In my previous post, I showed you how other math geeks look at Halloween. Now, we're going to put your Halloween smarts to the test!

When I would come home for trick-or-treating with friends, instead of diving into our candy, we'd start trading. For example, I might trade my M&Ms, a Mars bar, and a Reese's all for my friend's single Hershey bar.

It's a little tricky trying to trade candy over the internet, so I've got a puzzle version for you, in which we're going to trade letters for numbers. The object of this candy equation is to replace each letter with a number so that the equation is true.

The rules are as follows:
• Each letter represents only one digit, 0-9, throughout the problem.
• Different letters represent different digits, and no two letters represent the same digit.
• Any letter that represents the leftmost digit in a number never represents 0.
• When solved, the equation must be true.
• There is only one possible solution.


I'll post the answer to this candy puzzle in my next post.

Another part of the fun of Halloween is going out and seeing all the carved pumpkins. Flickr user freeflower has generously shared these pumpkins with us. These pumpkins are from a contest held every year in her neighborhood where you must vote on your favorite pumpkin before getting your treat.

Before you even get to see the pumpkins, though, you must unscramble them. Click or touch (yes, this works on mobile devices) on any block next to the empty space, and that block will slide into the empty space. The Shuffle button both mixes the pieces and resets the move counter, while the Original button shows the finished picture.


Looking for still more Halloween puzzles to strain your brain?

Check these out:
Halloween jigsaw puzzle gallery (JigZone)
Brain Game: Sweet Things (mental_floss)
Halloween Spot The Difference (Marion Cole)
A Hangman Haunting (pjnation)
Halloween World Ladder (sporcle)

Did you manage to solve any of these puzzles? If so, let me know how you did in the comments!

Halloween is just around the corner, so it's time for a little spooky fun, math geek style!

Here's a quick math joke to get us going: Why do mathematicians always confuse Halloween and Christmas? Because Oct 31 = Dec 25!

Edgar Allan Poe's classic poem, The Raven, has been a long-time Halloween favorite. It's very enjoyable, but what happens when you give it a mathematical twist?

Back in 1995, Mike Keith wondered what you would get if you crossed Poe with Pi. The result is his amazing work, Poe, E., Near a Raven. Here are the first two stanzas:

Poe, E.
Near a Raven

Midnights so dreary, tired and weary.
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
"This", I whispered quietly, "I ignore".

Perfectly, the intellect remembers: the ghostly fires, a glittering ember.
Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:
That inimitable lesson in elegance - Lenore -
Is delighting, exciting...nevermore.

How is it related to Pi? Take a close look at the title. Poe has 3 letters, followed by E., a single letter. We then have Near A Raven, a 4-letter word, followed by another single-letter word, concluding with a 5-letter word. In short, the number of letters gives us 3-1-4-1-5! Mike Keith explains more about “standard Pilish” in this article.

If you can remember that whole version, you can remember pi to 740 digits! Mike Keith took the same idea even further, when he used this poem as the opening to Cadaeic Cadenza. This is a story about classic poems being mysteriously re-written, and works as a mnemonic for 3,835 digits of Pi!

It seems as though Halloween, a time when post people are disguising themselves as something they aren't, geeks seem to feel more free to let their true self out more. For example, take Professor Weathers' 2009 Halloween lecture at Biola University:



With all the scary stuff in the video, such as the head-turning and psychos sneaking up on him, I can't help but wonder what would happen if a math professor did wind up facing the devil. It doesn't seem like a situation where intelligence or logic would help you prevail.

I'll wind up this post with just such a Halloween story for you, called I of Newton:

I've talked about magic squares quite a bit before, but the variety of ways people develop to present the magic squares never fails to amaze me.

As I mentioned in my previous post, this pastThursday would have been 96th Martin Gardner's birthday.

