Called Signs of Genius That Any Idiot Can Learn." Grey Matters readers will be familiar with many of these, if not all of them. Looking around the web, you can find many examples of such feats, including quickly multiplying by 9, dividing by 9, squaring numbers, and more!

* I'll wind up these snippets with some offbeat links. First, here's a very unusual magic square, featuring resistors that form a magic square if wired in parallel. If your resistor math is a little rusty, here's a short video to get you up to speed.

And because I'm a fan of iOS apps that help you train your brain, check out becomeananny.com's list of 10 iPhone apps that boost brain function.

November's snippets are here!

This time around, we're treating you to a little history, a little controversy, and some eye-opening mathematics. You shouldn't get too lost, as this journey will mostly take place through the magic of video.

• Here's a young man by the name of Ethan Brown, together with his uncle, wrestling commentator Joey Styles. Even as young as Ethan is, he performs an amazing magic square routine using his Uncle's birthday:



Would you like to learn how to do this? This routine, known as the Double Birthday Magic Square, was developed by Dr. Arthur Benjamin, who has made the entire routine available on his website as a free PDF!

• There's a classic challenge known as the Monty Hall paradox/dilemma/problem. I've written about it in 2006, and again in 2010 (among other times). Earlier this month, AsapSCIENCE posted a new video on it that explains it quite well:



If you read my post on Bayes' theorem, you should recognize the equations that were written at about the 2:00 mark in the video.

It turns out Bayes' theorem is an excellent tool for explaining the Monty Hall problem. Using the tree diagram approach from the Bayes' Theorem - Explained Like You're Five video, with help from Wolfram|Alpha and the Syntax Tree Generator, I put together and posted this visual explanation of the Monty Hall problem over at the Magic Cafe. If you've struggled with this paradox before, this explanation may help clear things up.

• Bayes' theorem really is powerful. For example, back in 2009, Air France Flight 447 disappeared off the radar and a long search began, not just for the plane and people, but for the reason as well. For 2 years, they searched for the airplane's flight recorder without luck, until they hired a team to use Bayes' theorem to narrow down a search area. After that, the flight recorders were recovered very quickly!

Even as powerful as Bayes' theorem is, it had a reputation as being bad mathematics through much of the 20th century. It's only recently that it's gained a wider respect. Sharon Bertsch McGrayne wrote a book on this history, called The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. She talks about the controversy in the following 32-minute lecture at Singularity Summit 2011:



There's also a 55-minute video of her Google talk available. If you're curious about her mentions of Alan Turing and the Enigma machine, I have a post from July all about Alan Turing.

• Numberphile has posted a number of good, enjoyable videos recently. Being interested in the Tau vs Pi fight (and let's not forget Eta), I enjoyed their Tau replaces Pi video:



Take the time to check the rest of their other recent videos out, as well. The explanations are always fun.

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.

In the past, Werner Miller has generously shared his creative brand of mathematical magic with Grey Matters readers numerous times.

He's just released two new books, Sub Rosa 1 and Sub Rosa 2, and has allowed Grey Matters readers to take a free peek at an effect from the book.

Sub Rosa is Latin for “in the strictest confidence.” In Werner Miller's case, it also means a multi-volume series of books containing a treasure trove of deceptive and original mathematical-based magic, none of which requires any difficult sleight-of-hand.

In Sub Rosa 2, there's an effect called Latin ESP square. You shuffle a set of standard ESP testing cards, and then have the spectator cut the cards. You then deal them out without looking at the faces, so that no row, column, or diagonal contains duplicates of any symbol!

The method is described below, and you can download this PDF at this link.



In the description above, you start by giving the deck a special type of shuffle. You can learn that shuffle in just a few minutes via this YouTube tutorial. When using it with just 25 cards, it's very deceptive. Although it looks like a very thorough mixing procedure, the only effect it has is the same as a cut.

Beyond just setting up an amazing pattern, this is also a great way to secretly set up a Latin square for a subsequent trick. Colm Mulcahy, whom I wrote about just last month, has some excellent trick involving Latin square set-ups, including Splitting the Pot and Amazon Arrays (Large Action). You can find much more about Latin squares on the Mathematical Association of America's website.

