If you enjoyed Marcus du Sautoy's 2008 documentary Story of Maths, you'll be happy to know that he's been chosen to host a few episodes of the BBC show Horizon.
His most recent appearance to date was on an episode called What Makes a Genius?, that should also be of interest to Grey Matters readers.
In this episode, Marcus continually asks the question, what makes the brain of a genius fundamentally different from the brain of someone who isn't a genius?
He starts off by demonstrating what most people think of as genius, with the help of Grey Matters favorite, Dr. Arthur Benjamin. Dr. Benjamin helps explain the difference between developing a skill, such as those he demonstrates, and the quality of creativity that is the true hallmark of genius.
From there, he goes back to the old arguments of nature vs. nurture, and we begin to learn how recent research is proving that both sides are more dynamic than was originally believed.
Here's the entire What Makes a Genius? episode of Horizon:
If you enjoyed Marcus du Sautoy's 2008 documentary Story of Maths, you'll be happy to know that he's been chosen to host a few episodes of the BBC show Horizon.
One of the best ways to draw and keep visitors at your site is with a puzzle. They're fun, challenging, and interactive. When you can customize them, such as using words or images to theme the puzzle, it makes the puzzle relevant, and can even help deliver your message.
Today, I'm going to share my favorite customizable online puzzles. All of these are available free, as well!
Since touch devices are becoming a larger and larger share of online users, I've gone the extra step of making sure all of these work with the greatest variety of touch devices.
• festisite's Rebus Generator:I figured it was best to start out with a simple static puzzle. To begin, make sure the Rebus button is clicked, and enter your chosen phrase. Select your font, font size, and paper size, and click the Layout Text button. You'll be taken to a page containing your custom rebus, which you can save as a screenshot and post to your website. Challenge your readers to post their answers in your comments!
• Alphametic Puzzles: Another simple puzzle you can include with little trouble on your site are alphametics, which are equations in which each digit is replaced with a letter. The classic example is SEND + MORE = MONEY (Answer: D=7, E=5, M=1, N=6, O=0, R=8, S=9, and Y=2 , so the equation becomes 9567 + 1085 = 10652). This site contains both a puzzle generator and puzzle solver. Once you've worked out a puzzle, you can include on your site with just some simple typing!
• Timed Quiz Generator: Yes, I'm including a shameless plug (Why not?). For those who don't already know, the Timed Quiz Generator lets you create quizzes of the How Many Xs Can You Name in Y Minutes? variety, such as this Dr. Seuss quiz. A tutorial video is included on the same page.
• DIY: linkbait in 5 minutes: If you like the idea of a timed quiz, but prefer more direct control over the program, check out 3n9.org's jQuery library that helps you develop timed quizzes. There are plenty of examples and documentation to help on this page.
• Scrabble Word Builder: This isn't so much a puzzle in and of itself, but an idea for the timed quizzes above. For example, if you run a candle site, challenge your readers to see how many words they can come up with using the letters from the word CANDLE. The timed quiz will take care of the timing and scoring, while the Scrabble Word Builder will provide you with the list of answers you'll need. You may want to limit the answers to a minimum of 3 letters.
• Word Search Generator: Here's another programming free puzzle for your site, an it's an impressive achievement. You start by entering a title and the words for your word search, and then select numerous options, such as grid size, colors, word directions, and more. You can even save your setting and reload them later! Once you actually create the puzzle, all you have to do is view the source, copy it, and save it as an HTML file, ready to upload. Still not impressed? Despite being programmed in 2005, the generated file worked flawlessly on the touch-based mobile devices on which I tested it!
• Mad Libs Word Game: Remember Mad Libs? They're a fun pastime where one person asks for several types of words, and then reads a funny story employing the given words. With this jQuery plug-in, you can engage your visitors with exactly the same type of silliness. Try out the online demo to see this in action for yourself.
• jQuizMe: This jQuery plug-in makes creating simple quizzes on your site very easy. You set up the questions and answers in an array, select a few options, and then your quiz is ready. Not only can you mix types of questions, but you can embed just about anything into the quizzer, including images and video.
Speaking of images, what about customizable puzzles using pictures?
• Memory/Concentration Game: This is the classic game where you start with a large group of hidden images, and you're only allowed to view 2 images at a time. The object is to pair up matching images. As an added bonus, this particular version also lets you reveal a message at the end. Check out the demo. If you can be sure that your site readers are using the latest browsers with HTML5 and CSS3 capabilities, you also might like to check out Branko Jevtic's version and Maykel Loomans' version.