Martin Gardner, of course, loved magic squares, and covered them numerous times in his writings. In honor of this, Richard Wiseman performed a magic square at a recent event known as the Martin Gardner Mind Party. Watch how he takes it beyond just the total, and relates everything to Martin Gardner's life:



This, like most presentations, was done with a 4 by 4 magic square. This is largely because an array of 16 numbers is large enough to be impressive, but also quick enough to perform that it remains engaging to an audience.

However, there are those venturesome souls out there who do like to take on larger magic squares. In the following video from Germany, performer Robin Wersig asks for a 3-digit number, getting 843 as a reply, and then asks for a starting square, for which D2 is given. He then proceeds to start at square D2, and creates an 8 by 8 magic square totaling 843 in every row and column, all while performing a Knight's Tour!



If that's still not large and impressive enough for you, how about a 25 by 25 magic square, totaling 7,825 in every direction? It really is incredible the levels to which people have taken magic squares.

To help untangle your brain from these mind-boggling magic squares, here are some squares that add up to different numbers in every direction, yet are still strangely impressive.

It's time for the October snippets, all of which come with a mathematical bent.

• Today would have been the late Martin Gardner's 96th birthday. In honor of his memory, check out one of his interesting surprises over at The Mathematical Tourist's post Martin Gardner’s Möbius Surprise.

Here's another mathematical surprise, courtesy of the Gardner fans over at G4G4:

Write out the alphabet in capital letters, starting with J:
JKLMNOPQRSTUVWXYZABCDEFGHI

Erase all letters that have left-right symmetry (such as A) and count the letters in each of the five groups that remain.

I won't spoil the surprise. Just try it.

• A while back, James Grime posted a video about the math behind an episode of Futurama. Unfortunately, because it used video from the show, it was taken down for copyright violations. Later, however, it was put back up, and wound up being removed. As of today, it's back up again.

If you click that link and it's down again, there's always the safe edit version. There's also plenty of Futurama-related math over at, strangely enough, Futurama Math.

• James Grime also has a nice article on non-transitive dice over at plus magazine. Basically, it's a way of making dice inherently unfair. The same approach can also be applied to coin flips. Interestingly, James Devlin points out that even a standard coin flip may be unfair.

• Since we've already covered dice and coins, how about winding up with playing cards? Over at the World of Playing Cards, you can easily lose an hour or a day just going through the different topics, including how playing card design and usage evolved in different countries!

No matter how well you think you know playing cards, you still might be hard pressed to figure out what's wrong with these playing cards.

I'll come full circle by bringing the topics of playing cards and math together by having you check out Johnny Ball’s ‘Two Wrongs Do Make a Right’ Trick. Try it out as in the description, and then read on to learn why it works!

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

–Benoît Mandelbrot, from the introduction to The Fractal Geometry of Nature

Before Benoît Mandelbrot, irregular shapes such as the ones discussed in the above quote, were considered by mathematicians to be unusual cases with properties that were too random to study for any sort of pattern. Mandelbrot was the first to develop the tools to explore what he called fractals.

What are fractals? Let's start with Martin Gardner's description of this snowflake curve (Graphic courtesy of António Miguel de Campos):



This is known as the Koch snowflake. In the above picture, the operation is repeated (or, as mathematicians like to say, iterated) only 7 times, but you can repeat the procedure as long as you like. Like many fractals, it repeats itself many times, and develops an amazing pattern.

One of the more interesting qualities of the Koch snowflake is that, no matter how much you magnify it, it will look so similar to any other magnification level that you will have trouble telling how far in (or out) you've zoomed!

The best way to start understand fractals is with a hands-on approach. Here's how to create Koch snowflake cupcake frosting. As an added bonus, Evil Mad Scientist Laboratories will also show you how to make cookies based on the Sierpinski carpet, and even Sierpinski triangle earrings. It's even possible to make a delicious fractal pizza!

Benoît Mandelbrot is known for creating a complex fractal known as the Mandelbrot set, described here in plain English. That explanation does skip over the nature of complex numbers, but fortunately BetterExplained.com explains complex numbers visually and simply. The Mandelbrot set exhibits a more starting complexity at every level than the previous examples, yet still retains an amazing self-similar quality as you zoom deeper.



