I've got plenty of news and free goodies to share with you today!

First, I have an old goodie. The Grey Matters Dashboard widget I mentioned in my last post has been posted on Apple's website, in their Dashboard Widgets Download section!

Speaking of widgets, this next announcement will interest those of you who are practicing the Day of the Week For Any Date feat. I've also created a Dashboard widget that generates random dates to quiz you on this feat, and will tell you whether you're right or wrong. You can set it to quiz you on dates in the 1900s, 2000s, or a full range of years from 1600-2399. You can download the Date Quiz widget at this link, from the Downloads section in the rightmost column, or from Apple's Dashboard Downloads section. Here's what the widget looks like:

Probably the most popular download on this site would have to be my list of Memory Effects (PDF, opens in new window). I just updated that list yesterday, and it is available at this link, in the Downloads section on the right, and at Scribd.com. As a matter of fact, thanks to Scribd.com's new iPaper feature, you can embed this file on your own website as easily as a YouTube video:

Read this doc on Scribd: MemoryEffects

With all these downloadable goodies, wouldn't it be great to keep track of all this site's downloads as they are updated or made available? I've made that much easier with the new downloads label! Here you'll find all the information on free downloads from this site. For example, many of you may know that I sell a memory training course called Train Your Brain and Entertain!, but did you know I offer a free lite version of the software for both Mac OS X and Windows? You can try it out and see if you would like the full course!

I even have some free goodies for you from other sites. Our old friends at Sporcle.com have been busy. They've added two new quizzes to their games section, one on Greek Gods and the other on Roman Gods. They've also created a new index page which lists games by their popularity that day. If you look closely on the page, you can also find some new (hidden?) quizzes, such as Best Picture Oscar winners from 1928-1969, Best Picture Oscar winners from 1970-2006 and a rather amusing quiz on the countries in Antarctica (which is surprisingly educational).

I'll wind up with an item for fans of the Knight's Tour. If you've learned it here, but maybe the Mental Gym's version, the iPhone version, the offline Mac OS X version, or the offline Windows version weren't exactly your style, then perhaps you might like the style of Evgeny Karataev's 3-D Knight's Tour! It's a little unusual at first, but it's easy to get used to, and you can use the techniques you learned here to win this version!

Enjoy these free goodies, and I'll see you again on Sunday, if they aren't still keeping you busy!

Just over two years ago, I introduced the Grey Matters Dashboard Widget, which displays the feed from this site. It's about time for an update, isn't it?

Thanks to the new intel-based iMac, and Dashcode, I've finally been able to update it! To be able to use it, you must be running Mac OS X 10.4 (Tiger) or 10.5 (Leopard).

The first change is the new look:

The other major change is the addition of a brief description for each post, so that you can quickly get a better idea of each post's topic. When you find a post that interests you, just click on the title, and it will open up in your default browser.

This widget is free, and is available for download by clicking this link. You'll also always be able to find it in the rightmost column under "DOWNLOADS", as the very first item of several free goodies! Once you click the link, the widget will download and activate itself in your dashboard. Click "Keep", and you're all set!

As I mentioned in the previous post, this is just the first of some great new changes for Grey Matters.

Sorry about the late post today, but big things are afoot here at Grey Matters Central!

Ever since I started Grey Matters, it's all been produced by an 800 MHz G4 iMac. As of this post, that is no longer true! The new system here is a 2.4 GHz Intel Core 2 Duo iMac!

This will allow me to create new projects and take Grey Matters in directions that were impossible or impractical before. What kind of projects? Well, you'll just have to wait and see!

The cover story of the March 2008 issue of Reader's Digest is an article featuring 20 quick tips to improve your memory!

The article itself, available for free online, is titled Retrain Your Brain: 20 Memory Techniques You'll Never Forget. These are some great, simple ideas you can use to remember things like people's names, where you left your glasses, to-do lists, website passwords, and words that are on the tip of your tongue.

The advice in the article comes from several well-known memory experts and authors, including Harry Lorayne (author of Ageless Memory: Secrets for Keeping Yourever Mind Young Forever), Gini Scott (author of 30 Days to a More Powerful Memory) and Carol Vorderman (author of Super Brain: 101 Easy Ways to a More Agile Mind). Who Remembers What? is an interesting sidebar article about the different nature of men's and women's memories.

In the past, Reader's Digest featured many other great articles on memory that are still available on their site. Beyond just standard memory techniques, there are some great ideas in Improve and Maintain Your Memory: 27 tricks to keep your brain in shape, such as listening to a book as you exercise, or snacking on grapes instead of cookies (yes, this really does help your memory!). Having trouble remembering people's names? Check out January 20008's article What's Your Name Again?.

If you prefer challenging your memory to just learning about it, try Memory Master. In this game, you're shown several pictures for a brief time. After that, you're shown twice as many pictures, and you have to remember which objects you were originally shown.

That's all for today's entry. Granted, this isn't one of my longer posts, but I figured that wouldn't be appropriate for an article about Reader's Digest.

