It's Sunday, and that means it's time for another of Werner Miller's devious magic square puzzles!

For those who are new, here are the basic rules of the puzzle:

You're given the same set of 4 starting numbers grouped together in similar locations in a 3 by 3, 4 by 4, and 5 by 5 grids. The object is to fill out the grids with other numbers in such a way that the following conditions are satisfied when the grid is completed:

1) Only positive whole numbers are used.

2) No number is duplicated in a single grid (the same number may be used once in each of the 3 grids, however).

3) Each row, each column, and both diagonals of a single grid must total the same sum. All 3 grids in a puzzle, however, are not required to have the same sum (and usually won't). In other words, each grid must be a magic square, but all 3 grids do not have to have the same magic square total.

4) Puzzles will be posted here each Sunday, with the answers to a given puzzle being made available the following Sunday.

3 by 3 grid:


4 by 4 grid:


5 by 5 grid:


Can you get the answers to these magic square puzzles? If you can solve it before next Sunday, let me know in the comments! You can either describe your solution there, as best as you can, or link to a graphic of your solution.
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Answers to last Sunday's puzzle:

3 by 3 grid:


4 by 4 grid:


5 by 5 grid:

For sometime now, I've been wanting to write a post on one of my major pet peeves: The old argument about memorization vs. understanding.

Let's start with the critics of memorization. Lisa VanDamme's article The Real Math Magic: Understanding vs Memorizing, while focused on mathematics, is typical of the arguments of understanding as opposed to memorization.

First, too many take “memorization” to mean simply rote memorization – just repeating things over and over again until your memory retains them. If this is what is truly being argued against, then I'll give the opposition their due.

However, between techniques like mnemonics, spiral learning, spaced repetition, and more, the techniques are much richer than simply repeating things to memorize them.

We've established that the techniques are richer than most people believe, but what about the benefits of memorizing? It's a little political, but Michael Knox Beran's article In Defense of Memorization does discuss the benefits of the classic form of scholastic memorization. Why Memorization Helps Kids to Learn, by Dr. Rick Bavaria, sums up the benefits of memorization more succinctly and directly.

Don't get me wrong, I'm not arguing for the elimination of understanding. One of my favorite sites, as regular readers know, is BetterExplained.com, where clarity of understanding is their goal. Indeed, their post A Gentle Introduction To Learning Calculus not only clearly explains the basis of calculus, but makes a good argument that calculus should be taught earlier, not later.

Dan Meyer, while arguing that math class needs a makeover, actually discusses exactly how to overcome the impediments to understanding that most students have:


When you boil the arguments down, it comes down to the fact that memorization and understanding should work together in learning. In talking about how to get users to RTFM, the Passionate Users Blog makes an interesting point: the more you can get someone to understand something, the required memory work decreases.

Note also, that the graph at the above article runs from “High” to “Low”, not “Everything” to “Nothing”. In other words, there will always be a need for some memorization, regardless of your level of understanding. For example, since mental models are rarely perfect, there will usually be a need to memorize the exceptions to the rules.

What are your thoughts and experiences on the classic memorization vs. understanding argument?

Time for another one of Werner Miller's devious magic square puzzles!

For those who are new, here are the basic rules of the puzzle:

You're given the same set of 4 starting numbers grouped together in similar locations in a 3 by 3, 4 by 4, and 5 by 5 grids. The object is to fill out the grids with other numbers in such a way that the following conditions are satisfied when the grid is completed:

1) Only positive whole numbers are used.

2) No number is duplicated in a single grid (the same number may be used once in each of the 3 grids, however).

3) Each row, each column, and both diagonals of a single grid must total the same sum. All 3 grids in a puzzle, however, are not required to have the same sum (and usually won't). In other words, each grid must be a magic square, but all 3 grids do not have to have the same magic square total.

4) Puzzles will be posted here each Sunday, with the answers to a given puzzle being made available the following Sunday.

