New Year - New Surprises!

Published on Thursday, December 30, 2010 in , , , ,

Happy New Year 2011!For the past several months, I've been working on a New Year's surprise for Grey Matters readers. It's almost done, and will be revealed on January 1st, 2011 (Saturday), and Sunday's post will focus on all the details.

To keep you going until then, I'm going to provide you with a few fun treats for the new year.

Since New Year's Day is coming up, you're probably working on your resolutions. Is it possible to make resolutions that have a better chance of being kept? Of course! Over at Gawker, check out How to Make Every New Year's Resolution Stick for Good. Richard Wiseman, over at his 59 Seconds blog, also has a great article called Achieve your New Year’s Resolutions!.

Don't forget to include improving your memory or math skills among your resolutions.

Enough of the serious stuff! We'll turn to some math, but nothing serious, of course. Maybe just some math doodling will fit the bill here:

Like that? The same author has several other math doodling videos you might enjoy:

Snakes + Graphs
Infinity Elephants
Sick Number Games

Have a Happy New Year!



Published on Sunday, December 26, 2010 in , , , ,

Andrew Mayne's Spirit PhotoHow many of you got that long-awaited techno-gadget phone or tablet during the holidays which you've been waiting for?

Since you probably built it up into a magical level in your mind before you got it, how about taking it to magical levels now that you have it?

To start, go over to Andrew Mayne's iPod Tricks site. Despite the name, most of the tricks are most image- or video-based, so they're easily adaptable to any handheld mobile device. In EZ Money, you can pull real money off an image of a dollar bill, with a comedic ending. Weapon X is more of an interlude in which you x-ray your arm, to show your built-in claws, just like Wolverine.

Among the more unusual tricks on that site, there's Spirit Photo, a supernatural-themed trick done with another person's mobile-device (must have internet access for the routine). iPhone Trick works specifically only with iPhones and iPod Touches, and makes it appear that you're damaging another person's device. The final one, iPunk'd, isn't done with the phone itself. It's a cut-and-restored earphone cord, which is a nice way to update a classic for the 21st century.

If you're familiar with John Bannon's classic packet effect, Twister Sisters, there's now a way to bring it into the 21st century, as well. Go to http://superpixel.com/tsr/, and the people can make their choices using the phone. The effect remains basically the same as the original, but with the added mystery of choices on the phone affecting real playing cards. There is no explanation, as you must be familiar with Twisted Sisters to use it in the first place. The developer features a nice presentation for this version over at the Magic Cafe.

Greg Rostami's iForce is already a classic among magic tricks for mobile devices, and now it's available for both the iOS and the Android platforms! Due to its versatility, there are already many ideas on the developer's own site, as well as the Magic Cafe.

The bigger news from Greg Rostami, though, is the release of iPredict! It's a little hard to describe, so check out the video below. Note that someone else's phone is used to dial the number!

This is hardly an exhaustive list of tricks for mobile devices. Do you have any favorites I didn't mention. Let's hear about them in the comments!


12 Days of Christmas (Repost)

Published on Thursday, December 23, 2010 in , , , , , , ,

12 Days of Christmas TreeNote: The holidays are keeping me busy, so I'm reposting my 12 Days of Christmas post from 2007. Enjoy!

Since the focus of this blog is largely math and memory feats, it probably won't be a surprise to learn that my favorite Christmas carol is The 12 Days of Christmas. After all, it's got a long list and it's full of numbers!

On the extremely unlikely chance you haven't heard this song too many times already this holiday season, here's John Denver and the Muppets singing The 12 Days of Christmas:

The memory part is usually what creates the most trouble. In the above video, Fozzie has trouble remembering what is given on the 7th day. Even a singing group as mathematically precise as the Klein Four Group has trouble remembering what goes where in their version of The 12 Days of Christmas (Their cover of the Straight No Chaser version):

Just to make sure that you've got them down, I'll give you 5 minutes to correctly name all of the 12 Days of Christmas gifts. Those of you who have been practicing this quiz since I first mentioned it in last Sunday's post will have an advantage.