When you watch your favorite crime drama, what is happening? Yes, of course, they're trying to catch the murderer, but do they just follow the evidence to the criminal, and end with their answer? No, usually they're looking not only to find the murderer, but to understand the events and evidence that led to the motive, the means and the opportunity.

Mathematics is very similar to this in many respects. Mathematics seeks to understand relationships in a precise a manner as possible, and seeks not just the answers themselves, but an understanding of the answers. Just like any good crime drama, this search has led to a few places that seem to be dead ends, but actually open up new directions of study.

Back in 1985, the PBS program NOVA focused on the history of these mathematical brick walls, in a program called Mathematical Mystery Tour. It's just under 54 minutes long, and is presented in 8 parts below (just click the right arrow to go to the next part). It's a fascinating look at the challenges that mathematicians have faced and even some that they still face.

This special doesn't have time to go into too much detail on many of its subjects, so I'm including some links below where you can go for a closer look:

* Better Explained (Two especially relevant articles are How to Develop a Mindset for Math and A Quirky Introduction To Number Systems)

* Euclid's Elements

* Fermat's Last Theorem (video on how it was finally solved in 1994)

* Imagining The Tenth Dimension (with YouTube link)

* Millenium Problems (unsolved mathematical problems)

About a year ago, I pondered whether free flashcard memorization would be the killer app of 2007. OK, it wasn't the killer app of 2007, but there was plenty of development in that field. Since it's been a year, I figured now was a great time for an update on the free flashcard scene.

All the programs and sitess I list below employ the idea of spaced repetition that was first developed by Sebastian Leitner, which helps lock in the information for the long term.

Anki: This offline program is made to run on Windows, Mac OS X and Debian, with the Linux source code available. It's easy to add and edit cards, and it even charts your progress!

DingsBums: A Java-based offline, program, this unusually-named flashcard program is flexible and easy to use!

Flashcard Exchange: This is a huge site with over 6 million flashcards already. While it is free, if you pay an additional fee, it will employ Leitner's spaced-repetition.

Flashcard Friends: What do you get when you cross MySpace or Facebook with a flashcard site? This!

Fresh Memory: This is an offline, platform-independent program. You can not only test your knowledge via flashcards, but via multiple choice questions, as well.

Mnemosyne: Named after the Greek goddess of memory, this offline program that runs on Windows, Linux and Mac OS X will support graphics, unicode (great for learning languages like Japanese or Arabic!), LaTeX (for mathematical formulas) and even three-sided cards!

Rememberize: This is a very straightforward flashcard site. There's few frills, but it will help you learn. There's a great tour of the site to help get you familiar with it.

Study Cell: If you have a Java-enabled cell phone, you can create quizzes and flashcards that will let you test yourself on the go!

Study Tag: On this free site, you organize facts into lessons, and lessons into complete courses. The best way to get an idea of this site is to take its free 6-part introductory course.

You Know The Drill: This is a free flashcard website that will allow you to quiz yourself on any type of information, but has tools specially geared towards languages.

If you're interested in more information on free flashcard software, check out my previous articles, Quizlet, 10 Online Memory Tools...For Free!, Review: Ebbinghaus 1.5 (Mac OS X), Create Free Flashcards and Quizzes, Free Flashcards Galore! and New Features at ProProfs!.

Did you ever receive that e-mail that claims that some state's legislature (usually Alabama) has recently passed a law redefining Pi? As Snopes notes, this is false. However, 111 years ago this week, the Indiana state legislature came very close to doing just that.

The change wasn't done to make Pi consistent with the Bible, but rather to deal with the age-old problem of trying to construct a square with the same area as a given circle, otherwise known as squaring the circle. Because Pi is transcendental, we now know that this is impossible.

However, what if you were working before the 19th century, before this was proved, and before calculus? There are numerous ways to get very, very close to the circle, and it's not unreasonable to think that just a few more adjustments would be needed to perfect the process. Let me show you how the madness takes hold. One of the best approaches I've found for seemingly squaring the circle was detailed by Stu Savory in April, 2005 (26 days too late if you ask me). He does a good job explaining, but I'd like to break it down a little differently for my readers. It uses only a compass and a straightedge.

We start by working on the line that will become the circle's diameter. Draw a horizontal line, and mark the rightmost end as R. Open the compass to a width of an inch or so, place the spike of the compass at R, and mark point T on the line to the left of R. Now place the spike on T, and mark a point to the left of T (not labeled). Place the spike on this unlabeled point, and mark one more point to its left, which will be labeled O. In short, OT should be exactly twice as long as RT.

Now, open your compass to length OR, with the spike at O. Use the compass to draw a circle. Mark as P the point where the horizontal line intersects with the circle on the left. Our final step concerning the diameter line itself (POR) is to bisect OP (Java required), marking OP's midpoint as H.

Next, we're going to make some measurements above POR. Construct a vertical line from point T. Mark the point at which this line intersects the circle as Q. Using the same length as QT, draw a chord starting at R. Mark the other end of this chord as S (length of QT=length of RS). Draw a line from S to P. Construct a line parallel to RS through point T (Java required), marking the point where it intersects SP as N. then construct another line parallel to RS through point O, marking the point where it intersects SP as M.