Basically, Benoît Mandelbrot developed a new set of tools that allowed us to see our current world in a new way, as well as see a whole new world. Thank you, Benoît Mandelbrot, for adding to our understanding of the world.

Even while I was putting together my new Memorize USA Facts section, it occured to me that it was missing a section on the US state flags. Since I couldn't find a ready-made resource that helped remember the state flags, I decided I had to develop one myself.

The result is the new State Flags section of the Memorize United States of America Facts page.

I create simple mnemonics for each flag designed to come to mind quickly, as you look at their design. To help in quick identification, I also tried to avoid requiring too much attention to detail to the flags.

Even with simple mnemonics, learning 50 flags can still seem overwhelming. To overcome this, I broke the flags up into various groups, each with some aspect in common, such as unusual shapes or simple designs. This resulted in a group of 7 videos, each teaching less than 10 videos, to make it easier to learn at your own pace.

Here's the shortest video of the group, focusing on states whose flags reflect state features, to give you an idea of how this works:



To makes sure you can come up with the mnemonics under tim pressure, I've included sporcle's US state flag quiz.

I'd love to hear any comments you have about this new section.

Happy 10/10/10! It's one of those silly number holidays, much like last year's 9/9/09. What can we do for 10/10/10?

I think it's a perfect time to examine the powers of 10. That's never been done better than in Charles and Ray Eames' classic educational film, Powers of 10:



Did anybody else ever notice that the narrator specifically mentions that the picnic is happening in October? Did they choose this because of the 10th month? He does say it's taking place “early one October“, so maybe it's a subtle 10/10 joke.

If the film got you thinking, you can experience it online in a more hands on manner, at powersod10.com. This site is very well developed, and really helps bring the experience to life.

On a geekier note, if you use binary, 101010 equals 42. This is silly, fun, and geeky, which seems to go perfectly with the day. As a matter of fact, many of us geeks are celebrating today as 42 Day.

This seems like the perfect time to bring up what I believe should be called Douglas Adams' identity: 1337% of Pi = 42.

Are you doing anything silly or special for 10/10/10? Let's hear about it in the comments!

One of the hardest things to convey in math is the sudden realization you get from seeing all the parts of the picture come together. It is however, one of the most satisfying moments, and one when you realize what math is truly supposed to be about.

Martin Gardner even wrote two books on various types of these experiences, which he calls an aha! solution, now combined into one volume, aha!: a two volume collection. To start getting an aidea of just how satisfying such an experience can be, take a look through it online for free here.

Joe Hurd, in his 5-minute Visual Mathematics lecture, gives some more examples of this:



There are many more excellent examples of this type of thinking at BetterExplained, one of my favorite sites. I've also written my own posts on visualizing Pi, math, and scale. Thanks to StumbleUpon, I've managed to find many great great sites about visualization.

James Tanton brings home the importance of seeing mathematics in pictures in his presentation, Five Principles of Math Genius Thinking:

Part 1


Part 2


I'd love to hear any aha!-type experiences you'd care to share in the comments!

Here in the US, the kids are now back in school, and we have an election coming up in less than a month, so it seems like a good time to learn about memorizing the basic facts about the United States of America.

Instead of writing a post about memorizing states, capitals and presidents, though, I decided it was time to add a new page to the Mental Gym, called Memorize United States of America Facts.

In this new section, you'll learn to memorize all the US Presidents, the US states and their capitals, as well as the US Constitution Preamble and all the amendments.

Each of these sub-sections includes links, videos, quizzes, and more to provide you with several different ways to learn, and help make it easier.

I'm a big proponent of memorization together with understanding, so there is also a Learn More... section, with links where you can learn more about USA history and geography.

Try out the new section, and let me know what you think! Any suggestions you have are appreciated, too.