Tag Archives: puzzles

Math Awareness Month, Hexapawn, and Plane Puzzles

Welcome to this week’s Math Munch!

April is Mathematics Awareness Month. So happy Mathematics Awareness Month! This year’s theme is “Mathematics, Magic, and Mystery”. It’s inspired by the fact that 2014 would have marked Martin Gardner’s 100th birthday.


A few of the mathy morsels that await you this month on mathaware.org!

Each day this month a new piece of magical or mysterious math will be revealed on the MAM site. The mathematical offering for today is a card trick that’s based on the Fibonacci numbers. Dipping into this site from time to time would be a great way for you to have a mathy month.

It is white

It is white’s turn to move. Who will win this Hexapawn game?

Speaking of Martin Gardner, I recently ran across a version of Hexapawn made in the programming language Scratch. Hexapawn is a chess mini-game involving—you guessed it—six pawns. Martin invented it and shared it in his Mathematical Games column in 1962. (Here’s the original column.) The object of the game is to get one of your pawns to the other side of the board or to “lock” the position so that your opponent cannot move. The pawns can move by stepping forward one square or capturing diagonally forward. Simple rules, but winning is trickier than you might think!

The program I found was created by a new Scratcher who goes by the handle “puttering”. On the site he explains:

I’m a dad. I was looking for a good way for my daughters to learn programming and I found Scratch. It turns out to be so much fun that I’ve made some projects myself, when I can get the computer…

puttering's Scratch version of Conway's Game of Life

puttering’s Scratch version of Conway’s Game of Life

Something that’s super cool about puttering’s Hexapawn game is that the program learns from its stratetgy errors and gradually becomes a stronger player as you play more! It’s well worth playing a bunch of games just to see this happen. puttering has other Scratch creations on his page, too—like a solver for the Eight Queens puzzle and a Secret Code Machine. Be sure to check those out, too!

Last up, our friend Nalini Joshi recently travelled to a meeting of the Australian Academy of Science, which led to a little number puzzle.


What unusual ways of describing a number! Trying to learn about these terms led me to an equally unusual calculator, hosted on the Math Celebrity website. The calculator will show you calculations about the factors of a numbers, as well as lots of categories that your number fits into. Derek Orr of Math Year-Round and I figured out that Nalini’s clues fit with multiple numbers, including 185, 191, and 205. So we needed more clues!

Can you find another number that fits Nalini’s clues? What do you think would be some good additional questions we could ask Nalini? Leave your thoughts in the comments!


A result from the Number Property Calculator

I hope this post helps you to kick off a great Mathematics Awareness Month. Bon appetit!

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!


Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike's octahedron.

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ’em out!

Evelyn's Schwartz lantern.

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A "nicely" triangulated cylinder.

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert Torrence and his Lights Out puzzle.

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!

Linking Newspaper Rings, Pascal’s Colors, and Poetry of Math

Welcome to this week’s Math Munch!

Here’s something that sounds impossible: turn a single newspaper page into two rings, linked together, using only scissors and folding. No tape, no glue– just folding and a few little cuts.

Want to know how to do it? Check out this video by Mariano Tomatis:

On his website, Mariano calls himself the “Wonder Injector,” a “writer of science with the mission of the magician.” And that video certainly looked like magic! I wonder how the trick works…

Mariano’s website is full of fun videos involving mathe-magical tricks. I like watching them, being completely baffled, and then figuring out how the trick works. Here’s another one that I really like, about a fictional plane saved from crashing. It’s a little creepy.

How does this trick work???

Next up is one of my favorite number pattern — Pascal’s Triangle. Pascal’s Triangle appears all over mathematics– from algebra to combinatorics to number theory.

Pascal’s Triangle always starts with a 1 at the top. To make more rows, you add together two numbers next to each other and put their sum between them in the row below. For example, see the two threes beside each other in the fourth row? They add to 6, which is placed between them in the fifth row.

Pascal’s Triangle is full of interesting patterns (what can you find?)– but my favorite patterns appear when you color the numbers according to their factors.

That’s just what Brent Yorgey, computer programmer and author of the blog “The Math Less Travelled,” did! Here’s what you get if you color all of the numbers that are multiples of 2 gray and all of the numbers that aren’t multiples of 2 blue.

Recognize that pattern? It’s a Sierpinski triangle fractal!

If you thought that was cool, check out this one based on what happens if you divide all the numbers in the triangle by 5. The multiples of 5 are gray; the numbers that leave a remainder of 1 when divided by 5 are blue, remainder 2 are red, remainder 3 are yellow, and remainder 4 are green. And here’s one based on what happens if you divide all the numbers in the triangle by 6.

See the yellow Sierpinski triangle below the blue, red, green, and purple pattern? Why might the pattern for multiples of two appear in the triangle colored based on multiples of 6?

If you want to learn more about how Brent made these images and want to see more of them, check out his blog post, “Visualizing Pascal’s Triangle Remainders.”

Finally, I just stumbled across this collection of mathematical poems written by students at Arcadia University, in a class called “Mathematics in Literature.” They’re the result of a workshop led by mathematician and poet Sarah Glaz, who I met this summer at the Bridges Mathematical Art Conference. Sarah gave the students this prompt:

Step1: Brainstorm three recent school or other situations in your

present life – you can just write a few words to reference them.

Step 2: List 10-20 mathematical words you’ve used in class in the
past month.

Step 3: Write about one of the previous situations using as many
of these words as possible. Try to avoid referencing the situation
directly. Write no more than seven words per line.

Here’s one that I like:

ASPARAGUS, by Sarah Goldfarb

An infinity of hunger within me
Dividing a bunch of green
Snap and sizzle,
Green parentheses in a pan
The aromatic property
Simplifying my want
Producing a need
Each fraction of a second
Dragging its feet impatiently as I wait
And when it is distributed on my plate
It is only a moment before zero
Units of nourishment remain.

Maybe you’ll try writing a poem of your own! If you do, we’d love to see it.

Bon appetit!

Isomorphisms in Five, Parquet Deformations, and POW!

Welcome to this week’s Math Munch!

Here’s a catchy little video. It’s called “Isomorphisms in Five.” Can you figure out why? The note posted below the video says:

An isomorphism is an underlying structure that unites outwardly different mathematical expressions. What underlying structure do these figures share? What other isomorphisms of this structure will you discover?

One of the reasons I LOVE this video is because I really like how the shapes change with the music– which is played in a very interesting time signature. I also love how you can learn a lot about the different growing shape patterns by comparing them. Watch how they grow as the video flips from pattern to pattern. What do you notice? What does the music tell you about their growth?

This video is by a math educator from North Carolina named Stuart Jeckel. The only thing written about him on his “About” page is, “The Art of Math”– so he’s a bit of a mystery! He has three more beautiful videos, all of which present little puzzles for you to solve. Check them out!

(Five-four isn’t a common time-signature for music, but it makes some great pieces. Check out this particularly awesome one. Anyone want to try making a growing shape pattern video to this tune?)


Here is an example of one of my favorite types of geometric patterns– the parquet deformation. To make one, you start with a tessellation. Then you change it- very gradually- until you’ve made a completely different tessellation that’s connected by many tiny steps to the original one.

I love to draw them. It’s challenging, but full of surprises. I never know what it’s going to look like in the end.

2012_10_31-par5Want to try making your own? Check out this site by the professors/architects Tuğrul Yazar and Serkan Uysal. They had one of their classes map out how some different parquet deformations are made. They mostly used computers, but you could follow their instructions by hand, if you like. The image above is a map for the first deformation I showed.

Click on this link to see some awesome deformations made out of tiles. Aren’t they beautiful? And here’s one made by mathematical artist Craig Kaplan. It has a great fractal quality to it:


Finally, here’s something I’ve been meaning to share with you for ages! Do you ever crave a good puzzle and aren’t sure where to find one? Look no farther than the Saint Ann’s School Problem of the Week! Each week, math teacher Richard Mann writes a new awesome problem and posts it on this website. Here’s this week’s problem:

For November 26, 2013– In the picture below, find the shaded right triangle marked A, the equilateral triangle marked B and the striped regular hexagon marked C. Six students make the following statements about the picture below: Anne says “I can find an equilateral triangle three times the area of B.”  Ben says I can find an equilateral triangle four times the area of B.” Carol says, “I can find a find a right triangle triple the area of A.” Doug says, “I can find a right triangle five times the area of A.” Eloise says, “I can find a regular hexagon double the area of C.” Frank says, “I can find a regular hexagon three times the area of C.” Which students are undoubtedly mistaken?

30- 60-90

If you solve this week’s problem, send us a solution!

Bon appetit!

Numenko, Turning Square, and Toilet Paper

Welcome to this week’s Math Munch!

Have you ever played Scrabble or Bananagrams? Can you imagine versions of these games that would use numbers instead of letters?

Meet Tom Lennett, who imagined them and then made them!

Tom playing Numenko with his grandkids.

Tom playing Numenko with his grandkids.

Numemko is a crossnumber game. Players build up number sentences, like 4×3+8=20, that cross each other like in a crossword puzzle. There is both a board game version of Numenko (like Scrabble) and a bag game version (like Banagrams). Tom invented the board game years ago to help his daughter get over her fear of math. He more recently invented the bag game for his grandkids because they wanted a game to play where they didn’t have to wait their turn!

The Multichoice tile.

The Multichoice tile.

One important feature of Numenko is the Multichoice tile. Can you see how it can represent addition, subtraction, multiplication, division, or equality?

How would you like to have a Numenko set of your own? Well, guess what—Tom holds weekly Numenko puzzle competitions with prizes! You can see the current puzzle on this page, as well as the rules. Here’s the puzzle at the time of this post—the week of November 3, 2013.

Can you replace the Multichoice tiles to create a true number sentence?

Challenge: replace the Multichoice tiles to create a true number sentence.

I can assure you that it’s possible to win Tom’s competitions, because one of my students and I won Competition 3! I played my first games of Numenko today and really enjoyed them. I also tried making some Numenko puzzles of my own; see the sheet at the bottom of this post to see some of them.

Tom in 1972.

Tom in 1972.

In emailing with Tom I’ve found that he’s had a really interesting life. He grew up in Scotland and left school before he turned 15. He’s been a football-stitcher, a barber, a soldier, a distribution manager, a paintball site operator, a horticulturist, a property developer, and more. And, of course, also a game developer!

Do you have a question you’d like to ask Tom? Send it in through the form below, and we’ll try to include it in our upcoming Q&A!


The level editor.

Say, do you like Bloxorz? I sure do—it’s one of my favorite games! So imagine my delight when I discovered that a fan of the game—who goes by the handle Jz Pan—created an extension of it where you can make your own levels. Awesome, right? It’s called Turning Square, and you can download it here.

(You’ll need to uncompress the file after downloading, then open TurningSquare.exe. This is a little more involved than what’s usual here on Math Munch, but I promise it’s worth it! Also, Turning Square has only been developed for PC. Sorry, Mac fans.)

The level!

The level I made!

But wait, there’s more! Turning Square also introduces new elements to Bloxorz, like slippery ice and pyramids you can trip over. It has a random level generator that can challenge you with different levels of difficulty. Finally, Turning Square includes a level solver—it can determine whether a level that you create is possible or not and how many steps it takes to complete.

Jz Pan is from China and is now a graduate student at the Chinese Academy of Sciences, majoring in mathematics and studying number theory. Jz Pan made Turning Square in high school, back in 2008.

Jz Pan has agreed to answer some of your questions! Use the form below to send us some.

If you make a level in Turning Square that you really like, email us the .box file and we can share it with everyone through our new Readers’ Gallery! Here is my level from above, if you want to try it out.

Jz Pan has also worked on an even more ambitious extension of Bloxorz called Turning Polyhedron. The goal is the same, but like the game Dublox, the shape that you maneuver around is different. Turning Polyhderon features several different shapes. Check out this video of it being played with a u-polyhedron!

And if you think that’s wild, check out this video with multiple moving blocks!

Last up this week, have you ever heard that it’s impossible to fold a piece of paper in half more than eight times? Or maybe it’s seven…? Either way, it’s a “fact” that seems to be common knowledge, and it sure seems like it’s true when you try to fold up a standard sheet of paper—or even a jumbo sheet of paper. The stack sure gets thick quickly!

Britney Gallivan and her 11th fold.

Britney and her 11th fold.

Well, here’s a great story about a teenager who decided to debunk this “fact” with the help of some math and some VERY big rolls of toilet paper. Her name is Britney Gallivan. Back in 2001, when she was a junior in high school, Britney figured out a formula for how much paper she’d need in order to fold it in half twelve times. Then she got that amount of paper and actually did it!

Due to her work, Britney has a citation in MathWorld’s article on folding and even her own Wikipedia article. After high school, Britney went on to UC Berkeley where she majored in Environmental Science. I’m trying to get in touch with Britney for an interview—if you have a question for her, hold onto it, and I’ll keep you posted!

EDIT: I got in touch with Britney, and she’s going to do an interview!

A diagram that illustrates how Britney derived her equation.

A diagram that illustrates how Britney derived her equation.

The best place to read more about Britney’s story in this article at pomonahistorical.org—the historical website of Britney’s hometown. Britney’s story shows that even when everyone else says that something’s impossible, that doesn’t mean you can’t be the one to do it. Awesome.

I hope you enjoy trying some Numenko puzzles, tinkering with Turning Square, and reading about Britney’s toilet paper adventure.

Bon appetit!

PS Want to see a video of some toilet-paper folding? Check out the very first “family math” video by Mike Lawler and his kids.

Reflection Sheet – Numenko, Turning Square, and Toilet Paper

Celebration of Mind, Cutouts, and the Problem of the Week

Welcome to this week’s Math Munch!  We’re going to revisit the work of Martin Gardner, look at some beautiful mathematical art, and see if we can dig into a college’s “problem of the week” program.

Martin Gardner

Martin Gardner

Last October, I wrote about Martin Gardner. He is one of the great popularizers of mathematics, known for his puzzles, columns in Scientific American, and over 100 books. Around the time of his birthday, October 21st, each year, people around the world participate in a global “Gathering4Gardner” — a so-called Celebration of Mind.

Two Sipirals

One of Martin Gardner’s many puzzles

These are gatherings of two or more people taking time to dig into the kinds of mathematics that Martin Gardner loved so much. Below you can find lots of ways to participate and share with family, friends, or strangers.

First, If you want to learn more about Gardner himself, here’s a very detailed interview. You can also try solving some of Gardner’s great puzzles. We featured both of these last year, but I recently found a whole new page of resources and activities for the Celebration of Mind.

In the video on the left you can see a geometric vanish like those we’ve previously featured (Get off the Earth, and Chocolate). The second is a surprising play on the Möbius Strip which we’ve also featured before (Art and Videos + Möbius Hearts). I hope you’ll find some time this week to celebrate Martin Gardner’s love of math and help grow your own. (Though, I guess if you’re reading this, you already are!)

Up next, check out the work of artist Elena Mir. This video shows a series of artworks she created over the last four years. They feature stacks of cut paper to form geometric shapes, and they make me wonder what I could make out of cut paper. If you make something, please let us now.

