Tag Archives: interactive puzzle

Nice Neighbors, Spinning GIFs, and Breakfast

A minimenger.

A minimenger.

Welcome to this week’s Math Munch!

Math projects are exciting—especially when a whole bunch of people work together. One example of big-time collaboration is the GIMPS project, where anyone can use their computer to help find the next large prime number. Another is the recent MegaMenger project, where people from all over the world helped to build a giant 3D fractal.

But what if I told you that you can join up with others on the internet to discover some brand-new math by playing a webgame?

Chris Staecker is a math professor at Fairfield University. This past summer he led a small group of students in a research project. Research Experiences for Undergraduates—or REUs, as they’re called—are summer opportunities for college students to be mentored by professors. Together they work to figure out some brand-new math.

The crew from last summer's REU at Fairfield. Chris is furthest in the back.

The crew from last summer’s REU at Fairfield. Chris is furthest in the back.

The irreducible digital images containing 1, 5, 6, and 7 points.

The irreducible digital images containing 1, 5, 6, and 7 “chunks”.

Chris and his students Jason Haarmann, Meg Murphy, and Casey Peters worked on a topic in graph theory called “digital images”. Computer images are made of discrete chunks, but we often want to make them smaller—like with pixel art. So how can we make sure that we can make them smaller without losing too much information? That’s an important problem.

Now, the pixels on a computer screen are in a nice grid, but we could also wonder about the same question on an arbitrary connected network—and that’s what Chris, Jason, Meg, and Casey did. Some networks can be made smaller through one-step “neighbor” moves while still preserving the correct connection properties. Others can’t. By the end of the summer, the team had come up with enough results about digital images with up to eight chunks to write about them in a paper.

To help push their research further, Chris has made a webgame that takes larger networks and offers them as puzzles to solve. Here’s how I solved one of them:

NiceNeighbors

See how the graph “retracts” onto itself, just by moving some of the nodes on top of their neighbors? That’s the goal. And there are lots of puzzles to work on. For many of them, if you solve them, you’ll be the first person ever to do so! Mathematical breakthrough! Your result will be saved, the number at the bottom of the screen will go up by one, and Chris and his students will be one step closer to classifying unshrinkable digital images.

Starting with the tutorial for Nice Neighbors is a good idea. Then you can try out the unsolved experimental puzzles. If you find success, please let us know about in the comments!

Do you have a question for Chris and his students? Then send it to us and we’ll try to include it in our upcoming Q&A with them.

 

Next up: you probably know by now that at Math Munch, we just can’t get enough of great mathy gifs. Well, Sumit Sijher has us covered this week, with his Tumblr called archery.

Here are four of Sumit’s gifs. There are plenty more where these came from. This is a nice foursome, though, because they all spin. Click to see the images full-sized!

tumblr_mdv99p6WcP1qfjvexo1_500

How many different kinds of cubes can you spot?

This one reminds me of the Whitney Music Box.

This one reminds me of the
Whitney Music Box.

Whoa.

Clockwise or counterclockwise?

Clockwise or counterclockwise?

I really appreciate how Sumit also shares the computer code that he uses to make each image. It gives a whole new meaning to “show your work”!

Through Sumit’s work I discovered that WolframAlpha—an online calculator that is way more than a calculator—has a Tumblr, too. By browsing it you can find some groovy curves and crazy estimations. Sumit won an honorable mention in Wolfram’s One-Liner Competition back in 2012. You can see his entry in this video.

And now for the most important meal of the day: breakfast. Mathematicians eat breakfast, just like everyone else. What do mathematicians eat for breakfast? Just about any kind of breakfast you might name. For some audio-visual evidence, here’s a collection of sound checks by Numberphile.

Sconic sections. Yum!

Sconic sections. Yum!

If that has you hungry for a mathematical breakfast, you might enjoy munching on some sconic sectionsa linked-to-itself bagel, or some spirograph pancakes.

Bon appetit!

Weights, Crazy Geometry Game, and Pumpkin Polyhedra

Welcome to this week’s Math Munch!

Weighing puzzleHere’s a puzzle for you: You have 12 weights, 11 of which weigh the same amount and 1 of which is different. Luckily you also have a balance, but you’re only allowed to use it three times. Can you figure out which weight is the different weight?

You certainly can! I won’t tell you how, but you can figure it out for yourself while playing this interactive weight game. This puzzle is tricky, but definitely fun. If one weight puzzle isn’t enough for you, you’re in luck– there are many, many variations! Check out this site to try a similar puzzle with nine weights, ten weights, and 27 weights.

Circle two pack

My solution to the Circle Pack 2 challenge. Can you do it in only 5 moves?

Next up, if you like drawing challenges, this is the game for you. Check out this crazy geometry game, in which you have to draw different shapes (like perfect equilateral triangles, squares, pentagons, and groups of circles of particular sizes) using only circles and straight lines! Here’s my solution to one of the challenges, the Circle Pack 2. See the two smaller circles inside of the larger middle circle? That’s what I wanted to draw– but I had to make all of those other circles and lines to get there! I did the Circle Pack 2 challenge in 8 moves, but apparently there’s a way to do it in only 5…

Truncated icosahedron pumpkinFinally, it’s pumpkin season again! Every year I scour the internet for new math-y ways to carve pumpkins. We’re all in luck this year– because I found great instructions for how to carve pumpkin polyhedra from Math Craft!  Check out this site to learn how to carve all the basics– tetrahedra, cubes, octahedra, dodecahedra, and (my favorite) icosahedra– and a bonus polyhedron, the truncated icosahedron (also know as the soccer ball).

Pumpkin polyhedra

Pumpkin Platonic polyhedra!

 

Don’t forget to make pi with the leftover pumpkin! Oh, and, bon appetit!

 

 

Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

GirlsAngleCoverGirls’ Angle is a math club for girls. Since 2007 it has helped girls to grow their love of math through classes, events, mentorship, and a vibrant mathematical community. Girls’ Angle is based in Cambridge, Massachusetts, but its ideas and resources reach around the world through the amazing power of the internet. (And don’t you worry, gentlemen—there’s plenty for you to enjoy on the site as well.)

Amazingly, the site contains an archive of every issue of Girls’ Angle Bulletin, a wonderful bimonthly journal to “foster and nurture girls’ interest in mathematics.” In their most recent issue, you’ll find an interview with mathematician Karen E. Smith, along with several articles and puzzles about balance points of shapes.

There’s so much to dig into at Girls’ Angle! In addition to the Bulletins, there are two pages of mathematical videos. The first page shares a host of videos of women in mathematics sharing a piece of math that excited them when they were young. The most recent one is by Bridget Tenner, who shares about Pick’s Theorem. The second page includes several videos produced by Girls’ Angle, including this one called “Summer Vacation”.

Girls’ Angle can even help you buy a math book that you’d like, if you can’t afford it. For so many reasons, I hope you’ll find some time to explore the Girls’ Angle site over your summer break. (And while you’ve got your explorer’s hat on, maybe you’ll tour around Math Munch, too!)

I did a Google search recently for “regular tilings.” I needed a few quick pictures of the usual triangle, square, and hexagon tilings for a presentation I was making. As I scrolled along, this image jumped out at me:

hexspiral

What is that?! It certainly is a tiling, and all the tiles are the “same”—even if they are different sizes. Neat!

Clicking on the image, I found myself transported to a page all about spiral tilings at the Geometry Junkyard. The site is a whole heap of geometrical odds and ends—and a place that I’ve stumbled across many times over the years. Here are a few places to get started. I’m sure you’ll enjoy poking around the site to find some favorite “junk” of your own.

Spirals

Spirals

Circles and spheres

Circles & spheres

Coloring

Coloring

Last up this week, you may have seen this coin puzzle before. Can you make the triangle point downwards by moving just three pennies?triangleflip

There are lots of variants of this puzzle. You can find some in an online puzzle game called Coins. In the game you have to make arrangements of coins, but the twist is that you can only move a coin to a spot where would it touch at least two other coins. I’m enjoying playing Coins—give it a try!

I solved this Coins puzzle in four moves. Can you? Can you do better?

I solved this Coins puzzle in four moves. Can you? Can you do better?

That’s it for this week’s Math Munch. Bon appetit!

 

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

thomas-edison

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!

MOVES, the Tower of Hanoi, and Mathigon

Welcome to this week’s Math Munch!

MOVESThe Math Munch team just wrapped up attending the first MOVES Conference, which was put on by the Museum of Math in NYC. MOVES is a recreational math conference and stands for Mathematics of Various Entertaining Subjects. Anna coordinated the Family Activities track at the conference and Paul gave a talk about his imbalance problems. I was just there as an attendee and had a blast soaking up wonderful math from some amazing people!

AnnaMOVES PaulMOVES JustinMOVES

Who all was there? Some of our math heroes—and familiar faces on Math Munch—like Erik Demaine, Tanya Khovanova, Tim and Tanya Chartier, and Henry Segerman, just to name just a few. I got to meet and learn from many new people, too! Even though I know it’s true, it still surprises me how big and varied the world of math and mathematicians is.

SuMOVES

Suzanne Dorée at MOVES.

One of my favorite talks at MOVES was given by Suzanne Dorée of Augsburg College. Su spoke about research she did with a former student—Danielle Arett—about the puzzle known as the Tower of Hanoi. You can try out this puzzle yourself with this online applet. The applet also includes some of the puzzle’s history and even some information about how the computer code for the applet was written.

danielle

Danielle Arett

A pice of a Tower of Hanoi graph with three pegs and four disks.

A piece of a Tower of Hanoi graph with three pegs and four disks.

But back to Su and Danielle. If you think of the different Tower of Hanoi puzzle states as dots, and moving a disk as a line connecting two of these dots, then you can make a picture (or graph) of the whole “puzzle space”. Here are some photos of the puzzle space for playing the Tower of Hanoi with four disks. Of course, how big your puzzle space graph is depends on how many disks you use for your puzzle, and you can imagine changing the number of pegs as well. All of these different pictures are given the technical name of Tower of Hanoi graphs. Su and Danielle investigated these graphs and especially ways to color them: how many different colors are needed so that all neighboring dots are different colors?

Images from Su and Danielle's paper. Towers of Hanoi graphs with four pegs.

Images from Su and Danielle’s paper. Tower of Hanoi graphs with four pegs.

 

Su and Danielle showed that even as the number of disks and pegs grows—and the puzzle graphs get very large and complicated—the number of colors required does not increase quickly. In fact, you only ever need as many colors as you have pegs! Su and Danielle wrote up their results and published them as an article in Mathematics Magazine in 2010.

Today Danielle lives in North Dakota and is an analyst at Hartford Funds. She uses math every day to help people to grow and manage their money. Su teaches at Ausburg College in Minnesota where she carries out her belief “that everyone can learn mathematics.”

Do you have a question for Su or Danielle—about their Tower of Hanoi research, about math more generally, or about their careers? If you do, send them to us in the form below for an upcoming Q&A!

UPDATE: We’re no longer accepting questions for Su and Danielle. Their interview will be posted soon! Ask questions of other math people here.

MathigonLast up, here’s a gorgeous website called Mathigon, which someone shared with me recently. It shares a colorful and sweeping view of different fields of mathematics, and there are some interactive parts of the site as well. There are features about graph theory—the field that Su and Danielle worked in—as well as combinatorics and polyhedra. There’s lots to explore!

Bon appetit!

Prime Gaps, Mad Maths, and Castles

Welcome to this week’s Math Munch!

It has been a thrilling last month in the world of mathematics. Several new proofs about number patterns have been announced. Just to get a flavor for what it’s all about, here are some examples.

I can make 15 by adding together three prime numbers: 3+5+7. I can do this with 49, too: 7+11+31. Can all odd numbers be written as three prime numbers added together? The Weak Goldbach Conjecture says that they can, as long as they’re bigger than five. (video)

11 and 13 are primes that are only two apart. So are 107 and 109. Can we find infinitely many such prime pairs? That’s called the Twin Prime Conjecture. And if we can’t, are there infinitely many prime pairs that are at most, say, 100 apart? (video, with a song!)

Harald Helfgott

Harald Helfgott

Yitang "Tom" Zhang

Yitang “Tom” Zhang

People have been wondering about these questions for hundreds of years. Last month, Harald Helfgott showed that the Weak Goldbach Conjecture is true! And Yitang “Tom” Zhang showed that there are infinitely many prime pairs that are at most 70,000,000 apart! You can find lots of details about these discoveries and links to even more in this roundup by Evelyn Lamb.

What’s been particularly fabulous about Tom’s result about gaps between primes is that other mathematicians have started to work together to make it even better. Tom originally showed that there are an infinite number of prime pairs that are at most 70,000,000 apart. Not nearly as cute as being just two apart—but as has been remarked, 70,000,000 is a lot closer to two than it is to infinity! That gap of 70,000,000 has slowly been getting smaller as mathematicians have made improvements to Tom’s argument. You can see the results of their efforts on the polymath project. As of this writing, they’ve got the gap size narrowed down to 12,006—you can track the decreasing values down the page in the H column. So there are infinitely many pairs of primes that are at most 12,006 apart! What amazing progress!

Two names that you’ll see in the list of contributors to the effort are Andrew Sutherland and Scott Morrison. Andrew is a computational number theorist at MIT and Scott has done research in knot theory and is at the Australian National University. They’ve improved arguments and sharpened figures to lower the prime gap value H. They’ve contributed by doing things like using a hybrid Schinzel/greedy (or “greedy-greedy”) sieve. Well, I know what a sieve is and what a greedy algorithm is, but believe me, this is very complicated stuff that’s way over my head. Even so, I love getting to watch the way that these mathematicians bounce ideas off each other, like on this thread.

Andrew Sutherland

Andrew Sutherland

Click through to see Andrew next to an amazing Zome creation!

Andrew. Click this!

Scott Morrison

Scott Morrison

Andrew and Scott have agreed to answer some of your questions about their involvement in this research about prime gaps and their lives as mathematicians. I know I have some questions I’m curious about! You can submit your questions in the form below:

I can think of only two times in my life where I was so captivated by mathematics in the making as I am by this prime gaps adventure. Andrew Wiles’s proof of Fermat’s Last Theorem was on the fringe of my awareness when it came out in 1993—its twentieth anniversary of his proof just happened, in fact. The result still felt very new and exciting when I read Fermat’s Enigma a couple of years later. Grigori Perelman’s proof of the Poincare Conjecture made headlines just after I moved to New York City seven years ago. I still remember reading a big article about it in the New York Times, complete with a picture of a rabbit with a grid on it.

This work on prime gaps is even more exciting to me than those, I think. Maybe it’s partly because I have more mathematical experience now, but I think it’s mostly because lots of people are helping the story to unfold and we can watch it happen!

fig110u2bNext up, I ran across a great site the other week when I was researching the idea of a “cut and slide” process. The site is called Mad Maths and the page I landed on was all about beautiful dissections of simple shapes, like circles and squares. I’ve picked out one that I find especially charming to feature here, but you might enjoy seeing them all. The site also contains all kinds of neat puzzles and problems to try out. I’m always a fan of congruent pieces problems, and these paper-folding puzzles are really tricky and original. (Or maybe, origaminal!) You’ll might especially like them if you liked Folds.

Christian's applet displaying the original four-room castle.

Christian’s applet displaying the original four-room castle.

Finally, we previously posted about Matt Parker’s great video problem about a princess hiding in a castle. Well, Christian Perfect of The Aperiodical has created an applet that will allow you to explore this problem—plus, it’ll let you build and try out other castles for the princess to hide in. Super cool! Will I ever be able to find the princess in this crazy star castle I designed?!

Crazy star castle!

My crazy star castle!

And as summer gets into full swing, the other kind of castle that’s on my mind is the sandcastle. Take a peek at these photos of geometric sandcastles by Calvin Seibert. What shapes can you find? Maybe Calvin’s creations will inspire your next beach creation!

Bon appetit!

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Folds, GIMPS, and More Billiards

Welcome to this week’s Math Munch!

First up, we’ve often featured mathematical constructions made of origami. (Here are some of those posts.) Origami has a careful and peaceful feel to it—a far cry from, say, the quick reflexes often associated with video games. I mean, can you imagine an origami video game?

heartfolds

One of Fold’s many origami puzzles.

Well, guess what—you don’t have to, because Folds is just that! Folds is the creation of Bryce Summer, a 21-year-old game designer from California. It’s so cool. The goal of each level of its levels is simple: to take a square piece of paper and fold it into a given shape. The catch is that you’re only allowed a limited number of folds, so you have to be creative and plan ahead so that there aren’t any loose ends sticking out. As I’ve noted before, my favorite games often require a combo of visual intuition and careful thinking, and Folds certainly fits the bill. Give it a go!

Once you’re hooked, you can find out more about Bryce and how he came to make Folds in this awesome Q&A. Thanks so much, Bryce!

gimpsNext up, did you know that a new largest prime number was discovered less than a month ago? It’s very large—over 17 million digits long! (How many pages would that take to print or write out?) That makes it way larger than the previous record holder, which was “only” about 13 million digits long. Here is an article published on the GIMPS website about the new prime number and about the GIMPS project in general.

What’s GIMPS you ask? GIMPS—the Great Internet Mersenne Primes Search—is an example of what’s called “distributed computing”. Testing whether a number is prime is a simple task that any computer can do, but to check many or large numbers can take a lot of computing time. Even a supercomputer would be overwhelmed by the task all on its own, and that’s if you could even get dedicated time on it. Distributed computing is the idea that a lot of processing can be accomplished by having a lot of computers each do a small amount of work. You can even sign up to help with the project on your own computer. What other tasks might distributed computing be useful for? Searching for aliens, perhaps?

GIMPS searches only for a special kind of prime called Mersenne primes. These primes are one less than a power of two. For instance, 7 is a Mersenne prime, because it’s one less that 8, which is the third power of 2. For more on Mersenne primes, check out this video by Numberphile.

Finally, we’ve previously shared some resources about the math of billiards on Math Munch. Below you’ll find another take on bouncing paths as Michael Moschen combines the math of billiards with the art of juggling.

So lovely. For more on this theme, here’s a second video to check out.

Bon appetit!