US$700 billion, the proposed amount of the financial institution bailout, is a big number to wrap your head around. Just how much is US$700,000,000,000?

First, let's think of it in terms of grains of salt. A tablespoon of salt contains roughly 100,000 grains of salt. A 1/2 cup of salt contains about 1 million grains of salt. How much is 700 billion grains of salt? Picture an average public school classroom filled up about 70% (between 2/3 and 3/4) of the way with salt! Even on senior prank day, that would result in immediate suspension.

Even more fun, think of getting this $700 billion in pennies - that's 70 trillion (with a t) pennies!

First of all, getting 70 trillion pennies would be impossible, as if you were to total all the various pennies minted by the US government since 1787, that would "only" come to 300 billion pennies (by most estimates). About a third to a half of those are either lost, destroyed by the mint, or are otherwise no longer accepted as legal tender (and probably in the hands of numismatists). That leaves about 140-200 billion pennies still in circulation, or $1.4 to $2 billion dollars in pennies. The US would have to be in existence for another 513 centuries (not years, centuries) before we'll have minted 70 trillion pennies!

Back to our main question, how many is $700 billion in pennies?

According to the MegaPenny project, it would take a stack of two trillion, six hundred twenty-three billion, six hundred eighty-four million six hundred and eight thousand pennies (that's 2,623,684,608,000 pennies or $26,236,846,080 - just over $26 billion) to build a life-size replica of Chicago's Sears Tower, ignoring the antennas on top.

Think about this: You could use $700 billion in pennies to create 26 complete, full-size replicas of Chicago's Sear's tower! And you would still have more than US$17 billion dollars (US$17,842,001,920 to be exact) in pennies left over!

According to the official US population clock, the US population at this writing is slightly less than 305.3 million people. If you divide up the remaining US$17.8 million up among all those people, you could send them more than $58 each!

What would happen if we simply divided up the full $700 billion among the current US population, without creating Sears Towers replicas? Each person would get US$2,292.82!

Now, since the proposed $700 billion bailout is money being taken from you, not given, this last amount is an even more interesting number to remember.

Think of it this way: Instead of just letting these companies fail, having stockholders remove the people who made the bad decisions in the first place, and have the companies slowly try to rebuild their reputation among stockholders and customers, everyone in the US, for only about $2,300 each, can keep the people who made the poor decisions in place, and give them completely new money with which to make those same bad decisions!

Gee, what a bargain.

Update (Oct. 2, 2008): You can find even more visualizations of 700 billion in Part II of this post.

Many people think of formulas as one of the most boring parts of math, when they really should be considered one of the most interesting parts. When you get right down to it, a formula is the mathematical expression of a pattern. Plug in the particular numbers you need, and the formula will show you the results you need.

Often, it isn't immediately apparent why a particular formula works, with the result that plugging numbers into the formula can seem like magic. If there's a formula for something that doesn't seem to have a coherent pattern, such as the Day of the Week For Any Date feat, the feeling of wonder may be increased even more.

Learning why a particular formula works, since it requires learning about and understanding a pattern, can also be eye-opening. For example, we all now that the area of a circle can be calculated with the formula pi*r², but why does that work? In this BetterExplained article, you'll be walked through the process of discovering the formula. You might just accidentally introduce yourself to calculus along the way, too.

Here's a question for you that needs a formula for an answer. At what temperature are Fahrenheit and Celsius the same number? To get Fahrenheit, you're usually taught to the Celsius temperature, multiply it by ⁹/₅, and then add 32. To go the other way, you're taught to subtract 32, then multiply by ⁵/₉.

Given this approach, the answer to the question isn't readily apparent, and would require some trial and error to work out. However, there's actually a much simpler approach that makes it immediately clear!

Try this much simpler formula on for size: (F + 40) = 1.8 × (C + 40). One look at this formula, and it's not hard to understand that the only temperature at which Fahrenheit and Celsius are the same number is -40 (-40 is the only number that turns both sides into 0).