3 by 3 grid:


4 by 4 grid:


5 by 5 grid:


Can you get the answers to these magic square puzzles? If you can solve it before next Sunday, let me know in the comments! You can either describe your solution there, as best as you can, or link to a graphic of your solution.
––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Answers to last Sunday's puzzle:

3 by 3 grid:


4 by 4 grid:


5 by 5 grid:

Martin Gardner, the father of modern recreational mathematics and the man who coined the term mathemagician, has passed away. He was 95.

He's the only person ever to have his own category on Grey Matters. I respect him and his work greatly, and will miss him much, even though I never had an oppotunity to meet the man in person.

James Randi has already penned a heartfelt tribute to him.

The Martin Gardner documentary I first ran across in December 2008 is now up on Vimeo, and is a wonderful example of why so many regarded him so highly.



Update (May 23, 2010): James Grime, known to YouTube users as singingbanana has just posted a perfect tribute to Martin Gardner. It teaches how to make one of the master's greatest works, a flexagon:



Forgive me for keeping this blog post short, but it's the hardest one I've ever had to write.

Most methods of memorizing a stack, as a quick look through the Memorized Deck Online Toolbox will show, require intermediary mnemonics. Here's an approach I've been developing that uses no mnemonics!

This method is based on the spiral learning approach used in J. J. Hayes' approach to memorizing poetry, but modified for the needs of stack memorization.

1) Create a written list of your stack, containing both position (1 to 52) and card information. If you're using the Tamariz stack, you'd list 1-4C, 2-2H, 3-7D, and so on. Important: When making the list, add the cards and positions from 1 through 4 at the bottom, too. This is because many memorized deck tricks often require you think of the top card as being “after” the bottom card.

2) Look at the first card, and read it out loud. I usually say the position then the card, then the position again, such as “1, Four of Clubs, 1”. The cover it up the card, and repeat it from memory in the same way. If you made a mistake, look at the card and state it out loud again, then cover it up and try repeating it from memory. If you got the card correct, move to the next card and repeat this process. Do this for each card and position 1-52.

3) Next, you're going to repeat this process from the beginning of the stack, but two cards at a time. Start by reading positions 1 and 2 out loud from your list, such as “1, Four of Clubs, 1, 2, Two of Hearts, 2”, covering them up and repeating them from memory. Again, if you make a mistake, repeat with the same cards, and if you didn't make a mistake, move on to the next 2 card. Go through the whole deck. Don't forget to go beyond the cards at positions 51 and 52 by finishing with positions 1 and 2.

4-7) Go through the whole process again, but 3 cards at a time (finishing with positions 52, 1 and 2 as the last triplet), then 4 cards at a time (finishing with positions 49, 50, 51 and 52, then 1, 2, 3, and 4), then 5 cards at a time (finishing with positions 51, 52, 1, 2, and 3 as the last set), and finally 6 cards at a time (finishing with positions 49, 50, 51, 52, 1, and 2 as the last set).

8) Just before you go to bed at night, try and recall the entire stack by position and card.

9) As mentioned by J. J. Hayes, sleep on it! Sleeping will give your brain a chance to effectively “download” the memorized stack into long-term memory.

Once you can go through the stack 6 cards at a time, you'll be surprised how well you have the cards and their positions memorized. What this approach does is both emphasize the exact information you need to know, as well as strongly making the associations of each card by creating, breaking, then remaking associations of cards with each other. Also, you should never have to work through any external associations for the positions and cards. One should automatically bring up the other.

At this point, though, you're only half done. As Dennis Loomis points out in his Memorized Deck Mastery article, you should not only know the cards and their positions, but also what cards come before and after each other.

To achieve this, you're going to repeat the above process, but without the card positions. Just as before, you'll make a list of cards, remembering to include the top 4 cards at the bottom of the list, as well. You'll then go through 1 card at a time, and continue until you've gone through the whole process 6 cards at a time, reaclling the stack just before bed, and then sleeping on it.

Once you've gone through both of these processes, you'll not only be able to go through the whole deck from memory, but you also won't need to recall any mnemonics. From this point on, regular review and testing is really all you should need. Tools that aid in spiral learning, such as Loop&Learn and/or Verbatim, can help with both the initial learning and the regular review.

This approach is a great way to memorize any large set of information that you have to know perfectly. Besides things like card stacks and poems, you could learn any detailed list this way. For example, the Knight's Tour path taught in The Chrysalis of a Polymath could also be learned this way. Just as with the cards above, you'd make sure to include the first few positions again at the end of the list, since the cyclic nature is important here, too.

If you try this approach out, I'd love to hear your comments about it! Did it work better or worse for you than other methods? What other uses did you come up with?

This isn't today's full blog post. It's more of a quick note about an upcoming release that Grey Matters will be interested in. Another full post is coming later today.

L&L Publishing is offering a special pre-release price on their upcoming Barrie Richardson DVDs!

Barrie Richardson's two major works, Theatre of the Mind and Act Two, are already regarded as vital works in magic, mentalism and presentation.

Grey Matters readers already know of them because of not only the amazing math and memory feats contained within them, but also the skill and ingenuity in developing presentations that make an audience respond.

The contents of the DVDs are listed here, and you'll note that Volume 3 is called MENTAL AEROBICS: MEMORY AND METAPHOR. Yep - it's chock full of the kind of stuff we here at Grey Matters enjoy, like magic squares and memory feats.

Below is a video excerpt of Barrie performing Dollar Divination and Thoughts With Wings. Before you view it it, thought, I recommend reading the excellent post New Barrie Richardson DVDs: Pre-Review over at Mentalism Magazine, as it explains some very important points about the whole point of the DVDs.

Earlier this week, I received an interesting e-mail from Werner Miller, whom I've discussed before on Grey Matters, and whose books push the boundaries of mathematical magic in amazing ways.

The e-mail concerned a new puzzle he's developed that relates to magic squares.

You're given the same set of 4 starting numbers grouped together in similar locations in a 3 by 3, 4 by 4, and 5 by 5 grids. The object is to fill out the grids with other numbers in such a way that the following conditions are satisfied when the grid is completed:

1) Only positive whole numbers are used.

2) No number is duplicated in a single grid (the same number may be used once in each of the 3 grids, however).

3) Each row, each column, and both diagonals of a single grid must total the same sum. All 3 grids in a puzzle, however, are not required to have the same sum (and usually won't). In other words, each grid must be a magic square, but all 3 grids do not have to have the same magic square total.

He sent me several of these puzzles, and I've decided to post 1 each Sunday, with the answers to follow on the following Sunday. For example, here's today's puzzle, the answers to which will appear next Sunday (March 23, 2010).

3 by 3 grid:


4 by 4 grid:


5 by 5 grid:


Can you get the answers to these magic square puzzles? If you can solve it before next Sunday, let me know in the comments! You can either describe your solution there, as best as you can, or link to a graphic of your solution.

The May snippets are now ready!

• I'm adding some new stand-alone pages to Grey Matters. The first two of these pages are already set-up: Downloads (both free and commercial!) and Social Links. Click on either of these pages, and then click the [+/-] icon to open up the contents of the page. Many of these are new or updated links. Eventually, these will replace the accordion menus in the rightmost column.

• Jack Dutton, of irontree software, has just released ForeDate through Lybrary.com. ForeDate is highly-customizable Windows software that allows you to practice the Day of the Week For Any Date feat.

• While I'm discussing memory feat-related software, there are a few iPhone/iPod Touch apps I've recently discovered. First, there's MnemeCards (iTunes Link) to help training in competition-style deck memorization. Next, there's Tweeti Pi (iTunes Link), which tests your ability to memorize pi, then allows you to tweet your results. We wind up this section with How To Memorize Anything (iTunes Link), which is an app with memory tips and games.

• There's an upcoming online magic event called the Essential Magic Conference. There are numerous benefits to signing up, but at the moment, I'd like to draw your attention to their videos. Many of these focus on the psychology and presentation in magic on a scale I've rarely seen before.

I'd especially like to point out the interview with Apollo Robbins, of istealstuff.com. Among other things, he discusses the question of whether your magic is giving or taking from the audience. With a pickpocketing act like Apollo's, it would be all too easy to turn it into a show-off type of routine. Memory and mathematical feats, like the ones here on Grey Matters, could also easily suffer from the same type of presentation. Thankfully, Apollo discusses the process of overcoming this potentially fatal flaw.

Last year, I provided you with some great geeky reading in my Geeky Library post. With summer almost upon us, I thought it was a good time to revisit suggestions for an updated geeky library.

Caroline Taggart gives us a good start with two books. In i before e (except after c), there's a whole host of mnemonics for a wide variety of subjects. If you enjoyed Every Good Boy Deserves Fudge: The Book of Mnemonic Devices, which I reviewed a few years back, I think you'll also enjoy i before e (except after c).

Caroline Taggart's other book is I Used to Know That: Stuff You Forgot From School. In this one, you revisit all those rules and facts that teachers told you were important. It's quite fun to look back and ask yourself how important those things actually were, above and beyond knowing them for the test. If you want to remember any of these, the two books mentioned in the previous paragraph would be perfect companions.

Closely related to I Used to Know That is Michael Powell's Stuff You Should Have Learned At School. The subjects here are divided up by subject: English, Mathematics, Science, History, Geography, The Classics, and Music. It's funny how much reading these facts brings back memories of when you were first learning them in school.

Turning to numbers, Derrick Niederman has released a numeric encyclopedia called Number Freak: From 1 to 200- The Hidden Language of Numbers Revealed. While there are online sites with content similar to this book, such as The Number Correlation Database, Number Dictionary, Number Gossip, Numeropedia, Prime Curios, What's Special About This Number?, and Zoo of Numbers, the book itself is quite engaging by using not only examples, but many fun mathematical puzzles and challenges, as well. You can learn more about this book via Math Blog's interview with Derrick Niederman.

It's tough to talk about math, and not bring up Martin Gardner, of course. I could mention just about any book, but I'd like to draw particular attention to The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, where he takes some of the most popular topics from his previous columns, and brings them up to date in a huge volume of mathematics that it nonetheless interesting for everyone to enjoy! Thanks to Google Books, you can even browse through the book here.

Coffee table books are often designed to catch the eye, and engage the reader as they look through it. It's tough to imagine applying this style of work to math, but Clifford A. Pickover has managed to do just that with his simply titled The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Each subject is only one page long, and they're arranged chronologically, so as you read it, you can get a sense of the types of mathematical knowledge that each era in history held.

Recently, I added a new section to the Mental Gym that teaches how to solve the 15 puzzle. While developing this section, The Famous 15 Puzzle proved to be a very big help. I learned about variations I'd never even heard of, and the true story behind the puzzles development (nope, it didn't originate with Sam Loyd). Jerry Slocum has more about this book on his own site, and even Java-based versions of some of the puzzles in the book! Interestingly, the puzzle that comes with the box isn't on tracks, which was an important feature of earlier versions.

The last 3 books I'll mention in this entry are dedicated to particular areas of knowledge that geeks are sure to find interesting.

First, there's The Elements: A Visual Exploration of Every Known Atom in the Universe. What The Math Book above is to math, this book is to the periodic table of the elements. Try browsing through it yourself to see one of the most beautiful and engaging looks at the elements that make up our universe. The authors have even gone a step beyond just a mere book, by developing an Elements app (iTunes Link) specifically for the iPad.

Bruce Bueno de Mesquita gave an interesting TED talk about predicting Iran's future (YouTube Link). It was based on his work about predicting outcomes based on self-interest and influence. His book, The Predictioneer's Game: Using the Logic of Brazen Self-Interest to See and Shape the Future talks more about this as a whole. It's a field that is relatively new, but fascinating to examine nonetheless. Browsing through the book, and reading through the site, you begin to wonder about the possibilities of a burgeoning field, which is often exciting.

I'll wind up this entry with The Logic of Life: The Rational Economics of an Irrational World, Tim Harford's book on game theory applied to the modern world, with a particular focus on why things in daily life so often seem to defy logic. Over at the author's YouTube account, there are some fascinating examinations of things like racial segregation, overpaid bosses, and economics in dating. It's almost funny to see these complex ideas broken down, and see how simple they can really become.

Do you have any suggestions of your own? Have you read any of these books, and would like to give your own review? Let's hear about it in the comments!