Now that we've got the memory part down, I'll turn to the math. What is the total number of gifts are being given in the song? 1+2+3 and so on up to 12 doesn't seem easy to do mentally, but it is if you see the pattern. Note that 1+12=13. So what? So does 2+11, 3+10 and all the numbers up to 6+7. In other words, we have 6 pairs of 13, and 6 times 13 is easy. That gives us 78 gifts total.

As noted in Peter Chou's Twelve Days Christmas Tree page (whose icon is at the top left of this post), the gifts can be arranged in a triangular fashion, since each day includes one more gift than the previous day. Besides being aesthetically pleasing, it turns out that a particular type of triangle, Pascal's Triangle, is a great way to study mathematical questions about the 12 days of Christmas.

First, let's get a Pascal's Triangle with 14 rows (opens in new window), so we can look at what it tells us. As we discuss these patterns, I'm going to refer to going down the right diagonal, but since the pattern is symmetrical, the left would work just as well.

Starting with the rightmost diagonal, we see it is all 1's. This represents each day's increase in the number of presents, since each day increases by 1. Moving to the second diagonal from the right, we see the simple sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12, which can naturally represent the number of gifts given on each day of Christmas.

The third diagonal from the right has the rather unusual sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91. This is a pattern of triangular numbers.

But what can triangular numbers tell us about the 12 days of Christmas? If you look at where the 3 in this diagonal, it's southwest (down and to the left) of the 2 in the second rightmost diagonal. If, on the 2nd day of Christmas, you gave 2 turtle doves and 1 partridge in a pear tree, you would indeed have given 3 gifts, but does the pattern hold? On the 3rd day, you would have given 3+2+1 (3 French hens, 2 turtle doves and a partridge in a pear tree) or 6 gifts total, and sure enough, 6 can be found southwest of the 3! For any of the 12 days, simply find that number, and look to the southwest of that number to see how many gifts you've given by that point! Remember when figured out that the numbers 1 through 12, when added, totaled 78? Look southwest of the 12, and you'll find that same 78!

Let's get really picky and technical about the 12 days of Christmas. It clearly states that on the first day, your true love gave you a partridge in a pear tree, and on the second day your true love gave you two turtle doves and a partridge in a pear tree. You would actually have 4 gifts (counting each partridge and its respective pear tree as one gift) by the second day, the first day's partridge, the second day's partridge and two turtle doves. By the third day, you would have 10 gifts, consisting of 3 partridges, 4 turtle doves and 3 French hens.

At this rate, how many gifts would you have at the end of the 12th day? Sure enough, the pattern of 1, 4, 10 and so on, known as tetrahedral numbers (Java required, opens in new window), can be found in our Pascal's Triangle as the 4th diagonal from the right.

If you look at the 2nd rightmost diagonal, you'll see the number 2, and you'll see the number 4 two steps southwest (two steps down and to the left) of it, which tells us you'll have 4 gifts on the second day. Using this same method, you can easily see that you'll have 10 gifts on the 3rd day, 20 gifts on the 4th day, and so on. If you really did get gifts from your true love in this picky and technical way, you would wind up with 364 gifts on the 12th day! In other words, you would get 1 gift for every day in the year, not including Christmas itself (also not including February 29th, if we're talking about leap years)!

If you're having any trouble visualizing any of this so far, Judy Brown's Twelve Days of Christmas and Pascal's Triangle page will be of great help.

One other interesting pattern I'd like to bring up is the one that happens if you darken only the odd-numbered cells in Pascal's Triangle. You get a fractal pattern known as the Sierpinski Sieve. No, this won't tell you too much about the 12 days of Christmas, except maybe the occurrences of the odd days, but it can make a beautiful and original Christmas ornament! If you have kids who ask about it, you can always give them the book The Number Devil, which describes both Pascal's Triangle and Sierpinski Sieve, among other mathematical concepts, in a very kid-friendly way.

There's another 12 Days of Christmas calculation that's far more traditional: How much would the 12 gifts actually cost if you bought them? According to the Motley Fool, the 12 gifts of Christmas would cost a total of US$78,100! If you wanted to cut your costs and buy just one of each, you would still have to shell out US$19,507!