Taking a closer look, though, and you can make the formula even easier. First, note that regardless of whether you're starting from Celsius or Fahrenheit, the first step will always be adding 40, and the last step will always be subtracting 40. To multiply by 1.8, simply multiply by 2 and subtract 10% of the result. To divide by 1.8, divide by 2 and add 10% (the actual percentage would be 11.11111%). If you think of Fahrenheit as the “bigger” number of the two, it's easy to remember that you need to multiply to go from Celsius to Fahrenheit, and divide when you're going from Fahrenheit to Celsius.

For example, what is the equivalent of 23 degrees Celsius? Add 40, to get 63. Double that, giving 126. Subtract 10% of that number, which is 126-12.6, which gives 113.4. Don't forget to subtract 40! This last step gives 73.4 degrees Fahrenheit. That's much easier, isn't it? Even if you round the 10% estimate (13 instead of 12.6), you'd still get 73 degrees!

We're going to move on to a very unusual formula now. Assuming every page in a given book is numbered starting with 1, how many digits total are required to number that book?

With a book of up to 9 pages, it would just be the number itself. A book with 10 numbered pages, however, requires 11 digits to number. There's the 9 digits from 1 through 9, plus the 2 digits in the number 10. If you have a book with n pages, and n is a number with k digits, then the formula for the number of digits required to number that book is kn-[((10^k-1)/9) - k].

That seems complicated, doesn't it? It's much simpler than it appears. Let's find out how many digits would be required to number a 20-page book, and I'll explain it along the way.

First, we multiply 2 by 20, because we have 20 pages, and 20 is a 2-digit number. So far, we have 40. Next, we subtract 11. Why 11? For this step, you need a number consisting of all 1s, and has the same number of digits (k), as the original number in question. If we started with a 3-digit number in the first step, we'd subtract 111 here. If we started with a 4-digit number of pages, we'd subtract 1,111, and so on. So, 40-11 is 29. Finally, we add the number of digits the number of pages (2, as 20 is a 2-digit number), giving us 31. This means that a 20-page book would require 31 digits to number from 1 to 20!

As you practice, this gets easier to understand, too. How many digits would a 256-page book require? 256 × 3 = 768. 768 - 111 (remember why?) = 657. 657 + 3 (256 is a 3-digit numb = 660, so a 256-page book requires a total of 660 digits to number! If you need practice multiplying 3 digit numbers by 3 and 4 digit numbers by 4 in your head, I highly recommend Arthur Benjamin's book, Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks.

One of the classic uses for formulas and patterns is what are known as “divisibility tests”. These are tests to determine whether one number divided by another number will result in a whole number. Most numbers from 1-9 have easy divisibility tests. 1? Every number is divisibly by 1! 2? If the rightmost digit is 0, 2, 4, 6, or 8, it's divisible by 2. 3? Add up the digits of the number, and if their total is divisible by 3, then the number is divisible by 3. 4? Are the rightmost 2 digits divisible by 4? Then the whole number is wholly divisible by 4! 5? If the last digit is 5 or 0, you can evenly divide a number by 5. 6? It's a little tricky, but if it's an even number AND passes the divisibility test for 3, you can divide it by 6. 8? If the last 3 digits are divisible by 8, then the number is evenly divisible by 8. 9? Add up the digits of the number. If they total a multiple of 9, then you can divide the original number evenly by 9.

Notice that I skipped right over 7. That's because 7 doesn't offer divisibility tests as easy as any of the others. Let's say we want to find out whether 398,594,966 is evenly divisible by 7. How can we do this?

This first test is really bizarre. Starting from the rightmost digit, and moving to the left, you multiply the first digit by 1, the next digit by 3, the next digit by 2, the next digit by -1, the next digit by -3, the next digit by -2, the next digit by 1, and so on, continuing on in the pattern of 1, 3, 2, -1, -3, -2, and so on. If the sum of these numbers is 7, then the whole number will be divisible by 7. Applying this to 398,594,966, we get:

(1×6)+(3×6)+(2×9)+(-1×4)+(-3×9)+(-2×5)+(1×8)+(3×9)+(2×3)=6+18+18-4-27-10+8+27+6=42. Since we know 42 is divisible by 7, so is 398,594,966.

