I've talked briefly about Mental Math tricks before, but I've never really had a good source for a large variety of mental math feats.

I'm here to tell you that thanks to some like-minded fellow geeks, that problem has been solved.

First, there's BEATCalc at the Math Forum. BEATCalc's main incarnation is as a newsletter sent out regularly with a trick that will let you impress your friends by doing math quicker than they can do it with a calculator! At this writing, they list 518 math tricks in all. If you want to get started with some of the impressive ones right away, my personal favorites are "Adding a series of ten numbers" and "Adding a series of doubles".

Another great read on mental math is "The Psychology of Rapid Mental Calculators". While there aren't as many routines here, the caliber of the featured effects is much higher. The dice that are mentioned in the Magic Dice Trick routine, it should be noted, are available commercially and inexpensively.

There is also a good selection of some impressive mental math feats over at Mentat Wiki (which I also recommend for its memory feats, of course), including alternative many alternative techniques taught in the above Psychology article.

However, for a comprehensive course in amazing mental math is the Vedic Mathematics course. No, you don't have to change your religion, and it's not "new math". It's just the old math, taught with some amazing shortcuts. In Vedic mathematics, there are several "sutras", or general principles, which are taught in general, and then how to apply them in specific cases. If you actually take the time to study and practice each step, you'll be amazed how quickly you'll pick up the principles.

After going only a short way throught the Vedic Mathematics course, I couldn't help but shake my head and wonder why we aren't teaching our children things like this and memory techniques in our schools. On the other hand, the longer these tecniques are considered unusual, the longer you and I can amaze others with them!

I need to be more careful of what I joke about. In a previous post, I jokingly commented about finding a Sudoku Knight's Tour.

Guess what? They do exist! Over at the Border's Chess Club site, Dan Thomasson has created and posted a 25 by 25 Sudoku Knight's Tour. In each row, column and 5 by 5 square, the numbers 1-25 appear only once each. Also, a knight can tour any of the 5 by 5 squares and hit the squares in order (From square 1, he can move to square 2, then square 3, and so on up to 25).

Not impressed? He has also created a 36 by 36 Sudoku Knight's Tour with the same constraints for the numbers 1-36!

Attention Mac OS 10.4 Tiger users! You can now access the Grey Matters blog directly from the Dashboard!

You can download the Grey Matters Widget for free by clicking here. It has also been added to the "Downloads" section over on the side.

Mac OS X 10.4 Tiger is required. If you're using Safari, click the download link. When the widget download is complete, show Dashboard, click the Plus sign to display the Widget Bar and click the widget's icon in the Widget Bar to open it. If you're using a browser other than Safari, click the download link. When the widget download is complete, unarchive it and place it in /Library/Widgets/ in your home folder. show Dashboard, click the Plus sign to display the Widget Bar and click the widget's icon in the Widget Bar to open it.

Most of the widget is a scrollable list of all the entries on this blog. Each title entry is listed according to its posting date. Clicking on a title will open up a browser window showing that article on this site. If the title itself isn't enough to help you find what you're looking for, you can search this entire site by entering a term in the search window in the lower left of the widget. When you hit return, a browser window will open up, and show you the search results from this site.

Clicking on the highlighted "i" in the corner will flip the widget to its back, which will give more informatio about the site. Clicking on the title itself will open a browser window and display the Grey Matters Site Feed.

This started out as just a personal project, but I was satisfied enough with the results, that I decided to add it to my blog. I hope you enjoy it, and find it useful!


Update: You can now also download the Grey Matters Widget from Apple's Dashboard Download section! Thank you, Apple!

As with many other challenging feats, there are those who have to push the boundaries of Sudoku. If you're just starting out, you might want to refer to my earlier Sudoku introduction instead.

For those who are curious about where else Sudoku can go, read on!

The first obvious variation would be to use letters instead of numbers. This would allow a square of 25 by 25 to be built! If you've tackled standard Sudoku puzzles before, you can see whether you're ready for an alphabetical Sudoku puzzle here.

Since magic squares are brought to mind, here's a PDF file of a Sudoku puzzle with the added constraint that 1-9 can only appear once each in each of the main diagonals!

Is it perhaps the 2-dimensional nature of Sudoku that no longer challenges you? If so, try the Dion Cube (PDF), in which you solve not only the faces, but the insides of the puzzle, as well! Don't worry, though, as there is a solution available, but no peeking before you try it, OK?

If those don't pose a challenge for you, then I suggest reading Sudoku Variations over at MAA Online, which has enough variations to keep you busy for a long time. A personal favorite from this page is the Domino Sudoku puzzle which is pictured above.

Even magicians are realizing that Sudoku opens up a world of new presentations. Just as they did with Rubik's Cube, magicians are finding ways to capitalize on Sudoku's popularity. Magician Justin M. Monehen has a new routine available called Sudoku: The Ultimate Mental Workout. In this routine, a random Sudoku puzzle is chosen from a puzzle book, and the magician studies the puzzle for a few seconds, and is then able to solve it in seconds! Several variations are included, including one in which an audience member intentionally miswrites one of the numbers, and the magician is able to point out which one it was.

I think part of what fascinates me about these Sudoku variations is the creativity involved in coming up with them, as well as working the solutions out to make sure they're even possible. Until I hear of a Knight's Tour on a Sudoku board, that's all for now!

Just a quick note today. The cover story of the January 16, 2006 issue of TIME Magazine is called How To Sharpen Your Mind. Check it out!

Suuji wa dokushin ni kagiru!

Loosely translated from the Japanse, this means, "Each number remains single." Thankfully, this phrase has been shortened in one simple word: "Sudoku."

Anyone who knows me, even those of you just know me just from this blog, had to realize that it wouldn't be long before I became addicted to Sudoku.

Even though it's becoming the hottest thing to hit the puzzle world since the Rubik's Cube, I'll go over the basics for the one or two people that haven't heard of it yet. Sudoku is a puzzle played (usually) on a 9 by 9 grid, which contains nine 3 by 3 smaller squares within. As in the picture above, a puzzle will have some numbers already filled in. The object is to fill in all the empty boxes, so that each row, column and 3 by 3 box contain each of the numbers 1 through 9 once, and only once. If you're not clear, this visual explanation of Sudoku rules will be of some help.

Despite the numbers, Sudoku really isn't a mathematical puzzle. Rather, it is a logical one. Instead of numbers, you could replace them with letters, symbols or just about anything else, and the rules and strategies wouldn't change.

Figuring out which one of the various strategies you need, and exactly how it will apply is what makes Sudoku so fascinating and challenging.

There are only two strategies that allow you to deduce a single number, "Naked Singles" and "Hidden Singles".

However, just using these strategies alone may not help solve a Sudoku puzzle. Often, you'll need other techniques to help. These other strategies help reduce the number of candidates for each square. The more strategies you know, and the more advanced they are, the more difficult the Sudoku grid you can solve.

The toughest ones are the ones that require guesses, with some trial-and-error involved. For now, I suggest starting with the simpler Soduko puzzles, and familiarizing yourself with a solid, basic approach (PDF).

Fortunately, there's no shortage of puzzles. You can find them online, in books, on your OS X, Linux or Windows computer, and even in your shirt pocket!

By the way, if you're wondering...yes, theat one in the upper left corner is a real Sudoku puzzle, and it is solvable.

Having been a friend of Lew Brooks for years, I was only too glad to recommend his Stack Attack book. I've also mentioned it in previous entries here. So, it gives me great pleasure to announce the release of the Stack Attack DVD!

This isn't just any second rate DVD, either. It has been professionally produced by Bob Kohler Magic.

The DVD, as well as the book, are focused, as you can guess from the title, on the use of stack. Just that single statement will scare away any number of purists in card magic. However, Lew opens with his thoughts on why he uses stacks, and why they shouldn't be avoided. There are many in magic who, if they only thing they paid attention to in the DVD were these opening remarks, they would be far better off.

The presentations are all in the form of charming, amusing and engaging stories. These are presentations that really hook the audience's interest, and keep it through the entire routine. Too often, story tricks have a reputation as being long, boring affairs, but Lew has obviously overcome those hurdles, as well.

One particular trick that I should mention is the "Four Second Stack". Lew recently performed this routine for Alan Ackerman, who took it as a completely legitimate demonstration of card stacking! You don't fool someone as knowledgeable as Alan Ackerman without thinking strongly about what you're doing. You may even find yourself being taken in at several points in the video.

One of the best things about this video, is that every aspect of the video is strong. You learn great routines, ingenious and subtle methods, taught well, and in great detail. Every trick on this video is well within the reach of the average magician, yet magicians of any skill level will walk away with a new perspective on their magic. For example, highly skilled magicians who use a memorized deck will find some very valuable tools on this DVD.

Will you learn the latest hottest moves on this video? No. Will you learn amazing routines that are easy to do, and be able to perform them in a way that will lock your performance in an audience's memory? Yes.