• jqPuzzle: If you've been to the Solving the 15 Puzzle section of the Mental Gym, you've already seen this in action. It makes putting together a custom 15 puzzle almost as easy as using an ordinary image tag. The demo page gives you a great idea of exactly what is possible. Once you've set jqPuzzle up, it's a sure way to make your readers take notice of your site!
• JigZone: JigZone is a jigsaw puzzle site that allows you to upload your own images, and create an impressive array of jigsaw puzzles from them. Besides the usual shape of pieces, they include shapes like bulbs, lizards, USA states, and more! The people behind this site have not only made it possible to embed your custom puzzles on your site, but they've gone the extra mile and made sure that the puzzles are playable on touch-based mobile devices, too!
This is by no means an exhaustive list. Do you have any favorite customizable online puzzles I missed? Let me know about them in the comments!
Performer Bill Fritz, known to Magic Cafe denizens as Mr. Mindbender, has done some incredible work on a performance magic square. Today, he is very generously sharing it for free with Grey Matters readers!
Bill Fritz' new work, Magic Squares for the Mathematically Challenged, consists of notes and a series of 8 videos detailing a performance piece in which you create a magic square for any number from 35 to 100, as given by a spectator.
Before going through the notes and videos, you need to be familiar with what is referred to as the Foundation Magic Square. Regular Grey Matters readers will be familiar with the Instant Magic Square taught here, but it has a few flaws. First, if you use this approach for several people at the same event, they can compare their squares and quickly figure out how you're doing it.
Even if you only perform this for one person, if they choose a number like 99, the resulting magic square can still suggest the method:
8 11 79 1 78 2 7 12 3 81 9 6 10 5 4 80As you can see, the numbers 78 through 81 stand out, and it's not hard to figure out that this has to do with the method.
The Foundation Magic Square solves this by allowing for adjustment to all 16 numbers, instead of just 4. If you look at a magic square for 99 using the Foundation Magic Square approach, you get results like this:
25 22 19 33 20 32 26 21 31 17 24 27 23 28 30 18See the difference? All the numbers are in the same range, so none of the numbers stand out.
You can have a Foundation Magic Square automatically generated for you on this page. Try it out with different numbers, and see if you can start determining the pattern, using the green magic square up in the corner as a clue to the starting point. The method behind the magic square's construction is detailed here.
You can also start understanding the Foundation Magic Square concept by using your favorite spreadsheet, and seeing the effects different numbers have on the square. Those familiar with the Peg System will have a simple way to memorize the basic magic square.
Mentally finding the adjustment numbers and the remainder can be tricky during actual performance, especially when you have to keep the presentation moving as you do so. Isn't there an easier way?
Yes! It is precisely this challenge that inspired Bill Fritz to create Magic Squares for the Mathematically Challenged! Whether you're up to the math or not, I think you'll find his approach to getting the required numbers easy to use. Watch how quickly Bill Fritz determines the needed numbers in the first screencast, which I recommend viewing in fullscreen mode:
Intrigued? I thought so. Without further ado, here are the Magic Squares for the Mathematically Challenged notes themselves (downloadable at link), which I also recommend viewing in fullscreen:
Here are the complete set of links to the screencasts, as they're referred to in the notes (viewing in fullscreen recommended):
SCREENCAST #1: The Challenge
SCREENCAST #2: Introduction
SCREENCAST #3: The Concept
SCREENCAST #4: Step 1
SCREENCAST #5: Step 2
SCREENCAST #6: Exceptions
SCREENCAST #7: Examples
SCREENCAST #8: Presentation Tips
Don't skip the Presentation Tips video, as there are some very handy and impressive handling tips that further help the work. If the Post-It Note Magic Square presentation sounds familiar, it may be because I linked to Bill's original description of it at the end of June's snippets post.
Bill, thank you very much for sharing your incredible work on the Foundation Magic Square for free here on Grey Matters!
I love learning about history, as evidenced by my work annotating James Burke's documentaries. My favorite of James Burke's show would have to be Infinitely Reasonable, as it actually discussed the history development of science and mathematical thought.
I always thought a longer discussion of the history of math would be interesting. It seems that the BBC and Oxford Mathematics Professor Marcus du Sautoy agree, as they produced and aired a 4-part documentary back in 2008, called The Story of Maths.