It doesn't quite “click”, does it? There are more methods that are almost as weird for 7, such as the X+10Y+9Z method and the 3X+Y method. Few of the divisibility tests for 7 can be turned into something you can do in your head, but there are a few that are interesting enough that they're fun and easy enough to work out on paper.

The first of these divisibility tests involves splitting the rightmost digit off from the rest of the number, doubling it, and then subtracting that result from the rest of the number. With 398,594,966 (better written as 398594966 for this example), we break off the rightmost 6, double it to get 12, and then subtract that 12 from 39859496, giving 39859484. Now, it still isn't apparent as whether 39859484 is a multiple of 7, but we can apply the test again to this new number! Performing this over and over until we get a number that's easily recognizable as a multiple of 7, we get:

39859484 --> 3985948 - (2×4) = 3985948 - 8 = 3985940
3985940 --> 398594 - (2×0) = 398594 -0 = 398594
398594 --> 39859 - (2×4) = 39859 - 8 = 39851
39851 --> 3985 - (2×1) = 3985 - 2 = 3983
3983 --> 398 - (2×3) = 398 - 6 = 392
392 --> 39 - (2×2) = 39 - 4 = 35

35 is a known multiple of 7, so we know that 398,594,966, and all the other numbers in that left column, are multiples of 7! As an aside, instead of doubling the number and subtracting the result, you can multiply the number by 5 and add the result, but the doubling and subtracting method is usually preferred, as it continually decreases the number.

That last method takes a while, though, as it only eliminates 1 digit from the number at each step. Wouldn't it be nice to eliminate several digits in one step, while still testing for even divisibility of 7? Yes, that is also possible!

You'll love this approach. First, break the number in question into groups of 3. Thanks to commas, this is usually very easy. Starting from the rightmost trio, we subtract the next group, add the group after that, and continue on adding and subtracting groups of 3, until you've used up the whole number.

In our example, 398,594,966 becomes 966 (the rightmost trio) - 594 + 398, which totals 770. It's easy to see that 770 is a multiple of 7, so it passes the test in one step!

Let's try that again. What about 258,894,048? That becomes 48 - 894 + 258, which is -588. For this test, you can ignore a minus sign, so we just focus on 588. Just at a glance, it's not easy to see if 588 is a multiple of 7, so what do we do? This is where the previous test comes in handy! With the previous test, 588 becomes 58 - (2×8) = 58 - 16 = 42. In just two steps, instead of the 9 steps that would otherwise be required, we've determined the number is a multiple of 7!

Finally, let's try this approach on 832,089,053. That becomes 53 - 89 + 832, or 796. 796 becomes 79 - (2×6) = 79 - 12 = 67, which is most definitely NOT a multiple of 7 (It's between 63 and 70). Now we know that 832,089,053 isn't a multiple of 7 with only two steps.

Of course, the quickest divisibility test for 7 these days is simply to enter the number in a calculator, and divide it by 7. If you get a whole number, it's evenly divisible by 7! However, if you're in a situation where you can't use a calculator, these tests can come in handy!

Back in March 2007, as the movie The Number 23 was being released, I posted an entry about special properties of the number 23, and where to find properties of just about any number. These online “number museums” have . . . well, multiplied since then!

Among the number museums I initially mentioned were What's Special About This Number?, Number Gossip, and the Java-powered site, The Secret Lives of Numbers. To completely catch up with the original post, it should be noted that Notable Properties of Specific Numbers is still around, but it now has a new home.

Archimedes' Laboratory, a site I've mentioned many a time before, has their list, called the Zoo of Numbers. This number features unique and interesting facts for numbers as high as 715, and as low as “NaN”, which many computer programmers will recognize as the symbol for “Not a Number” They've even gone so far as to arrange for you to shop for items featuring your favorite number!

Compared to “Zoo of Numbers,” the Database of Number Correlations sounds much drier and stuffier. However, don't let that fool you, as this site features more detail than most of the other sites. They include not only mathematical properties, but films with that number in the title, famous lists of the given number, and even things that happened that year, if applicable! For example, the entry for number 9 not only discusses facts like 9 is a square number and that it's the sum of the first 3 odd numbers, but also brings up the 9 Christian fruits of the spirit, films like 9 1/2 Weeks, 9 Songs, and the Whole 9 Yards, and even the fact that Roman emperor Vespasian was born in 9 A.D.!

There's one such “number museum” that you've probably visited, yet never thought of it as such. Which site is this? I'm talking about the well-known Wikipedia, where you can find individual entries on a mind-boggling amount of integers! Wikipedia's editability by anyone makes this especially rich as a number-property resource. Where else could you learn bizarre number trivia such as, in the entry on number 8, that Michael Phelps won 8 of a possible 8 medals in the Olympic Games that began on 8/8/08, at exactly 8:08:08 PM local time? If you don't want to be limited to integers, look through the categories of rational numbers, real numbers, or any kind of numbers!

Just as in any field, collectors of number properties want their work to stand out by giving special focus to their work. A prime example of this is the Prime Curios! site. The only properties that this site will even consider listing for any given number must somehow relate to prime numbers. Take the humble number 42, for example. 42 itself is obviously not a prime number, yet it has the interesting prime properties such as being immediately between two primes, and being the total of the two least consecutive primes ending in 1 (11 + 31 = 42).

An even more impressive specialty can be found at The On-Line Encyclopedia of Integer Sequences. As the name suggests, it doesn't focus on individual numbers like all the sites above, but rather sequences of whole numbers. The one thing that makes this site incredibly useful is the fact that you can enter any integer sequence into their search engine, when then quickly provides all the entries of which that sequence could be a part. This site can really surprise you sometimes. If you search for the classic sequence 1, 1, 2, 3, 5, 8, 13, you might think that it would only return the entry on Fibonacci numbers. If so, you'll be surprised when it returns more than 80 different possible number sequences!

As you can see, The On-Line Encyclopedia of Integer Sequences is the perfect site If you ever wanted to cheat like crazy when challenged with things like Idiot World's Math Godeek Test or number sequence puzzles from Mental Floss' Brain Games column. Of course, any true math geek realizes that, given any sequence of numbers, the next number can be mathematically proven to be ANYTHING, if you employ polynomials!

Due largely to the recent economic situation, I'm having trouble making ends meet. So, I'll be using today's post to ask for your help in continuing Grey Matters.

True, posting on blogspot, hosting images on photobucket, and using freehostia for the Mental Gym doesn't cost a cent, and that does help keep costs down. However, my internet access itself, and the storage space that goes along with it does cost money.

Were I to lose that access, this site could no longer be updated. I could no longer develop new sections to the Mental Gym, or add new timed quizzes to the popular How Many Xs Can You Name In Y Minutes? post. Indeed all the feeds including the timed quizzes feed would no longer be available, and the various widgets associated with them would no longer work. For that matter, none of my downloads, RSS feeds, the iPhone Metal Gym, and the the Presentation section. The blog and the Mental Gym would remain online, but updates to them would cease.

I apologize for having to use my regular post for money. However, if you've found Grey Matters enjoyable, and/or even valuable to you, I'd like to ask you to show you appreciation in any one of the following ways:

• Purchase 1 or more items from my selection of original products and/or downloadable products. Popular items have included the full memory course Train Your Brain and Entertain (also available as a download), and the 2009 Day for any Date feat calendar.

• Bid on the eBay auction where I'm selling David Britland's limited-edition classic magic book, The Mind and Magic of David Berglas.

• If you like books, but find that the eBay auction is asking little too much for you tastes, you can both help this site out and find some great deals by buying an item from my Zlio store. If you enjoy this site, you'll find many books and other items here, and many even offer a comparison of prices at various stores, so you can get the best deal.

• If you've find this site even useful, simply click the donate button below, and donate a dollar or two. Any amount will be welcome and appreciated.

Working on my new Scott's Puzzles blog has proved to be an interesting challenge. It's one thing to conceive of the idea that you're going to provide a variety of online puzzles that are related to current events, and quite another to actually do it.

One result, though, is that I've found many sites I didn't know about before that let you create custom puzzles. I'd like to share them with you:


PUZZLES WITH WORDS

• Official Scrabble® Dictionary Search - On it's own, you won't develop an online game with it. However, Quizicon (formerly Codebox Software) used it to develop a “Word Mix” puzzle, in which you're given a word, and asked to list as many words that can be made from its letters as you can. Since it's basically a specialized How Many Xs Can You Name In Y Minutes?-style quiz, you can generate the words at the dictionary, and turn it into an online puzzle with the Timed Quiz Generator!

• Do It Yourself: Linkbait - besides my own Timed Quiz Generator, this another great way to create online timed quizzes! It's powered by jQuery, a powerful Javascript framework, and is meant for programmers who are familiar with CSS, HTML, and Javascript.

• Eric Harshbarger's Applet Depot - This is a collection of free-to-use Java applets, many of which are puzzles. The seek-a-word puzzle is easy to use, with the word list being input as a parameter. Even with the same words, it will generate a different puzzle everytime!


PUZZLES WITH PICTURES

• jqPuzzle - This is a jQuery plug-in that takes a picture, and automatically turns it into a slider puzzle, like the classic 15 puzzle! The picture itself can be put into a post with a simple HTML image source tag (as long as you've loaded the plug-in and jQuery earlier), yet is also very customizable.

• Eric Harshbarger's Applet Depot - Yes, I mentioned this site already, but the FlipSwitch puzzle on this site means it deserves another mention. FlipSwitch is unlike any other puzzle I've seen, and you can see it in action in my Google Chrome post. You're presented with a grid that features parts of several pictures, and the object is to return it to a single picture. What makes this so hard is that when you click on a square, it advances not only its own picture one image forward, but also that of every square in the same row and column! I've only managed to solve it myself a handful of times.

• JigZone - This site might be considered as YouTube for jigsaw puzzles. You can upload your own pictures, which are then automatically turned into jigsaw puzzles. You can choose the cut (try their “lizard” cut, where, except for the edge pieces, every piece is exactly the same shape!) and even embed it into your blog. At first, I though I wouldn't be able to use this on my puzzle site, since my blog didn't have enough room to display even the smallest puzzle. However, thanks to FancyBox, another jQuery plug-in (I'm quickly learning how flexible and useful jQuery is!), I was finally able to embed it in a Lightbox-style window. My first (and, at this writing, only) jigsaw puzzle using this approach was my R.I.P. Don LaFontaine (1940-2008) post.

• Mazesmith - From the beginning, I've been wanting to include mazes on Scott's Puzzles, but this has been a challenge. I needed a Javascript program that generated a maze (10s of millions available), AND allowed people to solve it online (this narrows it down to the millions), AND is available as freeware (we're down in the 100,000s now), AND would allow the resulting mazes to be posted online for free (down to the 1,000s), AND could be posted in a blog post (we're down to less than 100 now), AND, since I need this to be about current events, would allow me to post mazes of custom shapes. That list narrows it down to 1 program, and Mazesmith is that program!

Despite that exhaustive list of features I required, Mazesmith not only came through, but offers an amazing list of features I hadn't even considered, such as custom icons for the maze-player and goal, custom colors, and even several ways of mathematically influencing the nature of the maze itself! Even with a custom shape, you'll get a different maze each time, which really helps.

If you want to create your own custom-shaped maze, it helps greatly to be familiar with both a graphics program (Photoshop/GIMP/GraphicConverter, etc.), and HTML, CSS, and Javascript. Custom-shaped mazes are encoded as a long string of 1s and 0s, with the 1s representing where the maze is allowed, and 0s where no piece is allowed. What I've found works best for me is to find an online picture, get rid of as much background as possible (if needed), and shrink the picture to about 40 to 45 pixels across. I then convert it to a black-and-white bitmap (where the black areas will be filled with the maze later on), taking into consideration both recognizability of the subject and enough black areas to provide a substantial and challenging maze.

Once I have a small, yet recognizable, black and white bitmap of the subject that will provide a challenging maze, I turn the picture into text via ToolPanel.org's ASCII Generator to turn it into text (all W's, to be specific). I copy the generated into a text editor, change all the spaces to 0s, all the Ws to 1s, and then remove the returns. This results is the specific long unbroken string of 1s and 0s I need to create a maze of my custom shape! I can then put it into the program, as described in the software and its instructions.