Since my Christmas spending is done, I'm going to have to forgo the expensive version, in favor of Miss Cellania's internet-style version of The 12 Days of Christmas. Happy Holidays!


Hunting the Hidden Dimension

Published on Sunday, December 19, 2010 in , ,

Wolfgang Beyer's Mandelbrot set renderingBack when I wrote about the late Benoît Mandelbrot's fractal work, I really only covered the basics about the nature of fractals themselves.

Mathipedia recently alerted me to a documentary on fractals that gives a more complete picture.

The documentary is called Hunting the Hidden Dimension, and was an episode of NOVA back in 2008. It begins with the 19th century discovery of number sets with such unusual qualities, they were referred to as “monsters”, but regarded as little more than curiosities.

Once computers were developed, and maverick mathematicians like Mandelbrot were able to examine them, it was found that there was far more to fractals than anyone had expected. After explaining much about their nature, Hunting the Hidden Dimension examines the unexpected effects they've had on communications, medicine, entertainment, and much more. Without knowing it, you've probably taken advantage of the unique qualities of fractals without even knowing it!

If the video above ceases to work, you can also watch the complete documentary at the Hunting the Hidden Dimension homepage. The home page also has many more interactive features, such as playing with the scale and iterations of classic fractals.

Even if you don't have the Flash plug-in, features such as the interview with Benoît Mandelbrot himself are well worth checking out.


Periodic Table of Videos

Published on Thursday, December 16, 2010 in , , ,

Antonio Delgado's Periodic Table of the ElementsBack in mid-2008, I discussed the chemical elements, mainly from the standpoint of memorizing them.

In this update, we turn our attention to understanding the elements. As always, I've tried to find the best resources that help you obtain that understanding as efficiently as possible.

Ever since I first saw this explanation of the atom on TV, I was astounded at how simple and direct it was! It's from the original show WKRP in Cincinnati, from an episode where Venus is asked by a friend to help convince her son to return to school. To do this, Venus bets the son he can explain the basics of the atom to him in 2 minutes:

This is a great basis on which to build. The Chem4Kids introduction to the periodic table, mentioned in my 2008 elements post, is still a great way to go from there.

Even if you understand the basics of the atom, and why the periodic table is grouped as it is, it's quite another thing to understand each element, especially when there are 118 of them! Fortunately, the science dept. of the University of Nottingham has found a great and clear way to teach about each elements.

The approach is taught on their site, The Periodic Table of Videos. The main menu is set up just like the classic periodic table, and clicking on an element takes you to a corresponding video featuring experiments and explanations. All of the videos are also available on the periodicvideos YouTube channel.

Here's a sample of the videos on the site, focusing on hydrogen:

Like the high school chemistry classes you may have forgotten, the experiments really help drive home the main points. However, they do have two advantages over the old science class. First, you can go back and review as many times as you want. Second, the videos are often updated, especially when an element is in the news. This video examining NASA's recent claim of arsenic as a building block of life is a perfect example.

If you go through the videos a few at a time, even just 1 or 2 per day, and work through the mnemonic methods I discuss back in the 2008 post at the same rate, you could quickly and effectively develop a better knowledge of the elements than probably 90% of the general public!

These resources are a great example of using your imagination to help break down complex topics into manageable parts, which is a lesson you can take beyond chemistry.

Update [12/17/10]: One great related link I've been sent is to Mr. Edmond's YouTube channel, where he teaches science concepts to music. Since this is a chemistry column, here's one on the difference between ionic and covalent bonds:


Fun and Simple Mnemonics

Published on Sunday, December 12, 2010 in , ,

Knuckle Mnemonic for Month LengthsAlmost all the posts on here since October have focused on mathematics. Perhaps it's time to get back to memory work.

Let's wade back into memory work with some mnemonics.

I've certainly talked enough about memorizing Pi, but what about memorizing the constant e? Robert Talbert teaches a different mnemonic approach that helps you memorize e simply by looking at a US$20 bill! Since I believe that memorization and understanding go hand-in-hand, understanding the nature of e is a big help. Also, knowing your US presidents will help here, strangely.