In this post, you can watch all 4 complete episodes, as long as they're available on YouTube. If you have any trouble viewing the videos below, you can still probably find the episodes on YouTube, or via a Google video search.
The first episode, The Language of the Universe, begins with the importance of math in our modern life, before going on to explore the origin and development of math in the ancient western countries of Egypt, Mesopotamia, and then Greece.
After the fall of Greece, scientific and mathematical development came to an unfortunate halt. However, in the eastern countries, math continued to expand and develop. In The Genius of the East, we learn about the influence of China, India, and the Middle East. It also discusses the effect of these developments on western countries as the Renaissance began.
With a renewed European interest in mathematics, we see great gains over the next few centuries in modeling and analysis from Minds like Descartes, Leibniz, Newton, Euler, and many others. In the third episode, we see how the work of men long ago helped take us to The Frontiers of Space.
What could possibly be left? Not only does mathematics itself continue to develop, but there are still numerous problems from the past that still haven't been solved. The effect these have on the current state of math, and their future potential, is the subject of the final episode, To Infinity and Beyond.
Rod Serling may have been more accurate than he knew when he stated in the Twilight Zone opening, There is a fifth dimension, beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. Perhaps it's time to take a closer look at dimension itself.
The desire to examine the nature of dimension is hardly new. Back in 1884, Edwin Abbott wrote a novel called Flatland, a short adventure in 22 chapters and less than 80 pages. It's now in the public domain, and is downloadable for free from sources such as Google Books and Project Gutenberg.
Flatland is a strange tale of beings in a two-dimensional beings, and the curious events that cause them to question whether a third dimension exists. In this world, to even suggest that there are 3 dimensions is tantamount to heresy!
Recently, there have been a couple of versions released. Below is a short 34-minute version featuring the voices of Martin Sheen, Kristen Bell, and Michael York, among others:
You have to love the 2-dimensional touches, such as refrigerators, briefcases, pitchers, and squaricles (the 2-dimensional version of cubicles). The references to Area 33H are a subtle joke, as 33 in hexadecimal is equivalent to 51 in decimal, so it's an Area 51 joke.
There's also another version that came out in 2007 (click for playlist), interestingly the same year as the one above. This one has a 95-minute run.
The main point of the original book, and both movies, is how hard it is to imagine dimensions outside of our own. Notice that perceptions are constantly being turned upside down. Arthur meets the king of Pointland, and the king of Lineland, and has trouble getting them to understand a 2nd dimension. Spherius has the same problems getting Arthur to understand the 3rd dimension. The final twist in the movie drives this point all too well.
To help you understand Flatland's conclusion, here's a wonderful segment from Carl Sagan's Cosmos explaining a tesseract, with help from the Flatland analogy:
Charles Hinton would extend on Abbott's work in 1907, releasing An Episode on Flatland: Or How a Plain Folk Discovered the Third Dimension. In this version, the Flatlanders lived on a large circle, as opposed to Abbott's infinite plane. The book itself is longer and more involved, and also involves political satire.
When Martin Gardner covered these works in his Scientific American column, which was later reprinted in his books The Last Recreations (Available on Amazon) and The Colossal Book of Mathematics (Also available on Amazon), it set the stage for a renewed interest in the topic.
A. K. Dewdney had written a monograph in 1979 called Two-Dimensional Science and Technology, which was reviewed in Gardner's column. With the response he received from that review, he was inspired to write the book Planiverse, which was released in 1984 (Preview the book here, and buy the full version is available on Amazon).
Planiverse begins in our world, with students in a classroom using a computer to try and work out how a 2-dimensional world would function. During this process, they wind up making contact with 2-dimensional beings, and begin learning about such a world in unexpected ways. I believe the modern twist of using a computer for communication with a 2D world makes this version of a 2D world easier to understand for most readers.
Many of the above resources refer to the difficulty of envisioning a 4th dimension. It's even joked about in the Flatland edition of XKCD. The video game mentioned in the comic, Miegakure, is currently under development. It will be interesting to see if it truly can help make the 4th dimension easier to understand.
If you still think the 4th dimension is hard to picture, what about 10 dimensions? Below is a handy guide to help you to imagine all 10 dimensions. It's not long, but you still may want to go through it in small portions, to help you better understand each stage:
To further explore the ideas here, check out the multi-media version (Flash required) and the text version at the author's website. He also maintains a 10-Dimension video blog on his YouTube channel.
