Tag Archives: arithmetic

We Use Math, Integermania, and Best-of-Seven

Welcome to this week’s Math Munch!

astronaut“When will I use math?” Have you ever asked this question? Well, then you are in for a treat, because the good people of We Use Math have some answers for you! This site was created by the Math Department at Brigham Young University to help share information about career paths that are opened up by studying mathematics. Here’s their introductory video:

The We Use Math site shares write-ups about a wide range of career opportunities that involve doing mathematics. I was glad to learn more about less-familiar mathy careers like technical writing and cost estimation. Also, my brother has studied some operations management in college, so it was great to read the overview of that line of work. In addition, the We Use Math site has pages about recent math discoveries and about unsolved math problems. Check them out!

Next up is one of my long-time favorite websites: Integermania!

Perhaps you’ve heard of the four 4’s problem before. Using four 4’s and some arithmetic operations, can you make the numbers from 1 to 20? Or even higher? Some numbers are easy to make, like 16. It’s 4+4+4+4. Some are sneakier, like 1. One way it can be created is (4+4)/(4+4). But what about 7? Or 19? This is a very common type of problem in mathematics—which math objects of a certain type can be built with limited tools?

swilson21-e1315080873212

Steven J. Wilson

Integermania is a website where people from around the world have submitted number creations made of four small numbers and operations. It’s run by Steven J. Wilson, a math professor at Johnson County Community College in Kansas. (Steven has even more great math resources at his website Milefoot.com)

There are many challenges at Integermania: four 4’s, the first four prime numbers, the first four odds, and even the digits of Ramanujan’s famous taxicab number (1729).

Here are some number creations made of the first four prime numbers. Can you make some of your own?

Here are some number creations made of the first four prime numbers.
Can you make some of your own?

One of my favorite aspects of Integermania is the way it rates number creations by “exquisiteness level“. If a number creation is made using only simple operations—like addition or multiplication—then it’s regarded as more exquisite than if it uses operations like square roots or percentages. I also love how Integermania provides an opportunity for anyone to make their mark in the big world of mathematical research—it’s like scrawling a mathematical “I wuz here!” After years of visiting the site, I just submitted for the first time some number creations of my own. I’ll let you know how it goes, and I’d love to hear about it if you decide to submit, too.

Here are recaps of all the World Series since 1903 from MLB.com

Here are recaps of all the World Series since 1903 from MLB.com

Now coming to the plate: my final link of the week! Monday was the first day of the new Major League Baseball season. I want to share with you a New York Times article from last December. It’s called Keeping Score: Over in Four About a Fifth of the Time. The article digs into the outcomes of all of the World Series championships—not so much who won as how they won. It takes four victories to win a seven-game series, and there are 35 different ways that a best-of-seven series can play out, put in terms of wins and losses for the overall winner. For instance, a clean sweep would go WWWW, while another sequence would be WWLLWW. The article examines which of these win-loss sequences have been the most common in the World Series.

(Can you figure out why there are 35 possible win-loss sequences in a seven-game series? What about for a best-of-five series? And what if we tried to model the outcome of a series by assuming each team has a fixed chance of winning each game?)

worldseriesstats

A clip of the stats that are displayed in the Times article. Click through to see it all.

I was curious to know if the same results held true in other competitions. Are certain win-loss sequences rare across different sports? Are “sweeps” the most common outcome? After sifting through Wikipedia for a while, I was able to compile the statistics about win-loss sequences for hockey’s Stanley Cup Finals. This has been a best-of-seven series since 1939, and it has been played 73 times since then. (It didn’t happen in 2005 because of a lockout.) You can see the results of my research in this document. Two takeaways: sweeps are also the most common result in hockey, but baseball more frequently requires the full seven games to determine a winner.

It could be a fun project to look at other best-of-seven series, like the MLB’s League Championship Series or basketball’s NBA Finals. If you pull that data together, let us know in the comments!

Batter up, and bon appetit!

******

UPDATE (4/4/13): My first set of five number creations was accepted and are now posted on the Ramanujan challenge page. Here are the three small ones! Can you find a more exquisite way of writing 47 than I did?

myintegermania

Sandpiles, Prime Pages, and Six Dimensions of Color

Welcome to this week’s Math Munch!

Four million grains of sand dropped onto an infinite grid. The colors represent how many grains are at each vertex. From this gallery.

We got our first snowfall of the year this past week, but my most recent mathematical find makes me think of summertime instead. The picture to the right is of a sandpile—or, more formally, an Abelian sandpile model.

If you pour a bucket of sand into a pile a little at a time, it’ll build up for a while. But if it gets too tall, an avalanche will happen and some of the sand will tumble away from the peak. You can check out an applet that models this kind of sand action here.

A mathematical sandpile formalizes this idea. First, take any graph—a small one, a medium sided one, or an infinite grid. Grains of sand will go at each vertex, but we’ll set a maximum amount that each one can contain—the number of edges that connect to the vertex. (Notice that this is four for every vertex of an infinite square grid). If too many grains end up on a given vertex, then one grain avalanches down each edge to a neighboring vertex. This might be the end of the story, but it’s possible that a chain reaction will occur—that the extra grain at a neighboring vertex might cause it to spill over, and so on. For many more technical details, you might check out this article from the AMS Notices.

This video walks through the steps of a sandpile slowly, and it shows with numbers how many grains are in each spot.

A sandpile I made with Sergei’s applet

You can make some really cool images—both still and animated—by tinkering around with sandpiles. Sergei Maslov, who works at Brookhaven National Laboratory in New York, has a great applet on his website where you can make sandpiles of your own.

David Perkinson, a professor at Reed College, maintains a whole website about sandpiles. It contains a gallery of sandpile images and a more advanced sandpile applet.

Hexplode is a game based on sandpiles.

I have a feeling that you might also enjoy playing the sandpile-inspired game Hexplode!

Next up: we’ve shared links about Fibonnaci numbers and prime numbers before—they’re some of our favorite numbers! Here’s an amazing fact that I just found out this week. Some Fibonnaci numbers are prime—like 3, 5, and 13—but no one knows if there are infinitely many Fibonnaci primes, or only finitely many.

A great place to find out more amazing and fun facts like this one is at The Prime Pages. It has a list of the largest known prime numbers, as well as information about the continuing search for bigger ones—and how you can help out! It also has a short list of open questions about prime numbers, including Goldbach’s conjecture.

Be sure to peek at the “Prime Curios” page. It contains intriguing facts about prime numbers both large and small. For instance, did you know that 773 is both the only three-digit iccanobiF prime and the largest three-digit unholey prime? I sure didn’t.

Last but not least, I ran across this article about how a software company has come up with a new solution for mixing colors on a computer screen by using six dimensions rather than the usual three.

Dimensions of colors, you ask?

The arithmetic of colors!

Well, there are actually several ways that computers store colors. Each of them encodes colors using three numbers. For instance, one method builds colors by giving one number each to the primary colors yellow, red, and blue. Another systems assigns a number to each of hue, saturation, and brightness. More on these systems here. In any of these systems, you can picture a given color as sitting within a three-dimensional color cube, based on its three numbers.

A color cube, based on the RGB (red, green, blue) system.

If you numerically average two colors in these systems, you don’t actually end up with the color that you’d get by mixing paint of those two colors. Now, both scientists and artists think about combining colors in two ways—combining colored lights and combining colored pigments, or paints. These are called additive and subtractive color models—more on that here. The breakthrough that the folks at the software company FiftyThree made was to assign six numbers to each color—that is, to use both additive and subtractive ideas at the same time. The six numbers assigned to a given number can be thought of as plotting a point in a six-dimensional space—or inside of a hyper-hyper-hypercube.

I think it’s amazing that using math in this creative way helps to solve a nagging artistic problem. To get a feel for why mixing colors using the usual three-coordinate system is such a problem, you might try your hand at this color matching game. For even more info about the math of color, there’s some interesting stuff on this webpage.

Bon appetit!

Fractions, Sam Loyd, and a MArTH Journal

Welcome to this week’s Math Munch!

Check out this awesome graph:

What is it?  It’s a graph of the Farey Fractions, with the denominator of the (simplified) fraction on the vertical axis and the value of the fraction on the horizontal axis, made by mathematician and professor at Wheelock College Debra K. Borkovitz (previously).  Now, I’d never heard of Farey Fractions before I saw this image – but the graph was so cool that I wanted to learn all about them!

So, what are Farey Fractions, you ask?  Debra writes all about them and the cool visual patterns they make in this post.  To make a list of Farey Fractions you first pick a number – say, 5.  Then, you list all of the fractions between 0 and 1 whose denominators are less than or equal to the number you picked.  So, as Debra writes in her post, for 5 the list of Farey Fractions is:

As Debra writes, there are so many interesting patterns in Farey Fractions – many of which become much easier to see (literally) when you make a visualization of them.  Debra has made several awesome applets using the program GeoGebra, which she links to in her post.  (You can download GeoGebra and make applets of your own by visiting our Free Math Software page.)  These applets really show the power of using graphs and pictures to learn more about numbers.  To play with the applet that made the picture above, click here.  Check out her post to play with another applet, and to read more about the interesting patterns in Farey Fractions.

Next, check out this website devoted to the puzzles of puzzlemaster Sam Loyd.  Sam Loyd was a competitive chess player and professional puzzle-writer who lived at the end of the nineteenth century.  He wrote many puzzles that are still famous today – like the baffling Get Off the Earth puzzle.  Click the link to play an interactive version of the Get Off the Earth puzzle.

The site has links to numerous Sam Loyd puzzles.  Check out the Picture Puzzles, in which you have to figure out what object is described by the picture, or the Puzzleland Puzzles, which feature characters from the fictional place Puzzleland that Sam created.

Snow MArTH, made by MArTHist Eva Hild and others at a snow sculpture event in Colorado. From the Spring, 2011 Hyperseeing.

Finally, take a look at some of the beautiful pictures and fascinating articles in this journal about mathematical art (a.k.a., MArTH) called Hyperseeing.  Hyperseeing is edited by mathematicians and artists Nat Friedman and Ergun Akleman.  Hyperseeing is published by the International Society of the Arts, Mathematics, and Architecture, which Nat founded to encourage education connecting the arts, architecture, and math – which we here at Math Munch love!  In one of his articles, Nat defines hyperseeing as, “Interdisciplinary education… concerned with seeing from multiple viewpoints in a very general sense.  Hyperseeing is a more complete way of seeing.”

There are so many beautiful images to look at and interesting articles to read in Hyperseeing.  Among other things, each edition of Hyperseeing features a mathematical comic by Ergun.  Here are some of my favorite Hyperseeings from the archives:

This edition of Hyperseeing features art made from Latin Squares and “organic geometry” art, among many other things.

This edition of Hyperseeing features crocheted hyperbolic surfaces (which we featured not long ago in this Math Munch!) and sculptures made with a 3-D printer, among many other things.

This is the first edition of Hyperseeing. In it, Nat describes the mission of Hyperseeing and the International Society of the Arts, Mathematics, and Architecture.

Bon appetit!

P.S. – You may have noticed a new thing to click off to the right, below the Favorite Munches.  This is our For Teachers section.  The Math Munch team has put together several pages to describe how we use Math Munch in our classes and give suggestions for how you might use it, too.  Teachers and non-teachers alike may want to check out our new Why Math Munch? page, which gives our mission statement.

P.P.S. – The Math Munch team is going to Bridges on Thursday!  Maybe we’ll see you there.

A Sweater, Paper Projects, and Math Art Tools

Sondra Eklund and her Prime Factorization Sweater

Welcome to this week’s Math Munch!

Check out Sondra Eklund and her awesome prime factorization sweater! Sondra is a librarian and a writer who writes a blog where she reviews books. She also is a knitter and a lover of math!

Each number from two to one hundred is represented in order on the front of Sondra’s sweater. Each prime number is a square that’s a different color; each composite number has a rectangle for each of the primes in its prime factorization. This number of columns that the numbers are arranged into draws attention to different patterns of color. For instance, you can see a column that has a lot of yellow in it on the front of the sweater–these are all number that contain five as a factor.

You can read more about Sondra and her sweater on her blog. Also, here’s a response and variation to Sondra’s sweater by John Graham-Cumming.

Next up, do you like making origami and other constructions out of paper? Then you’ll love the site made by Laszlo Bardos called CutOutFoldUp.

Laszlo Bardos

A Rhombic Spirallohedron

A decagon slide-together

Laszlo is a high school math teacher and has enjoyed making mathematical models since he was a kid. On CutOutFoldUp you’ll find gobs of projects to try out, including printable templates. I’ve made some slide-togethers before, but I’m really excited to try making the rhombic spirallohedron pictured above! What is your favorite model on the site?

Last up, Paul recently discovered a great mathematical art applet called Recursive Drawing. The tools are extremely simple. You can make circles and squares. You can stretch these around. But most importantly, you can insert a copy of one of your drawings into itself. And of course then that copy has a copy inside of it, and on and on. With a very simple interface and very simple tools, incredible complexity and beauty can be created.

Recursive Drawing was created by Toby Schachman, an artist and programmer who graduated from MIT and now lives in New York City and attends NYU.  You can watch a demo video below.

Recursive Drawing is one of the first applets on our new Math Art Tools page. We’ll be adding more soon. Any suggestions? Leave them in the comments!

Bon appetit!

Rice, Rectangles, and Mathmagicland

Welcome to this week’s Math Munch!

Want to practice your math facts?  Want to help feed hungry people around the world?  Well, with Free Rice you can do both at once!  Every time you answer a question correctly, the website donates 10 grains of rice through the UN’s World Food Programme.  You can work on multiplication or pre-algebra, as well as vocabulary, flags of the world, and other subjects.  It’s good practice for a good cause!  What do you say?  Will you help?

Up next, meet Edmund Harriss.

I found him through his fantastic math blog, Maxwell’s Demon, but he’s also a visiting professor at the University of Arkansas and a mathematical artist to boot.  We’re going to take a look at his recent blog post, “the 2×1 rectangle and domes.”  I seriously encourage you to read the entire thing, but I’ll share a few highlights.  The 2×1 rectangle is called a domino, and when you cut one in half along the diagonal, you get a lovely triangle with nifty tiling capabilities!

Also, standard plywood comes in the same proportion (8’x4′), and they can be easily combined to make several types of domes, as you can see below.  Click here to see how a hexayurt is built.  Edmund goes on to talk about the truncated octahedron, and how we can use its shape to design these domes.  How amazingly clever!

Finally, let’s take a look at a classic Disney film, from 1959 – Donald in Mathmagicland.  Donald Duck, on some sort of hunt, finds himself in a very strange place, surrounded by numbers, shapes, and patterns.  The trees even have square roots!  Mr. Duck meets “The True Spirit of Invention,” a mysterious voice that leads him (and us) on an adventurous trip through Mathmagicland.  If you skip to 16:48 in the video, you can learn about Billiards, a game played on the 2×1 rectangle!  How fitting!

Bon appetit!

Impossible, Impossible, Impossible

Welcome to this week’s Math Munch!

The Penrose Triangle is an “impossible figure” – or so claim many reputable mathematics sources.  It’s a triangle made of square beams that all meet a right angles – which does sound pretty impossible.  Penrose polygons features in some of M. C. Escher’s most confounding artwork, like this picture:

But, little do these mathematicians know… you can build your own Penrose Triangle out of paper!  Check out these instructions and confound your friends.

Want more optical illusions?  Check out these awesome ones by scientist Michael Bach.

Mathematicians also seem pretty sure that .99999999…. = 1.  Well, trust Vi Hart to show them what’s-what.  Here’s a video in which she tells us all that, in fact, .99999999999… is NOT 1.

Finally, did you know that 13×7=28?  Well, it does.  And here’s the proof:

Bon appetit!  Oh – and April Fools!

(Beat, Beat, Beat…)

Welcome to this week’s Math Munch!

What could techno rhythms, square-pieces dissections, and windshield wipers have in common?

Animation in which progressively smaller square tiles are added to cover a rectangle completely.

The Euclidean Algorithm!

Say what?  The Euclidean Algorithm is all about our good friend long division and is a great way of finding the greatest common factor of two numbers. It relies on the fact that if a number goes into two other numbers evenly, then it also goes into their difference evenly.  For example, 5 goes into both 60 and 85–so it also goes into their difference, 25.  Breaking up big objects into smaller common pieces is a big idea in mathematics, and the way this plays out with numbers has lots of awesome aural and visual consequences.

Here’s the link that prompted this post: a cool applet where you can create your own unique rhythms by playing different beats against each other.  It’s called “Euclidean Rhythms” and was created by Wouter Hisschemöller, a computer and audio programmer from the Netherlands.

(Something that I like about Wouter’s post is that it’s actually a correction to his original posting of his applet.  He explains the mistake he made, gives credit to the person who pointed it out to him, and then gives a thorough account of how he fixed it.  That’s a really cool and helpful way that he shared his ideas and experiences.  Think about that the next time you’re writing up some math!)

For your listening pleasure, here’s a techno piece that Wouter composed (not using his applet, but with clear influences!)

Breathing Pavement

Here’s an applet that demonstrates the geometry of the Euclidean Algorithm.  If you make a rectangle with whole-number length sides and continue to chop off the biggest (non-slanty) square that you can, you’ll eventually finish.  The smallest square that you’ll chop will be the greatest common factor of the two original numbers.  See it in action in the applet for any number pair from 1 to 100, with thanks to Brown mathematics professor Richard Evan Schwartz, who maintains a great website.

Holyhedron, layer three

One more thing, on an entirely different note: Holyhedron! A polyhedron where every face contains a hole. The story is given briefly here. Pictures and further details can be found on the website of Don Hatch, finder of the smallest known holyhedron.  It’s a mathematical discovery less than a decade old–in fact, no one had even asked the question until John Conway did so in the 1990s!

Have a great week! Bon appétit!