Thanks again to Werner Miller to allow Grey Matters to share this great routine. For more of this ingenious thinking, check out Sub Rosa 1 and Sub Rosa 2, as well as his other works!

In the previous post, you learned about a disguised form of tic-tac-toe known simply as 15, and how to avoid losing if you go first.

In this post, you'll learn about the best strategy to use when you go second.

The game of 15 was written about repeatedly by Martin Gardner. His original Scientific American column on it is available in his book Mathematical Carnival. Gardner also discusses it briefly in Aha! Insight. In both sections, he also discusses other interesting ways to disguise tic-tac-toe.

You can also find out more variations of tuc-tac-toe, including 15, in the Games column of the August 1979 issue of OMNI magazine.

To start, you should understand that going second effectively puts you on defense. In this version of tic-tac-toe, as with regular tic-tac-toe, the first player has roughly twice as many opportunities to win than the second player does.

When going second in 15, the first move is simple. If the other player takes the 5 for their first move, your response is to take any even number. In this post, I'll assume you always take the 4, but the strategies can be adapted to any even number. If the other player takes anything EXCEPT the 5 for their first move, then you must take the 5.

Where do we go from here? Obviously, that depends on the other player's second move. We'll start by assuming they took something other than the 5.

Other person goes first, first move is anything EXCEPT a 5: The first thing you need to watch out for is whether that second card, combined with their first, can make a total of 15. If so, you need to block that potential win by taking that card. For example, if they took the 4 first, you took a 5, then they took a 3, you have to realize that their 4 and 3 can be a win with an 8, so you need to take the 8.

After the other player makes their third move, you need to check for a threatened win and block that, as well. If this move doesn't threaten a win, take any odd numbered card (except 5, of course). At this point, you'll have the 5 and an odd numbered card (such as the 9). If they're smart, they'll see this and block your win (Seeing your 5 and 9, they take the Ace, for example). If they do this, all you can do is block and draw. If they miss it, you've got a win!

In this post, you'll be introduced to a simple new game to play. Even better, it's a game you'll never lose.

It's not another version of Nim. This time, it's even sneakier!

Here's the rules of the game of 15:

1) Nine playing cards, with face values from Ace to nine, are face up on the table. The Ace always has a value of 1 in this game.

2) Players alternate taking turns, and on a given player's turn, they must take 1 card from the available group on the table. Neither player make take a card that has already been removed from the main pile.

3) The winner is the first person to obtain exactly 3 cards that add up to 15.

What kind of strategy would you use to win this game, or at least prevent losing?

You might be surprised to learn that you probably already know this game. It's a disguised version of (or in math terminology, it's isomorphic to) tic-tac-toe! How exactly does this relate to tic-tac-toe? Imagine the numbers 1 throught 9 arranged as a classic 3 by 3 magic square, so as to total 15 horizontally, vertically and diagonally:


This might seem like a hard arrangement to keep in memory, but it's easier if you picture the arrangements of even and odd numbers separately:



Now you can clearly see how the game relates to tic-tac-toe, and why it's played to 15. Since the other person doesn't realize what they're playing, this gives you an advantage.

In this version, however, the Ace through nine are laid out in a straight line in order, not the traditional crisscross pattern, so you can't see things as clearly as you would in a regular game. So, exactly what strategy should be used?

The proper strategy depends on whether you're going first or second. Let's start by assume you're going first. Following the classic strategies for X, as taught at chassandpoker.com and Wikihow, you'll want to take a corner square, which in this game equates to any even card (2, 4, 6, or 8).

When first learning this version of the game, always take a 4 when you go first. As you become more proficient in the game, you can start with any even card, but always starting with a particular card at first will help you get familiar with the essential.

There's only 2 different replies the other player can make:

1) They choose a 5: This is akin to taking the center square. You must reply by taking the 6 (the diagonally opposite corner). This might seem strange, as you'll have a 6 and 4 with no possibility of a 5, but you're setting a trap for them. If their 2nd move involves taking either the 8 or the 2 (a corner square, in other words), you've just won!

How? You take the sole remaining even number, which simultaneous blocks their possible win, and opens up 2 ways to win for you! When they block 1 way, you simply play the other to win.

Below is an animation of how the game looks in the standard form of tic-tac-toe. If you arrange the cards in the form of a magic square above, you'll be able to better follow along as I teach the strategies.