I hope you've enjoyed this mind-stretching post. Please share any thoughts you have in the comments!
November already?!? Didn't 2010 just start? I guess I'd better get the November snippets in before it's suddenly New Year's Eve.
• If you liked the Scott Flansburg videos, I have some videos of people who specialize in the day of the week for any date feat for you. First, check out 6-year old William Dam, who is a very young human calendar.
Ever wonder how fast the human mind can calculate the day of the week? Back in June, Freddis Reyes Hernandez was able to give the day of the week for 74 dates. This was a world record until Jan Van Koningsveld did 78 dates in 1 minute in September!
Speaking of the day of the week for any date feat, I should mention that MindMagician.org just updated their Day For Any Date quizzer for improved readability and compatibility across various browsers.
• I've recently run across some sites with interesting mathematical tidbits. In the snippets section of Nerd Paradise, I found this interesting method for determining how many zeros a factorial will have. For example, 10! (10 factorial) = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800, which has 2 zeroes. How many would say, 103! have?
Also, I've been enjoying Ben Vitale's number-focused blogs, Fun With Num3ers and More Fun With Num3ers. I was especially intrigued with this post, as I'd never realized that not only were all the odd square numbers 1 greater than a multiple of 8, but were also 1 greater than a multiple of 8 times a triangular number!
• If you've enjoyed my past posts about the Prisoner's Dilemma, check out JimBobJenkins' YouTube Channel, where William Spaniel offers a complete free course in game theory!
Don't forget to keep checking the Mind Your Decisions blog for more interesting game theory posts. Their most recent post concerns a Texas Hold 'Em paradox that will be very familiar to fans of Scam School's non-transitive scams.
That's all I have for now. Do you have any math- or memory-related links you'd like to share? Let's hear about them in comments!
Scott Flansburg is also known as the human calculator. He's been making the rounds on several TV shows lately, so you have an excellent opportunity to see why he's earned that name.
Recently, he appeared on Chicago's channel 7 to demonstrate some of his math skills. The anchors were armed with an iPad, while Scott was armed only with his mind:
Notice that he doesn't use math skills to put himself above everybody else, but rather to inspire others to do their best.
Indeed, many of the feats you'll see here are taught right here on Grey Matters. For example, when he was asked to divide 716 by 9, you can learn that feat in my Mental Division With Decimal Precision post. The calendar feat, which is taught here, is used as the perfect closer here.
You can learn these feats, but whether you wind up as good and as fast as Scott is up to you and your commitment. In his books Math Magicn and Math Magic for Your Kids, he goes on to describe many more impressive feats that you can do, as well as help you with everyday math.
If you're curious about what he meant when he referred to everything coming back to 9, check out The Magic of 9.
Back in August, Scott also had the honor of appearing on the first episode of Stan Lee's Superhumans:
In the full version of this segment, you can also see that quick multiples series feat (49 in the Chicago video, 48 in the above video), which has really become his trademark. As near as I can tell, it's unique to him, and he does it very well!
My favorite appearance, however, was from about 6 years ago, when he appeared on a similar program, called More Than Human. This clip can be also found on Scott Flansburg's own site:
Here, you see Scott extracting the cube root of 658,503. This is a perfect example of how strange the patterns in math can be. Not only are cube roots easier to calculate than you might think, but doing 5th roots is even easier than cube roots!
This shouldn't diminish the impressive levels to which he's taken these feats. Notice that, in this version of the quick multiples series feat, he not only outpaces the calculator and causes a mistake, but also can determine the nature of the mistake! That's not something you can teach, and that's what makes Scott Flansburg's performances so impressive.
If you like these, you can find more videos on Scott's own blog, as well.
(NOTE: Check out the other posts in The Secrets of Nim series.)
Back in my Secrets of Nim Wrap-up post, I briefly mentioned this Spiked Math cartoon. The stick figure (presumably visiting from XKCD) is challenged to a game of multi-pile standard Nim, and after just looking at the piles, he concludes that he's lost.
Being able to just look at a multi-pile Nim game like that would be amazing wouldn't it? Sure, you could always use the Nim Strategy Calculator, but is it really possible to look at an unfamiliar multi-pile Nim game and analyze it in your head? Surprisingly, the answer is yes!