Video of memory grandmaster Scott Hagwood can be seen over on FOX News's "FoxFan Central" page, talking about his new book, "Memory Power".

Sherman, it's time to set the way-back machine to the early 1980s, to remember a popular game called Simon.

Simon was game that tested your memory by challenging you to recall an ever-lengthening sequence of flashing lights. The original version looked like this. Over the years, it has had several makeovers, including a transparent look, on it's way to the current incarnation.

If you want to experience what it was like to play Simon, try this Flash version. Go ahead and try a few games, and come back after it drives you mad.

You're back? That was quick. Would you like to score better on Simon than you did before? Simple...write the sequence down! How's the computer going to know?

OK, anyone who has read more than one entry knows that I have a method up my sleeve for helping do the actual memory work.

First, you need to be familiar with the Link system, the most basic of memory techniques. Practice this until you can remember, say, a list of 20 common nouns and verbs, backwards and forwards.

Next, we need a simple way to make the colors more memorable. To do this, we're going to reduce the colors to their first letters: R(ed), G(reen), Y(ellow) and B(lue). To make this even easier, we're going to remember the colors in pairs, and give each of the 16 possible color pairs a unique, memorable image:

BB=BB (picture a BB Gun)
BG=Bag
BR=Bear
BY=Boy

GB=Goblin
GG=Gag (someone choking)
GR=Gear
GY=Guy

RB=Rabies (picture a dog foaming at the mouth)
RG=Rug
RR=Railroad
RY=Roy (Siegfried & Roy)

YB=Young & Beautiful (obvious)
YG=Yoga
YR=Year (Calendar)
YY=Yo-Yo

When the game begins, you simply think of the first color. When the next color comes up, you put it together with the first color to create your first image. To help remember the first image, I always picture whatever it is coming out of my mouth, which usually helps to make the image very memorable.

For example, imagine you're playing a game in which the sequence goes: RGBRYYGBBYRR...

Here's how you would picture this at each step? Like this:

1: Simply remember "Red"
2: RG="Rug", so you imagine a big long rug coming out of your mouth.
3: "Mouth, Rug, Blue"
4: "Mouth, Rug, Bear" (BR=Bear). At this step, you're picturing your mouth, out of which comes a big rug, which unrolls to reveal a bear.
5: "Mouth, Rug, Bear, Yellow" (Yellow Bear?)
6: "Mouth, Rug, Bear, Yo-yo" Hmm, the bear goes off to play with a yo-yo, maybe?
7: "Mouth, Rug, Bear, Yo-yo, Green"
8: "Mouth, Rug, Bear, Yo-yo, Goblin"
9: "Mouth, Rug, Bear, Yo-yo, Goblin, Blue"
10: "Mouth, Rug, Bear, Yo-yo, Goblin, Boy"
11: "Mouth, Rug, Bear, Yo-yo, Goblin, Boy, Red"
12: "Mouth, Rug, Bear, Yo-yo, Goblin, Boy, Railroad"

As you can see, at each of the odd steps, you simply use the color as a modifier, until you move on to the next step and get the full picture.

Also, you can start to see the true beauty of this mnemonic system. Even though memorizing our example list only required 6 words (not including "mouth", which didn't represent a color pair), it easily brings to mind a 12-light sequence! If you've practiced the Link system up to 15 words, a sequence of 30 lights should pose no challenge with some practice.

With a little work, you could adapt this to Simon², or even the more tactile Simon Stix.

Now, go back to that Flash version of Simon, and see if you don't start scoring better.

Sorry, but it can still drive you mad. No technique has been discovered to stop that.

For those who like the idea of remembering things, but not spending time practicing mnemonic techniques, the people at the Interesting Niche Ideas blog may have just the thing for you. Thanks to them, you can learn Five Ways to Remember Practically Everything quickly and easily!

Will this article help you win the World Memory Championship? No. Will it help you bring back all the correct groceries? Yes.

Happy New Year to one and all!

With a new year, comes a new feature to Grey Matters! If you take a look to the column on your right, between "Previous Posts" and "Archives", you'll see our new feature - "Posts by Category".

I've added this feature, to help you find posts on your favorite topics. Currently, there are 5 Categories:

Memory
Math (formerly "Mental Math")
Products
Puzzles
Site Features

Clicking on each of these links will show a list of links to all the posts relevant to that particular category. More categories will be added as needed, of course.

Here's to best wishes in 2006 to you all!


Update: I have added a 6th category, which is called Performance. I've changed Mental Math to simply Math. Finally, I've also divided the larger categories up into sub-categories so as to make posts of interest easier to find.