It reminds me of the work of Matt Shlian that we featured in our very first post.  You can watch Matt’s TED Talk or visit his website to see all sorts of cutouts and other paper sculptures, plus incredible videos like the one below. It might be my favorite video I’ve ever posted on Math Munch.

Finally, Macalester College in St. Paul, Minnesota has a weekly problem that they offer to their students, and the problem archives can be found online. These are for college students, so some of them are advanced or phrased in technical language, but I think we can find some that all of us can dig in to. Give these a try:

Have a mathematical week, and let us know if you do anything for the Celebration of Mind. Bon appetit!

God’s Number, Chocolate, and Devil’s Number

Welcome to this week’s Math Munch! This week, I’m sharing with you some math things that make me go, “What?!” Maybe you’ll find them surprising, too.

The first time I heard about this I didn’t believe it. If you’ve never heard it, you probably won’t believe it either.

Ever tried to solve one of these? I’ve only solved a Rubik’s cube once or twice, always with lots of help – but every time I’ve worked on one, it’s taken FOREVER to make any progress. Lots of time, lots of moves…. There are 43,252,003,274,489,856,000 (yes, that’s 43 quintillion) different configurations of a Rubik’s cube, so solving a cube from any one of these states must take a ridiculous number of moves. Right?

Nope. In 2010, some mathematicians and computer scientists proved that every single Rubik’s cube – no matter how it’s mixed up – can be solved in at most 20 moves. Because only an all-knowing being could figure out how to solve any Rubik’s cube in 20 moves or less, the mathematicians called this number God’s Number.

Once you get over the disbelief that any of the 43 quintillion cube configurations can be solved in less than 20 moves, you may start to wonder how someone proved that. Maybe the mathematicians found a really clever way that didn’t involve solving every cube?

Not really – they just used a REALLY POWERFUL computer. Check out this great video from Numberphile about God’s number to learn more:

Screen Shot 2013-10-02 at 2.48.01 PM

Here’s a chart that shows how many Rubik’s cube configurations need different numbers of moves to solve. I think it’s surprising that so few required all 20 moves. Even though every cube can be solved in 20 or less moves, this is very hard to do. I think it’s interesting how in the video, one of the people interviewed points out that solving a cube in very few moves is probably much more impressive than solving a cube in very little time. Just think – it takes so much thought to figure out how to solve a Rubik’s cube at all. If you also tried to solve it efficiently… that would really be a puzzle.

Next, check out this cool video. Its appealing title is, “How to create chocolate out of nothing.”

This type of puzzle, where area seems to magically appear or disappear when it shouldn’t, is called a geometric vanish. We’ve been talking about these a lot at school, and one of the things we’re wondering is whether you can do what the guy in the video did again, to make a second magical square of chocolate. What do you think?

infinityJHFinally, I’ve always found infinity baffling. It’s so hard to think about. Here’s a particularly baffling question: which is bigger, infinity or infinity plus one? Is there something bigger than infinity?

I found this great story that helps me think about different sizes of infinity. It’s based on similar story by mathematician Raymond Smullyan. In the story, you are trapped by the devil until you guess the devil’s number. The story tells you how to guarantee that you’ll guess the devil’s number depending on what sets of numbers the devil chooses from.

Surprisingly, you’ll be able to guess the devil’s number even if he picks from a set of numbers with an infinite number of numbers in it! You’ll guess his number if he picked from the counting numbers larger than zero, positive or negative counting numbers, or all fractions and counting numbers. You’d think that there would be too many fractions for you to guess the devil’s number if he included those in his set. There are infinitely many counting numbers – but aren’t there even more fractions? The story tells you about a great way to organize your guessing that works even with fractions. (And shows that the set of numbers with fractions AND counting numbers is the same size as the set of numbers with just counting numbers… Whoa.)

Is there something mathematical that makes you go, “What?!” How about, “HUH?!” If so, send us an email or leave us a note in the comments. We’d love to hear about it!

Bon appetit!