No, I don't have the headline backwards. StackView is still only available as a Windows program, but recent developments have actually made it easier and better for Macs to run StackView!

With the older PowerPC-based Macs, the only way to run Windows programs was via emulators like Virtual PC. Once the Macs switched to Intel-based chips, the emulators became even easier to set up.

The current version of the operating system, Mac OS X Leopard, even has a built-in program, called Boot Camp, that lets you boot your Mac up as a Windows computer. There are also programs like Parallels, that let you run Windows and OS X simultaneously.

These all worked well enough, but they all suffered from the same flaw. Whether the emulators were free or commercially available, they all required you to purchase a copy of Windows. If a Mac user just wanted to run StackView, this expense is a little hard to justify.

Back in September, Nick Pudar announced a way to run StackView on the Mac that keeps everything very inexpensive! The big secret is running Darwine, a version of Wine that runs on Intel-based Macs (via Darwin, thus the name).

The big advantage to Darwine is that it's free and you don't need to purchase Windows! It is important, however, to keep in mind that both Darwine and StackView are developed and maintained as personal projects, so they won't always work perfectly together. Also, the faster your Intel-based Mac is, the better experience you'll have with both programs.

You should download Darwine first (I highly recommend the most recent stable build - 1.0.1 at this writing), and setting it up first. From there, download StackView and set it up. Once that's done, you can run it an use it.

The PDF manual is good, but I also suggest accessing the StackView Musings blog and going through the tutorials, as well.

One last note: In the StackView on the Mac post itself, ignore the 2nd paragraph. That guy sounds like a real jerk. ;)

Today is just a short entry, as I've been spending my time working an upcoming puzzle for Scott's Puzzles.

If you'd like to be ready when it's posted early Wednesday morning:

1) Practice solving the 15 puzzle with these tips. If you don't have a 15 puzzle at home, play this one at the Mental Gym.

2) Once you're comfortable with the basic 15 puzzle, try 15 puzzles of varying sizes at Scott's Puzzles.

3) Learn to deal with the ingenious version known as Rate Your Mind, Pal. Click the link, then click on the Rate Your Mind, Pal link, and then press the S key to shuffle the puzzle. Knowing this version's history may also help.

That's all the clues you get for now!

Update (Oct. 29, 2008): The puzzle itself has now been posted!

This edition of snippets is dedicated to the humble deck of playing cards!

• If you took all the standard decks of 52 cards that have ever existed into consideration, has every possible arrangement of those cards existed at some point? As this proof shows, it's highly doubtful. There are more arrangements possible for a 52-card deck than there are seconds between now and the time the universe began, especially taking into consideration that humans have existed for only a very small fraction of that time, and the standard deck of 52 cards has only existed for a small fraction of humanity's existence.

• What exactly happens when you mix cards in various ways? This can be answered with the free Windows program StackView! I've mentioned this program numerous times before, but it's worth noting again for those who missed it previously. The related blog, StackView Musings, is also a great source of tricks (especially for those of you who like memorized deck tricks) and information about the program itself.

• Speaking of memorized deck tricks, many of them start with a named card. How do you practice a trick with truly random cards? Random.org's Playing Card Shuffler is an effective answer. Random.org is a site featuring what is a rarity in computer programming - a true random number generator!

• For those of you into both programming and playing cards, I have a few treats for you. If you've ever wondered how to work with playing cards in CSS, Brainjar has the answer! Not only can you use programming techniques to get a browser to understand cards better, but you can also use programming techniques to get your brain to remember cards better, too! For memory competitions, there are more effective techniques, but this method combined with a binary memory system could have some interesting and creative uses.

• Purely for fun, did you know that many cards and card combinations have their own nicknames? Many of the unusual and interesting stories behind them are linked on this Wikipedia page, but you'll have to Google the majority of them to discover the stories behind them.

Don't forget, for those interested in memorizing a stacked deck, I still maintain and update the Memorized Deck Online Toolbox.

Do any of you have more playing card resources you've found interesting and/or useful? Let me know about them in the comments!

With so many magicians interested in mathematical magic, it's not surprising that so many magicians have developed so many original mathematical principles.

One magician, P. Howard Lyons, who is known among magicians as the publisher of Ibidem magazine, developed an amazing procedure for divisibility by 7 that's different from the other such tests I described last month.

Lyons first revealed the following method in Martin Gardner's The Unexpected Hanging And Other Mathematical Diversions. Before learning this method, you should know your multiples of 7 up to 100 by heart.

As an example number, let's see if 3,846,532,658,790 is divisible by 7.

1) Break the number up into 2-digit pairs, working from right to left. Since our example contains 13 digits, the leftmost one will wind up as a solo:

3 84 65 32 65 87 90

2) Next, below each number pair, write the difference between that number pair and the nearest multiple of 7. If any of the numbers are below 7, just leave them as they are:

3 84 65 32 65 87 90
3  0  2  4  2  3  6

Make sure you understand this step. The 90 at the right has a 1 below it because the closest multiple of 7 below it is 84, and 90 is 6 greater than 84. 65? The closest multiple below it is 63, and 65-63 equals 2, so we write a 2 below it. If you're familiar with modular arithmetic, you're simply writing down the answer to each number modulo 7. Since all the numbers will range from 0 to 7, you'll wind up with a row of all single digits.

3) Now break the row of single digits into groups of 3, again from right to left, and put them below one another:

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3

4) With these numbers arranged, you're going to add them. However, you're not going to carry any numbers. If any series of numbers adds up to 10 or more, you're just going to write the whole number down.

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3
-------
2  5 13

5) For each number in this new “total”, you're going to cast out 7s, similar to the way you did in step 2:

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3
-------
2  5 13

2  5  6

In our example, the only number over 7 is 13, so we reduce that to 6, as 13-7=6.

6) The 3 single-digit numbers you get in step 5 will now be used to create two 2-digit numbers. The leftmost digit and the middle digit are paired to make the first number, and then the middle digit and the rightmost digit are paired to make the other one:

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3
-------
2  5 13

2  5  6

25  56

Yes, the middle digit is used twice.

7) Each number in this pair of numbers is, once again, subjected to casting out 7s as in steps 2 and 5.

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3
-------
2  5 13

2  5  6

25  56
 4   0

8) We're almost there. For this last step, you take the rightmost digit from this step, and subtract the leftmost digit from it. If the rightmost digit is lower than the rightmost digit, as in our example, add 7 to the rightmost digit. This will prevent having to work with negative numbers:

3 84 65 32 65 87 90
3  0  2  4  2  3  6

2  3  6
0  2  4
      3
-------
2  5 13

2  5  6

25  56
 4   0
 4   7

7 - 4 = 3

All this has reduced our original 13-digit number, 3,846,532,658,790, down to a 3. What does this mean?

If the single digit answer you get at the end is a 0, then your original number is evenly divisible by 7. If you get any other number, your original number isn't divisible by 7.

Now, in my previous post about divisibility by 7, there are shorter and quicker methods for determining if a number is evenly divisible by 7, so you may be wondering why I've taken the time to teach this particular method. Well, it offers a bonus that the other methods don't.

The resulting number in this method also tells you exactly how far over a multiple of 7 you are! Our example gave us a result of 3, so we know that 3,846,532,658,790 is 3 greater than a multiple of 7. If you start up your calculators, and work out (3,846,532,658,790-3)/7, you'll get 549,504,665,541 exactly.

With 7 or more digits, you really should use this method on paper. However, if you're given a number of 6 digits or less, you can do it in your head. This is because 6 digits will result in only 3 digits after step 3, and you'll be able to effectively skip steps 4 through 6.

If you look into the original article in Martin Gardner's The Unexpected Hanging And Other Mathematical Diversions, you'll also learn a great lightning calculation feat using this approach that will make you seem like a genius to your audiences!

Werner Miller is up to his, well, new tricks!

Over at Online Visions, Werner Miller has posted his newest routine, Symbol Pattern Square I. In this routine, you have a standard deck of 25 ESP cards (see graphic in the upper left), and you have the spectator give it a free cut. After the cut, you place the top card, face-down, on a 5 by 5 grid in a location of the spectator's choice, marking it as the chosen card and location. The remaining cards are mixed further, and then dealt out face down in a random pattern.

After all the cards are dealt out, the cards in the same diagonal as the chosen card (which itself is left face down) are turned over, and it is shown that they all have the same design (all stars in the example). The initial chosen card is turned up, and it proves to be the one remaining card of this same design! You can also turn the other cards face up to show that each row and each column contain exactly one of each design.

Not only will those who enjoy doing magic appreciate this routine, but also my readers who enjoy math and magic squares, too.