I hope you've enjoyed this look at free online puzzle creators and generators, and hope you find it useful! If you have any favorites I didn't list, please let me know about them in the comments!

This won't be much of a post today. I'm spending this day remembering September 11, 2001. I watching not only coverage of the memorials, but also the Internet Archive's September 11 video archive footage of the news coverage from September 11th, 12th, and 13th.

I went through both July and August without posting any snippets! It's time to rectify that:

• Back in May, I wrote a post on approaches to memorizing the chemical elements. It seemd I missed one (probably much more than one, but I digress). Julianksy20's Googlepage on memorizing the periodic table of the elements is impressive! It focuses on the chunking approach, which focuses on smaller groups and what they have in common. No, you won't learn which element goes with which number, but you will be able to name them all if you pratice this method!

• If you haven't taken a look at the free memory software Anki since I first mentioned it back in February, you should take another look. Not only does it have features like Leitner's spaced learning approach and online synchronization of software, along with a list of features that's longer than my arm (or browser window, for that matter), but it's available for just about every platform for free! (OK, there's not an iPhone version, but there's always iFlipr for that.)

• Here's a toy just for pure fun: Doermann's perfect magic square creator (page is in both German and English). Enter any number from 23 to 100, and you'll get a perfect magic square. There's plenty more about magic squares on this blog, if you get curious.

• Finally, for those that have mastered the Day of the Week For Any Date feat, we have some inspiration. This page describing uses for online date calculators also might stir up some new ideas about how to present your date abilities. Some might think it would be too difficult to do some of things on this page in your head, such as calculating the difference between 2 dates, but Dr. Thomas Harrington's book, The Magic of Memory: Phonetic Mnemonics has simple approach for doing just that! As a matter of fact, it's even easier than the standard date feat!

There we go, the snippet drought has ended! Just don't let me go so long without another snippet entry, OK?

I ended my previous post with the comment, Hmm . . . maybe I need to look at redesigning some site features at Grey Matters. If you look closely, you'll note that this has already begun!

Starting a month or two ago, I decided this site needed a new design. My first thought was that it needed a new look, as I had done before (check out this 2006 look for the site, this 2005 look for the presentation section, and even this pre-Grey Matters blog version of the Mental Gym! from 2001!). I examined numerous blog templates, and even found some great candidates, but I could never really find one that made me want to switch from the current design.

Some of it is boring, behind-the-scenes type of redesign, such as reducing my dependence on code from outside sites, fixing/removing broken links, setting up folders that automatically sync with the servers that power Grey Matters, and so on. However, I have a few visual goodies already up and running.

First, you'll notice the tabs that used to be at the top are now a solid block. If you run your pointer over this solid block, however, you'll notice it's not just a set of links, but rather a full-fledged menu bar! Instead of just going to the main pages of each section, you can now go to many of the sub-sections of each area, as well.

Want to go to the Knight's Tour instructions? Move your mouse-pointer over the Mental Gym area, and when that section drops down, go down to Mental “Feat”-ures. When the sub-menu drops down, click on Knight's Tour, and you'll be taken directly to it! The tabs only linked to the main pages of each site, so I'm hoping this change will encourage people to explore Grey Matters in more detail.

The other major visual change is that of changing the Site Feeds and Downloads sections in the rightmost column from simple lists into accordions. This not only makes them look better, but now they're clearer, better organized, and easier to use, all while taking up less space! The downloads section, for example, is no longer a hodge-podge of downloads for various systems. They're now organized by type of system: OS X, Windows, and iPhone.

One of the side effects of resdesigning the righmost column has been to remove or fix all the broken links. In my site feed subscription badges, for example, I had one badges for a feed site that hadn't been in business for over a year!

While many of the changes are employing Javascript, I'm also being very careful to make sure that any changes will still have at least some functionality if you're surfing with Javascript turned off.

I have more plans to improve the site. Some features you will see, some you won't see, and some are only experimental, and may not be seen on the site by you or me. If you have any suggestions on improving the appearance and function of the site, please let me know!