Speaking of memorizing USA facts, The Tutor Whisperer has some great and original mnemonics for learning the US states and their locations. The example only teaches from the Pacific coast states to the midwest states, but it wouldn't be hard to create similar mnemonics for the rest of the states.

Perhaps you only need to memorize the original 13 colonies? Then perhaps Venus, Mercury, Mars, Saturn, Neptune, and Pluto can help! No, you don't need to read your horoscope, just use this stately astro-mnemonic. This mnemonic will probably take more work than the others in this post, but it sure is handy!

As long as we're talking about outer space now, how do you go about remembering something like the speed of light? In miles per second, 186,000 isn't tough to remember, but what about in meters per second? Can we guarantee certainty, clearly referring to this light mnemonic? Not only can we, you just learned it without knowing that you did!

Ready-made mnemonics are great, but what about when there are no ready-made mnemonics for what you're studying? One solution for creating custom mnemonics I've mentioned before is JogLab. They've just made their site even more useful with a special introductory video about creating acrostic poems (It employs annotation links, so I'm not embedding it here).

Finally, those who use the Number Shape system of memory will find the following video handy. I've seen several variations of images for use with this system, but this video teaches the most distinct and easily-remembered images I've ever run across:

What are your favorite mnemonics? Post and explain them in the comments!


Yet Still More Quick Snippets

Published on Thursday, December 09, 2010 in , , , ,

LinksIt's time for December snippets. Since we've been doing so many math-related posts lately, we're dedicating this edition to math goodies.

• If you like the math-related videos I've been posting lately, and still can't get your fill, check out bachelorsdegree.org's list, 20 Incredible TED Talks for Math Geeks. This list alone with keep your mind and eyes open for a while!

• If it's more math instruction videos for which you're looking, check out Math Vids, My Math Shortcuts, and Lazy Maths. Each of these sites has a good selection of instructional videos that will help you to learn how to handle various types of math problems more efficiently.

• The classic Penney Ante scam is usually done with three coin flips. If you want a real challenge, though, try doing Penney's Game with 4 coin flips, as described in this Wolfram Blog post. Unlike the Scam School video, the Wolfram Alpha Blog article doesn't provide a nice, neat system that automatically generates a better flip than a given one. Can you find such a system for 4 coin flips?

• Rounding out this edition of snippets, here's a freebie you can use for the entire next year: The 2011 “Lightning Calculation” Calendar! Each month, you'll learn new mental math skills. By the end of 2011, you'll have developed many amazing mathematical abilities! It's free as a downloadable PDF or for $17.60 as a pre-printed calendar from Lulu.com.


Last To Be Chosen II

Published on Sunday, December 05, 2010 in , , , ,

James GrimeIn the previous post, I featured James Grime's Last To Be Chosen video, where we learned about the Kruskal Count, and discussed some variations.

Today, you'll see James Grime's explanation, and learn a little bit more about how this works. I'll also discuss variations after the video explanation itself.

If you haven't watched James Grime's video from the previous post, please watch that first. Here's his explanation of the Kruskal Count:

Beyond the explanation itself, let's start with the fact that about 58% of the time, every single one of the first 10 cards will always lead to the predicted card. This fact can come in handy!

For example, if you can reasonably work in a pre-arranged deck when presenting this, and you've checked that the first 10 cards all lead to the same card, you could have a prediction card ready. In an informal performance, however, you're better off using a shuffled deck and playing the percentages.

See how the average value affects the card? When jacks, queens, and kings are respectively counted as 11, 12, and 13, the odds of this trick working go down to 66% - only 2 times out of 3! Notice that the spelling variation shoots the odds up to 96.5%!

Why? If a card is randomly pulled out of the deck, its value is most likely to spell with four letters. 20 out of the 52 cards spell with four-letter values (four, five, nine, jack, and king), for a roughly 38.5% chance. The other 32 are split evenly between three-letter values (ace, two, six, and ten) and five-letter values (three, seven, eight, and queen), giving a roughly 30.75% chance of appearing.