RefresherBefore we move ahead, let's make sure we're on the same page. All the Nim terminology I'll be using is defined in Part 1 of this series.
You should understand standard multi-pile Nim, and the similarities and differences between that and multi-pile Misère Nim.
Pay special attention to the fact that multi-pile Misère Nim really only differs in the endgame. You should also know the moves by heart that will win multi-pile Nim, standard and Misère.
Finally, you'll need to know your powers of 2 up to 16: 1, 2, 4, 8, and 16. Sure, you could go on beyond these to 32, 64, 128, and so on, but very few Nim games will have rows of 32 or more objects.
Once you've mastered the strategies I'm about to describe, the best way to present this is to have the other person choose the number of piles used, and how many objects will be used. You then stipulate that you will decide who goes first.
Who Goes First?When faced with an unfamiliar layout, you first need to determine who should go first. Let's use a layout of 4, 6, and 7 objects as an example.
WYou start by determining the largest power of 2 involved. What's the largest power of 2 that we'll use here? 4, 6, and 7 are all less than 8, so the largest power we'll need to look at is 4. OK, so what do we do with the 4 we've decided to use?
What we're going to be doing is looking for pairs of each power of 2, in a process that will be described shortly, starting with the largest and then working with each power down to 1. In our above example, this means we'll be looking for pairs of 4s, pairs of 2s, and then pairs of 1s.
Ask yourself: How many rows have at least 4 objects in them? In our 4, 6, and 7 example, all three do! In other words, we have a pair of 4s, and an extra 4, as depicted below:
Anytime you have even a single unpaired power of 2, as we do in this case, you'll want to go first. If you examine all the powers of 2, and everything is paired up, you'll want the other person to go first. In this case, remember that you have a 4 (that unpaired 4).
However, for later use, you're going to want to analyze any and all upaired groups. For the next round, we need to mentally remove all those groups of 4 (effectivly ignoring the first 4 cards in each row), leaving just a group of 2 and 3:
After 4, we now have to deal with 2. With piles of 2 and 3, both have 2 objects in them:
The 2s are paired, so we don't have to think about 2s. Don't forget that unpaired 4 from earlier, though.
For the next round, we mentally remove those 2s, and that leaves us with a single pile of 1, which is unpaired. So, from a pile of 4, 6, and 7, we've narrowed down that we have an unpaired 4 and an unpaired 1.
As mentioned before, when you have 1 or more unpaired powers of 2 in the layout, you should decide to go first.
In the cartoon linked above, there were piles of 3, 5, and 6, and the player was told that he had to go first. Analyzed the same way, we see that there's a pair of 4s:
Leaving us with 3, 1, and 2 objects, and looking at 2s, we have a pair of 2s, as well:
Removing those obviously leaves us with a pair of 1s. So, every power of 2 we examined was paired up. That means there's no good move for the first player in this game, and the first player will lose, assuming all other moves are played perfectly.
What's My First Move?Going back to our example game of 4, 6, and 7, we found ourself left with an unpaired 4 and an unpaired 1. How do we use this to determine our first move?
The rule of thumb here is to remove enough items from the largest unpaired number (4 in our example) that all the other numbers (just 1 in our example) will be paired up.
For our example case, where we wound up with an unpaired 4 and an unpaired 1, we're going to pair up that 1 by trimming the 4 down. In other words, we're going to remove 3. But from which pile?
The removal of 3 we've determined should be done from the smallest pile where such a move is possible. In our piles of 4, 6, 7, this means the pile of 4. Removing 3 leaves us with piles of 1, 6, and 7 for the other player.
All this sounds complicated, but it's really just a way of applying the equals pairs and mirroring approach you learned in the earlier multi-pile Nim columns.
Do you have to keep analyzing powers of 2 to play the game? Not really. If you know the strategies taught earlier for piles of 3, 5, 7, once you get all the piles down to numbers equal to or less than those, you can start playing those safe moves from memory.
If you can memorize winning moves for larger games, perhaps using the Nim Strategy Calculator, then you'll have a larger set of practiced memorized moves, and can handle even larger games automatically!
What if you're given four piles, consisting of 13, 5, 1, and 9 objects? Do you think you could properly analyze it? Here's a video titled Classic Nim Game - Powers of Two, in which two high school girls analyze exactly this game. The video can be downloaded and watched offline if you don't have Flash installed.
Master this, and you can play Nim like a mathematical genius!
Answer to Halloween Candy Puzzle: