# World’s Oldest Person, Graphing Challenge, and Escher Sketch

On April 19th, Jiroeman Kimura celebrated his 116th birthday. He was – and still is – the world’s oldest person, and the world’s longest living man – ever. (As far as researchers know, that is. There could be a man who has lived longer that the public doesn’t know about.) The world’s longest living woman was Jeanne Calment, who lived to be 122 and a half!

Most people don’t live that long, and, obviously, only one person can hold the title of “Oldest Person in the World” at any given time. So, you may  be wondering… how often is there a new oldest person in the world? (Take a few guesses, if you like. I’ll give you the answer soon!)

Some mathematicians were wondering this, too, and they went about answering their question in the way they know best: by sharing their question with other mathematicians around the world! In April, a mathematician who calls himself Gugg, asked this question on the website Mathematics Stack Exchange, a free question-and-answer site that people studying math can use to share their ideas with each other. Math Stack Exchange says that it’s for “people studying math at any level.” If you browse around, you’ll see mathematicians asking for help on all kinds of questions, such as this tricky algebra problem and this problem about finding all the ways to combine coins to get a certain amount of money.  Here’s an entry from a student asking for help on trigonometry homework. You might need some specialized math knowledge to understand some of the questions, but there’s often one that’s both interesting and understandable on the list.

Anyway, Gugg asked on Math Stack Exchange, “How often does the oldest person in the world die?” and the community of mathematicians around the world got to work! Several mathematicians gave ways to calculate how often a new person becomes the oldest person in the world. You can read about how they worked it out on Math Stack Exchange, if you like, or on the Smithsonian blog – it’s a good example of how people use math to model things that happen in the world. Oh, and, in case you were wondering, a new person becomes the world’s oldest about every 0.65 years. (Is that around what you expected? It was definitely more often than I expected!)

Next, check out this graph! Yes, that’s a graph – there is a single function that you can make so that when you graph it, you get that.  Crazy – and beautiful! This was posted by a New York City math teacher named Michael Pershan to a site called Daily Desmos, and he challenges you to figure out how to make it!  (He challenged me, too. I worked on this for days.)

Michael made this graph using an awesome free, online graphing program called Desmos. Michael and many other people regularly post graphing challenges on Daily Desmos. Some of them are very difficult (like the one shown above), but some are definitely solvable without causing significant amounts of pain. They’re marked with levels “Basic” and “Advanced.” (See if you can spot contributions from a familiar Math Munch face…)

Here are more that I think are particularly beautiful. If you’re feeling more creative than puzzle-solvey, try making a cool graph of your own! You can submit a graphing challenge of your own to Daily Desmos.

If you’ve got the creative bug, you could also check out a new MArTH tool that we just found called Escher Web Sketch. This tool was designed by three Swiss mathematicians, and it helps you to make intricate tessellations with interesting symmetries – like the ones made by the mathematical artist M. C. Escher. If you like Symmetry Artist and Kali, you’ll love this applet.

Be healthy and happy! Enjoy graphing and sketching! And, bon appetit!

# Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

Last up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!

 Zachary Forest Johnson A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!

# TED, Bridges, and Silk

Welcome to this week’s Math Munch!

The Math Munch team at TEDxNYED

Marjorie Rice | click to watch her interview video

On Saturday, the Math Munch team gave a 16-minute presentation at TEDxNYED about Math Munch!  (Eventually there will be a video, and we’ll be sure to share it with you right away, but you’ll have to wait a month, maybe.)

We started with the story of Marjorie Rice, and in searching for a good picture of her, we came across this wonderful interview in a documentary about Martin Gardner.  It’s so neat to hear her speak about her discoveries.  You can see how proud she is and how much she truly loves math.  Feel free to watch the whole documentary if you like.  I haven’t gotten a chance yet, but I know it’s full of incredible stuff.

In the spirit of TED, I decided to share a few mathematical TED talks.  This one is absolutely fascinating.  In it, mathematician Ron Eglash describes how fractals underly the african designs.  You know how we love fractals.

If you’re hungry for another TED talk, here’s one about connections between music, mathematics, and sonar.

Up next, remember when we wrote about attending last year’s Bridges conference?  Well it happens every year, of course, and this year’s gallery of mathematical art is available online!  Click on one of those images and you get to more of the artists work.  I could easily spend hours staring at this art, trying to understand them, and reading the descriptions and artist statements.  Seriously, there is just way too much cool stuff there, so I’ve picked out a few of my favorites.  Also, I have great news to announce: Chloé Worthington (previously featured) had some of her art accepted to the exhibition!  Congratulations, Chloe!  If you look closely, you’ll see some of my art in there too.  :)

 Bjarne Jespersen Marc Chamberland Bob Rollings Chloé Worthington

By the way, if you ever create any mathematical art of your own, we’d love to see it!  Send us an email at mathmunchteam@gmail.com, and maybe we’ll feature your work in an upcoming Math Munch. (Only if you want us too, of course.)

Silk creator Yuri Vishnevsky

Finally, I know many of you like playing around with Symmetry Artist, which can be found on our page of Math Art Tools.  If you like that, then you’ll love Silk!  It’s much the same, but generates a certain kind of whispiness as you draw that looks really cool.  It also lets you spiral your designs toward the center, a feature which Symmetry Artist lacks.  You can download the Silk app for iPad or iPhone, if you like.  Silk was designed by Yuri Vishnevsky, with sound design by Mat Jarvis.  Yuri has agreed to do a Q&A for us, but we haven’t quite finished it just yet.  I’ll upload it as soon as possible, but for now, you can read an interview Mat and Yuri did with a website called Giant Fire Breathing Dragon.

Bon appetit!

# “Happy Birthday, Euler!”, Project Euler, and Pants

Welcome to this week’s Math Munch!

Did you see the Google doodle on Monday?

This medley of Platonic solids, graphs, and imaginary numbers honors the birthday of mathematician and physicist Leonhard Euler. (His last name is pronounced “Oiler.” Confusing because the mathematician Euclid‘s name is not pronounced “Oiclid.”) Many mathematicians would say that Euler was the greatest mathematician of all time – if you look at almost any branch of mathematics, you’ll find a significant contribution made by Euler.

Euler was born on April 15, 1707, and he spent much of his life working as a mathematician for one of the most powerful monarchs ever, Frederick the Great of Prussia. In Euler’s time, the kings and queens of Europe had resident mathematicians, philosophers, and scientists to make their countries more prestigious.  The monarchs could be moody, so mathematicians like Euler had to be careful to keep their benefactors happy. (Which, sadly, Euler did not. After almost 20 years, Frederick the Great’s interests changed and he sent Euler away.) But, the academies helped mathematicians to work together and make wonderful discoveries.

Want to read some of Euler’s original papers? Check out the Euler Archive. Here’s a little bit of an essay called, “Discovery of a Most Extraordinary Law of Numbers, Relating to the Sum of Their Divisors,” which you can find under the subject “Number Theory”:

Mathematicians have searched so far in vain to discover some order in the progression of prime numbers, and we have reason to believe that it is a mystery which the human mind will never be able to penetrate… This situation is all the more surprising since arithmetic gives us unfailing rules, by means of which we can continue the progression of these numbers as far as we wish, without however leaving us the slightest trace of any order.

Mathematicians still find this baffling today! If you’re interested in dipping your toes into Euler’s writings, I’d suggest checking out other articles in “Number Theory,” such as “On Amicable Numbers,” or some articles in “Combinatorics and Probability,” like “Investigations on a New Type of Magic Square.”

Want to work, like Euler did, on important math problems that will stretch you to make connections and discoveries? Check out Project Euler, an online set of math and computer programming problems. You can join the site and, as you work on the problems, talk to other problem-solvers, contribute your solutions, and track your progress. The problems aren’t easy – the first one on the list is, “Find the sum of all the multiples of 3 and 5 below 1000″ – but they build on one another (and are pretty fun).

Pants made from a crocheted model of the hyperbolic plane, by Daina Taimina.

Finally, if someone asked you what a pair of pants is, you probably wouldn’t say, “a sphere with three open disks removed.” But maybe you also didn’t know that pants are important mathematical objects!

I ran into a math problem involving pants on Math Overflow (previously). Math Overflow is a site on which mathematicians can ask and answer each other’s questions. The question I’m talking about was asked by Tony Huynh. He knew it was possible to turn pants inside-out if your feet are tied together. (Check out the video below to see it done!) Tony was wondering if it’s possible to turn your pants around, so that you’re wearing them backwards, if your feet are tied together.

Is this possible? Another mathematician answered Tony’s question – but maybe you want to try it yourself before reading about the solution. Answering questions like this about transformations of surfaces with holes in them is part of a branch of mathematics called topology – which Euler is partly credited with starting. A more mathematical way of stating this problem is: is it possible to turn a torus (or donut) with a single hole in it inside-out? Here’s another video, by James Tanton, about turning things inside-out mathematically.

Bon appetit!

P.S. – The Math Munch team will be speaking next weekend, on April 27th, at TEDxNYED! We’re really excited to get to tell the story of Math Munch on the big stage. Thank you for being such enthusiastic and curious readers and allowing us to share our love of math with you. Maybe we’ll see some of you there!

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

Welcome to this week’s Math Munch!

“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?

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?

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

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?)

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?

# Sam Loyd, Weight Problems, and Exercises

Welcome to this week’s Math Munch!

Chess composer, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

First up, remember Sam Loyd?  (We’ve featured him twice before.)   He was an american chess player and recreational mathematician who lived from 1841-1911.  He was also a chess composer, someone who writes endgame strategies and chess puzzles.  In fact, he wrote all sorts of puzzles, which his son published in a book called Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums.  (That link will take you to a scan of all 385 pages!)  By the way, those 5000 puzzles are only about half of the ones he wrote in his lifetime.  It’s no wonder Martin Gardner called him “America’s greatest puzzler.”  An interesting anecdote: Sam Loyd claimed until his death to have invented the 15 puzzle, but in fact he did not.  The actual inventor was Noyes Chapman, the Postmaster of Canastota, NY.

I wanted to show you some of Sam’s “Puzzling Scales” problems.  Why don’t you stop reading now and just solve them both?

These different weights balance because of the torque they apply

There are lots of puzzles like this, based on different weights balancing with each other.  A friend sent me this page of weight puzzles based on the idea of torque.  The farther out an object is placed, the more torque it applies to the balance, so it’s possible for a 1 pound weight to balance a 2 pound weight if you set them at the right distances.  The distance and wight multiply to give the torque applied.

These problems come from a massive bank of puzzles over on Erich’s Puzzle Palace.  If you like, you can also play this torque game I found.

 Place 1 through 5 to balance the weights. Place 1 through 6 to balance the weights.

I love problems like this, but I started to wonder, “what if the scales don’t balance?  Maybe you could make a puzzle out of that.”  I did exactly that, creating a series of imbalance puzzles.  Your job is to order the shapes by weight.  They start out easy, but there are some tricky ones.  I especially like #6.

### In each case, order the three objects by weight.

I’m also hosting an imbalance puzzle-writing contest.  My two favorite puzzlists will win a print of their choosing from my Stars of the Mind’s Sky series of mathematical art.  You should try your hand at writing one.  Just email it to Lost in Recursion.

Finally – we all love great problems and puzzles, but skill building is an important aspect of mathematics as well, and exercises help us build skill.  Exercises are often dull, but I found a website with some exercises I quite like, and I wanted to share them with you.  Check out the Coffee Break section over on StudyMaths.co.uk.

 Detention Dash Find the Primes Odd One Out

Detention Dash, for example, is just a timed multiplication chart, but typing the answers in on my computer really made me feel some of the patterns in the numbers.  You should try it.  Odd One Out also keeps you on your toes and makes you think about different kinds of numbers.  I find them surprisingly fun.  I hope you agree.

Bon appetit!

# Maths Ninja, Folding Fractals, and Pi Fun

Welcome to this week’s Math Munch!

First up, have you ever been stuck on a gnarly math problem and wished that a math ninja would swoop in and solve the problem before it knew what hit it?  Have you ever wished that you had a math dojo who would impart wisdom to you in cryptic but, ultimately, extremely timely and useful ways?  Well, meet Colin Beverige, a math (or, as he would say, maths) tutor from England who writes a fun blog called Flying Colours Maths.  On his blog, he publishes a weekly series called, “Secrets of the Mathematical Ninja,” in which the mathematical ninja (maybe Colin himself?  He’s too stealthy to tell)  imparts nuggets of sneaky wisdom to help you take down your staunchest math opponent.

For example, you probably know the trick for multiplying by 9 using your fingers – but did you know that there’s a simple trick for dividing by 9, too?  Ever wondered how to express thirteenths as decimals, in your head?  (Probably not, but maybe you’re wondering now!)  Want to know how to simplify fractions like a ninja?  Well, the mathematical ninja has the answers – and some cute stories, too.  Check it out!

A picture of a Julia set.

Next, I find fractals fascinating, but – I’ll admit it – I don’t know much about them.  I do know a little about the number line and graphing, though.  And that was enough to learn a lot more about fractals from this excellent post on the blog Hackery, Math, and Design by Steven Wittens.  In the post How to Fold a Julia Fractal, Steven describes how the key to understanding fractals is understanding complex numbers, which are the numbers we get when we combine our normal numbers with imaginary numbers.

Now, I think imaginary numbers are some of the most interesting numbers in mathematics – not only because they have the enticing name “imaginary,” but because they do really cool things and have some fascinating history behind them.  Steven does a really great job of telling their history and showing the cool things they do in this post.  One of the awesome things that imaginary numbers do is rotate.  Normal numbers can be drawn on a line – and multiplying by a negative number can be thought of as changing directions along the number line.  Steven uses pictures and videos to show how multiplying by an imaginary number can be thought of as rotating around a point on a plane.

A Julia set in the making.

The Julia set fractal is generated by taking complex number points and applying a function to them that squares each point and adds some number to it.  The fractal is the set of points that don’t get infinitely larger and larger as the function is applied again and again.  Steven shows how this works in a series of images.  You can watch the complex plane twist around on itself to make the cool curves and figures of the Julia set fractal.

Steven’s blog has many more interesting posts.  Check out another of my favorites, To Infinity… and Beyond! for an exploration of another fascinating, but confusing, topic – infinity.

Finally, a Pi Day doesn’t go by without the mathematicians and mathematical artists of the world putting out some new Pi Day videos!  Pi Day was last Thursday (3/14, of course).  Here’s a video from Numberphile in which Matt Parker calculates pi using pies!

In this video, also from Numberphile, shows how you only need 39 digits of pi to make really, really accurate measurements for the circumference of the observable universe:

Finally, it wouldn’t be Pi Day without a pi video from Vi Hart.  Here’s her contribution for this year:

Bon appetit!