As you'll see, it's based on the Knight's Tour approach to creating an odd-ordered (3 by 3, 5 by 5, 7 by 7, and so on) magic square. As a matter of fact, if you're not already familiar with that approach, you can learn that from this article first, and then work with the ESP cards later.

Once you read through and try this, you'll enjoy the method, but you may wonder if there's more to the effect. I can't tell you for sure, but I'm willing to bet there is. First, Werner Miller has a great knack for taking simple mathematical principles to highly entertaining levels, and second, it is called Symbol Pattern Square I.

Yes, I know I'm using the “link” graphic I usually use for my snippet posts, but this isn't a snippet post. At least, it's not a snippet post of mine.

Regular readers will remember my past mentions of BetterExplained here, here, and most recently, here. Their newest post has a very pleasant surprise!

I've always like their posts, and have often wished they could explain topics beyond just business, programming, and math. Thanks to their newest feature, links.betterexplained.com, they've gone and done just that! Thanks to slinkset, they now have an area where you can share your own favorite articles that helped you understand ANYTHING better, and learn from articles submitted by others.

As I was searching through the topics already up there, a familiar feeling hit me that I couldn't place. Finally, it hit me what the feeling was. It seemed to me that this was a more mature and detailed version of the lessons I learned from Tennessee Tuxedo's Phineas J. Whoopee as a kid. It also reminded of when I was learning about history from James Burke for the first time.

I guess you could say I've been an 'Aha' addict for quite some time now. Even after I'd gotten hooked on discoveries like these, I was thrilled to discover that Martin Gardner enjoyed 'Aha' moments, as evidenced by his books Aha! Insight and Aha! Gotcha.

I like this idea, as it not only widens the topics about which you can understand better, but you can learn about them from a wider range of people. I'm already combing through my bookmarks to see if I have any sites I can contribute, and I hope to see your ideas there, too!

On this site, I've shown you how to use math to do unusual things like determining the day of the week for any date or determining someone's age.

Those wizards at Google Labs, however, have figured how to use math for one of the most unusual purposes ever - keeping you from embarassing yourself!

This past Monday, Google announced a new feature for Gmail, called Mail Goggles. If you've ever sent an e-mail late at night, especially when you're not completely yourself, you'll appreciate this new feature.

Mail Goggles, when activated, checks to see if you're trying to mail during times when you're traditionally week, with the default being between 10 PM and 4 AM on Friday and Saturday, and if it is, it brings up a window that asks you to solve 5 arithmetic problems in under a minute, like this:

If you don't solve them all correctly in under 1 minute, the mail isn't sent, and is instead held in your draft box.

If you already have a GMail account, it's easy to set up. You simply click on Settings, and then click on the Labs tab. Scroll down to where it says Mail Goggles, click the Enable button, and then scroll down and click the Save Changes button.

Once that's done, you can go back to the General tab, and you'll see an Email Googles preferences are there, which looks like this:


From here, you can set on which days and which hours you want the Email Goggles to work, as well as the difficulty of the math problems you're given. The difficulty ranges from 1 to 5, with 1 being the easiest level, and 5 being the hardest. The blue window pictured above features level 5 math problems, to give you an idea of what you'd be up against. Click the Save Changes button if you made any changes, and you're all set.

That's all there is to Google's Email Goggles (easier to write than it is to say)! Of course, if you're a regular Grey Matters reader, and you've been practicing any of the feats on here, I suggest using the 5 settings, as you're probably sharp even when you're not at your best.

A while back, I asked for financial help for Grey Matters in any way you could help out. There was a very real possibility that I would've been unable to continue Grey Matters, but you, my readers, came through! Thank you very much!

As a thank you to my readers, who come here in the first place because they like to challenge their minds, I thought I'd share some of my favorite puzzles with you.

I'll start with a simple one as a warm up, courtesy of SharpBrains:

Question: The top two scales are in perfect balance. How many diamonds will be needed to balance the bottom set?

Think about it for a while. This is one of those puzzles that will suddenly “click”. You can find the answer here.

The rest of these will be a bit more interactive than the previous one, and they'll also be a little more challenging.

The first of these is an old puzzle, updated to use Flash. It's called SuPuzzle (No, sudoku fans, it's not a sudoku puzzle). All you have to do is connect each of three houses to water, gas, and electricity suppliers, without crossing any lines. It can done, but while you're working through this puzzle, you'll start believing otherwise.