This is why the clock version from the previous post works so reliably, despite having such a short chain. Even though eleven and twelve have more letters, the only chance you really have to spell them is on your first choice. After that, the process doesn't let you land on 11 or 12, so you don't get any other chance to spell them.

We discussed a few other places to apply this principle in the previous post, such as book and poem passages, as well as clocks. With a better understanding of the formulas and percentages, what other variations are there?

Try experimenting with the following idea. Since we know that spelling the values increases the odds so well that it can work with a short circular chain such as a clock, what would happen if you used just the 13 cards of 1 suit, and allowed spelling in a circular chain by moving 1 card to the bottom for each letter? What patterns can you find if they're in Ace through King order? How about King through Ace order? Are there any usable patterns if those 13 cards from the same suit are shuffled? Jim Steinmeyer's book Impuzzibilities contains a very nice presentation using an approach like this.

Imagine this: You remove two decks of cards in their cases, one red-backed and one blue-backed, and you set one aside, explaining that you've reversed one card in that one as a prediction. You then have the person shuffle the other deck, and go through the procedure as described. After they go through the entire deck, but haven't mentioned their card, you remove the reverse card from the other deck, reminding them that this prediction has been in the box since before they even shuffled the deck! You ask them what their last card chosen was, and it proves to be the same as the reversed card!

How is this possible? You need a tool from a magic shop known as a Brainwave deck. I won't reveal the secret here. The basic idea of a Brainwave deck is that it's specially prepared to allow you to show any card as the only face-down card in the deck (and even a different back color). Once you use the Kruskal count procedure to determined the likely card chosen, you know which card to show face down.

There is a similar tool known as the Invisible Deck, but the card would be shown face-up. For this routine, I think the added suspense of being able to keep the card face-down helps the routine. Those who've been into magic will probably also know of many variations of these routines (such as Kolossal Killer or Heirloom) that could be used here, as well.

I'd like to hear about any variations you develop, as well!

Update [12/17/10]: James Grime has just published a detailed paper on Kruskal's Count. It's an early draft, so he warns that there may be a few mistakes. Check it out!


Last To Be Chosen

Published on Thursday, December 02, 2010 in , , , , ,

James GrimeJames Grime, known to YouTube viewers as singingbanana, is back! This time, he's performing a rather interesting experiment/magic trick.

I've talked about this principle before, but I don't want to ruin it by giving the principle away before you've had a chance to enjoy it. The first 90 seconds of the following video are mainly to clarify the instructions. After that, the actual experiment begins:

Did he get your card? Remember, it's an experiment, so it may not have worked.

Whether or not that worked, here's a similar experiment to try. Instead of cards, this version uses the first sentences of the US Declaration of Independence. Go to that link, and try out Martin Gardner's presentation of this same principle.

Intrigued? How could these routines work a majority of the time? Indeed, with the right set-up, it is possible to guarantee that the principle works every time! Still, the question remains, how is it possible?

Both of these routines are based on a principle developed by physicist Martin David Kruskal, and is known as the Kruskal Count.

A simple and understandable explanation of exactly how this principle behind the Kruskal Count works is shown in the article, Up The Magician's Sleeve. You can also try this approach yourself with various coins or poker chips of various colors.

With a single deck in this trick, the probability of it working is roughly 84%, which is why the version of the video may not have worked for you. When the same trick is performed with two decks, the probability of the trick working goes up to roughly 95%!

Once you get an idea of how the Kruskal Count works, you start to realize that it will work with anything in a specific order. This is why it also works with the Martin Gardner version above that used the US Declaration of Independence.

It can also work with the opening paragraphs of your favorite book or poem, as well. In fact, I suggested this approach for memorized poems in one of my Memory Binder posts. Computer Science for Fun has a great Wizard of Oz version.

What other types of things involve some chain? How about a clock? Here's a great clock version, based on a similar routine that David Copperfield performed on his Unexplained Forces special:

This is an interesting approach, first developed by Jim Steinmeyer, that combines the Kruskal Count with a circular chain.

This seems like a good place to wrap up the post and let you start exploring. What other routines for the Kruskal principle can you find, with or without the circular idea from the clock trick above? I'd love to hear about it in the comments if you come up with a great version!