Back in March, I mentioned the Pi Day Magic Trick, brought to you by Brian Brushwood and James Grime. Finally the secret has been revealed!

OK, I've actually discussed the principle of this effect before, including in my Magic of 9 post and my age-guessing post.

However, Brian himself has finally discusses the exact nature of the secret in the video below, and afterwards talks to James Grime about this routine briefly:



I think it's interesting that James Grime was able to use this principle to fool several well-posted math professors. It just goes to show you the power of a simple principle when it's properly disguised.

Another way to disguise the principle is by using subtraction instead of multiplication. Check out 7-up's version of the same effect with Fido Dido, and see if you can discern how the same principle applies here.

Since he mentions that you can still experience this effect on Twitter, this seems like a good time to mention that you can now follow Grey Matters on Twitter at: http://twitter.com/grey_matter

(UPDATE - July 14, 2011: Verbatim has been updated and improved. Click here to learn about the update!)

Many of you know that I've been working on converting my Verbatim web app (What is a web app?) to a native app (The kind you download from the iTunes App Store). However, a new app has just been released that blows Verbatim out of the water!

It's called Loop&Learn (iTunes Link), and it's so new, I couldn't have included it in Thursday's iPhone post, because it wasn't available yet!

The focus of Verbatim is the poem memorization approach I first learned in J. J. Hayes' article, in which you memorize a poem (or monologues, presentations, scripture, scripts, song lyrics, speeches, etc.) by learning the lines in larger and larger groups each time you go through it. Loop&Learn is the first program I've run across that allows you to use this same approach for learning.

Had the Loop&Learn app just stopped there, I might just think of it as competition for Verbatim. However, I haven't even begun to scratch the surface! Much like Verbatim, you start by entering the text of what you want to remember. Besides things like poems and speeches, this could also be any sort of list, such as presidents, elements, and so on.

Here's where the differences begin. After giving each “loop” some basic information, such as title, tags, and description, you record a snippet of sound for each item on the list (determined by where the RETURNs and/or line ends are in the text you entered earlier). If you're running Loop&Learn on an iPod Touch, you'll need to purchase an external microphone (preferably ones with built-in earphones) to make us of this app. The sounds will be played back as you go through your loop later.

Optionally, you can also include graphics to go with each sound snippet. This is a very handy feature, as it not only allows you to have pictures of the things you're memorizing, but can also be used to depict mnemonic associations to further strengthen your ability to remember the contents of the loop!

When you play back your loop, the most basic way is to play the whole thing straight through. However, you can also play back individual segments individually, or in groups. You can also choose to either have it play the segments as if you're repeating them along with the app, or to pause, so you can repeat after the app. The ability to play segments in groups and to have a listen-and-repeat mode makes Loop&Learn perfect for the approach in the J. J. Hayes' article I mentioned earlier.

Already, this is an extraordinary app, but there's one other feature that's very important: it's social networking features! Once you've developed your own loops, you can upload them to the Loop&Learn library to share with others, and download loops developed by others, too. In addition, you can rate each loop, and announce loops via Twitter and e-mail., all within the app itself!

Below is a video to give you a better idea of the look, feel, and capabilities of Loop&Learn. You can find more about Loop&Learn in their Video Tour page and their YouTube channel.


Getting back to the Verbatim comparison, Verbatim does have a way of reviewing items that Loop&Learn doesn't offer. In Verbatim, you can review your pieces getting hints by either filling in blanks or seeing only the first letter. If you enjoy this feature, and would like to see it in a native app, iByMemory (iTunes Link) offers this functionality, and would make a good companion app.

If you go into Verbatim itself, click on Manual, then on Related resources, you'll find a whole host of resources that are useful for not only Verbatim, but Loop&Learn and iByMemory, as well!

All in all, I find Loop&Learn to be an incredible value, and highly recommend this app, especially because of the incredible multimedia and social features. It's easy to develop, and fun and effective to use.

On a related note, I've stopped development on a native app version of Verbatim. The Verbatim web app will remain available, but I am now free to pursue other native app projects that have been put off until now.