Finally, we're going to work below PQR, and get our square's base measurement. Set your compass open to length PM, with the spike at P. Swing the compass to mark the point K, below POR, on the circle's circumference, and draw the PK chord (PK=PM). Set your compass to the length of MN, and then construct a tangent at a right angle to POR, marking the bottom-most point of this tangent as L. Draw lines RL, RK and KL. Finally, construct a line parallel to LK, going through point C. Mark as D the point where this line intersects RL.

So what does this give us? RD is the base of our square we need. Let's work through this next part slowly.

The formula for the area of the circle, as we all learned in school, is Pi * r². The formula for a square with a base of x is x². Because of the way in which these measurements were done, RD will always be a straight line about 1.772453 times longer than RO, the circle's radius. As it happens, this is the square root of Pi!

In other words, the length of line RD equals the circle's radius times the square root of Pi ((sqrt(pi))*r). Using RD as the base of a square, we can get the square's area by multiplying this by itself! Squaring the square root of Pi gives us Pi, of course, and squaring r gives us r², or Pi * r² for the area of this square!

Everything seems right, but can you figure out why this isn't a true squaring of the circle? What actually winds up happening is that RD is the square root of 355/113, a common rational fraction used for Pi early on. 355/113=3.1415929204, which is accurate to 6 places after the decimal point! It's square root is 1.7724539262, which also accurate to 6 places after the decimal point. This gives a square and a circle that are so close in area that, using a pencil that could mark in 1/100 of an inch, you would have to create a circle more than 4 miles in diameter to see your error!

If you look at the bottom of Stu Savory's blog entry, he has a quicker and more interesting way of apparently squaring the circle with a compass, straightedge and a coin!

Are you beginning to see why so many people were obsessed with squaring the circle? When you don't know that Pi goes on forever, it seems to be a matter of just making a few more adjustments to make a historical mathematical breakthrough! This was actually so common that, at one point, the insane belief that one could square a circle was listed in medical texts as morbidus cyclometricus!

I hope you've enjoyed this rather odd look at a weird bit of mathematical history.

I have good news for owners of the iPhone and/or the iPod Touch. I've just doubled the size of the iPhone/iPod Touch Mental Gym!

In addition to the four challenges that were already there, four new brain-bending games have been added:

* 15 Puzzle: While there have been several versions of the 15 puzzle made for the iPhone, this is the first one I've found that both employs the classic numbered squares, and will properly scramble the puzzle for you. This makes it the perfect tool, not only for learning to solve this classic puzzle, but also for learning how to arrange a magic square on it, as in my video.

* Calc10: This is one of those games that's so simple, it proves surprisingly addictive. In this game, you're given four random numbers from 0-9, the four basic arithmetic operators (addition, subtraction, multiplication and division) and a memory function. Using each of the four numbers only once, and finishing with the memory function empty, you must figure out how to arrive at a total of 10 as quickly as you can. Sometimes, this is very easy, such as when you get the numbers 0, 8, 7 and 2. Here, you would simply hit 7 * 0, giving 0, then add 8 and add 2, resulting in 10. It can be get much harder, though, like the time I got the numbers 9, 6, 6 and 9. Can you make 10 out of that (remember, you must use all four numbers)? There's a reset function, which clears the calcuator and gives you back all your numbers, but doesn't reset the clock. Finally, the solution function can be used when you can't solve the puzzle. This is also very helpful in teaching you approaches you may not have previously considered.

* Lights Off: As you might guess from the name, this is a version of the puzzle best known as Lights Out. Any time you press a lighted button, it will turn off, and pressing a dark button will light it. The object is to turn off all the lights on the board, which would be simple if it weren't for the fact that touching any button will not only flip its own state, but that of the buttons directly above it, below it, to its right and to its left. Ideally, you want to shut all the lights off in under 15 moves. You can learn to do this in two steps, first by learning how to solve the puzzle, then by learning how to solve it in the minimum number of moves.

* Peggy: This is the classic Peg Solitaire game, played on the classic English board. The only way to move a peg is by jumping over another peg that it directly above it, below it, to its right or to its left into an empty space. When a peg is jumped, that peg is removed from the board. The challenge is to finish with only one peg left on the board, ideally in the same space that was initially left empty. While Wikipedia lists many great peg solitaire links, I would particularly like to draw your attention to George Bell's Peg Solitaire Page, for those who want to understand more about the game itself.

Please take some time to play and enjoy these games, and let me know if you other suggestions for the iPhone/iPod Touch Mental Gym.

Oh, and for those of you wondering about my 9, 6, 6, 9 challenge from the Calc10 section, here's the solution. First, you would click on 6, then divide it by 9, giving an answer of 2/3. This is stored in the memory function. Next, you add 6 + 9, which is 15. Now, you hit the multiplication button and memory. This is 15 times 2/3, which is 10!