Next, we move to Tom Jolly's Wriggle Puzzle (Java required). Actually Tom has many great variations of this puzzle, in which you try and lead a worm out of the grid, but there's only one playable one on this linked site.

These last two are meant to work through over a longer time. You probably won't work through these tonight. On the other hand, I do have some pretty sharp readers, so who knows?

Our first long-term puzzle is this set of 25 Bulbous Blob Puzzles. In these, you have to slide the pieces around, with the ultimate goal of getting the red blob into the hallway in the upper left. It's a little like those Rush Hour puzzles, but done with a diagonal twist.

This final one will stay with you even after you solve it, because you'll enjoy challenging others with it. It's an old puzzle called Petals Around the Rose. In this version, you're challenged by the website itself to figure out the method behind the scoring of each round. Once you figure it out, you're a Potentate of the Rose! You must keep the secret of the scoring, but if you can carry 5 dice with you, you can challenge others with it as well.

I hope you enjoy these, and I thank you again for helping Grey Matters to continue!

My previous post, How much is $700 billion?, proved so popular and generated so much discussion, that I'm able to bring you more visualizations of $700 billion.

In that last post, we finished with visualizing $700 billion in pennies, so let's start with dollars this time. This site will show you what $315 million looks like in $1 bills. Picture another stack of the same size right next to it, and then a third stack that's between 1/4th and 1/5th of the size of either of the others stacks, and you'll have $700 billion.

Next, let's move from real money, like pennies and dollar bills, to counterfeit money. What would it take if you decided to print $700 billion in counterfeit bills out on your inkjet printer? Frank Gibson describes this apporach in disturbing detail over at That Blog Frank Used to Have. While I don't suggest actually trying this, the process is fun to visualize.

700 billion of anything is large, so what about applying this to other units besides money?

Let's try it with distances. If you were 700 billion meters from Earth, you would get this view:

That bright spot in the middle of the picture is the sun, and those rings represent the orbits of Mercury, Venus, Mars and Earth.

If this picture looks familiar then you've probably seen the Powers of 10 video. Their photo of 1 trillion meters from Earth was edited down to 70% of the area to make that view. If that still doesn't impress upon you how far 700 billion meters is, start at the sun here, and scroll right until you get to Mars.

Take a sheet of writing paper, and fold it in half. Keep folding it in half until you can't make another fold. Most likely, you won't get beyond the 7th fold, because at that point, you're trying to fold 128 layers of paper in half. Back in 2005, Britney Gallivan actually worked out how to fold a piece of paper in half 12 times, resulting in 4,096 layers of paper. If you wanted to fold a piece of paper in half enough times to have more than 700 billion layers of paper, you would have to figure out how to fold it in half 40 times. Assuming 1 layer is a standard paper thickness of 1/10th of a mm, the resulting 1,099,511,627,776 layers would reach more than a quarter of the distance from the Earth to the moon!

Instead of distances, though, let's try time. How long ago was 700 billion seconds? 1 million seconds ago was just over 11.5 days ago. 1 billion seconds ago was roughly 31.68 years ago. That means that 700 billion seconds ago was 22,182 years ago (In 2008, 22,182 years ago would be 20,174 B.C.)! Humans were on Earth at that time, but the first civilizations wouldn't appear for another 8-10 millenniums (not years or centuries, millenniums)!

Coming around full circle to money again, have you ever played the lottery? I don't mean those scratch-off cards, but the games where you pick 6 different numbers from 1-50, and then hope those same numbers are picked in a drawing. Try this lottery simulator to get a feel for just how difficult this is. The probability of picking the exact right 6 numbers is 1 in 11,441,304,000 (greater than 1 in 11.4 billion).

Does that seem frustrating? Imagine that the lottery asked you to pick 7 numbers from 1-50 instead! The probability of that would be 1 in 503,417,376,000 (greater than 1 in 500 billion). Even playing Keno in Las Vegas, where you have to pick the correct 6 numbers in a range of 80 (1 in 216,360,144,000) offers a better chance than that!

People who want to take your money via gambling want your chances of winning to be as low as possible so they can bring in money, but still seem high enough to get potential players interested. Think about this: If people like that won't even offer a game whose probability of a player winning is even half of 1 in 700 billion, what does that tell you about the size of 700 billion?