Monthly Archives: March 2013

Sam Loyd, Weight Problems, and Exercises

Welcome to this week’s Math Munch!

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

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?

Puzzling Scales 1 Puzzling Scales 2
Torque puzzle 1

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.

Torque puzzle 1

Place 1 through 5 to balance the weights.

Torque puzzle 2

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.

Imbalance 1 Imbalance 2 Imbalance 3

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

Detention Dash

Detention Dash

Find the Primes

Find the Primes

Odd One Out

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!

ninjaFirst 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.

colin_bridgeFor 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 the Julia set.

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.

complex multiplicationNow, 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.

here comes the julia set

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!

Dots-and-Boxes, Choppy Waves, and Psi Day

Welcome to this week’s Math Munch!

And happy Psi Day! But more on that later.


Click to play Dots-and-Boxes!

Recently I got to thinking about the game Dots-and-Boxes. You may already know how to play; when I was growing up, I can only remember tic-tac-toe and hangman as being more common paper and pencil games. If you know how to play, maybe you’d like to try a quick game against a computer opponent? Or maybe you could play a low-tech round with a friend? If you don’t know how to play or need a refresher, here’s a quick video lesson:

In 1946, a first grader in Ohio learned these very same rules. His name was Elwyn Berlekamp, and he went on to become a mathematician and an expert about Dots-and-Boxes. He’s now retired from being a professor at UC Berkeley, but he continues to be very active in mathematical endeavors, as I learned this week when I interviewed him.

Elwyn Berlekamp

Elwyn Berlekamp

In his book The Dots and Boxes Game: Sophisticated Child’s Play, Elwyn shares: “Ever since [I learned Dots-and-Boxes], I have enjoyed recurrent spurts of fascination with this game. During several of these burst of interest, my playing proficiency broke through to a new and higher plateau. This phenomenon seems to be common among humans trying to master any of a wide variety of skills. In Dots-and-Boxes, however, each advance can be associated with a new mathematical insight!”

Elwyn's booklet about Dots-and-Boxes

Elwyn’s booklet about

In his career, Elywen has studied many mathematical games, as well as ideas in coding. He has worked in finance and has been involved in mathematical outreach and community building, including involvement with Gathering for Gardner (previously).

Elywn generously took the time to answer some questions about Dots-and-Boxes and about his career as a mathematician. Thanks, Elywn! Again, you should totally check out our Q&A session. I especially enjoyed hearing about Elwyn’s mathematical heros and his closing recommendations to young people.

As I poked around the web for Dots-and-Boxes resources, I enjoyed listening to the commentary of Phil Carmody (aka “FatPhil”) on this high-level game of Dots-and-Boxes. It was a part of a tournament held on a great games website called Little Golem where mathematical game enthusiasts from around the world can challenge each other in tournaments.

What's the best move?A Sam Loyd Dots-and-Boxes Puzzle

What’s the best move?
A Dots-and-Boxes puzzle by Sam Loyd.

And before I move on, here are two Dots-and-Boxes puzzles for you to try out. The first asks you to use the fewest lines to saturate or “max out” a Dots-and-Boxes board without making any boxes. The second is by the famous puzzler Sam Loyd (previously). Can you help find the winning move in The Boxer’s Puzzle?

Next up, check out these fantastic “waves” traced out by “circling” these shapes:

Click the picture to see the animation!

Lucas Vieira—who goes by LucasVB—is 27 years old and is from Brazil. He makes some amazing mathematical illustrations, many of them to illustrate articles on Wikipedia. He’s been sharing them on his Tumblr for just over a month. I’ll let his images and animations speak for themselves—here are a few to get you started!

A colored-by-arc-length Archimedean spiral.

A colored-by-arc-length Archimedean spiral.


A sphere-like degenerate torus.

A Koch cube.

A Koch cube.

There’s a great write-up about Lucas over at The Daily Dot, which includes this choice quote from him: “I think this sort of animated illustration should be mandatory in every math class. Hopefully, they will be some day.” I couldn’t agree more. Also, Lucas mentioned to me that one of his big influences in making mathematical imagery has always been Paul Nylander. More on Paul in a future post!

Psi is the 23rd letter in the Greek alphabet.

Psi is the 23rd letter in the Greek alphabet.

Finally, today—March 11—is Psi Day! Psi is an irrational number that begins 3.35988… And since March is the 3rd month and today is .35988… of the way through it–11 out of 31 days—it’s the perfect day to celebrate this wonderful number!

What’s psi you ask? It’s the Reciprocal Fibonacci Constant. If you take the reciprocals of the Fibonnaci numbers and add them add up—all infinity of them—psi is what you get.


Psi was proven irrational not too long ago—in 1989! The ancient irrational number phi—the golden ratio—is about 1.61, so maybe Phi Day should be January 6. Or perhaps the 8th of May—8/5—for our European readers. And e Day—after Euler’s number—is of course celebrated on February 7.

That seems like a pretty good list at the moment, but maybe you can think of other irrational constants that would be fun to have a “Day” for!

And finally, I’m sure I’m not the only one who’d love to see a psi or Fibonacci-themed “Gangham Style” video. Get it?

Bon appetit!


EDIT (3/14/13): Today is Pi Day! I sure wish I had thought of that when I was making my list of irrational number Days…

Collaborative Math, Petals, and Theseus

Welcome to this week’s Math Munch!

Collaborative MathematicsLet’s start with a great new blog – a place for you to do math – Collaborative Mathematics. It’s the pet project of mathematician, teacher, and juggler, Jason Ermer.  The idea is simple. Jason posts videos about a little mathematical idea, and he offers up a challenge question for viewers to solve. In fact, he has lots of ideas for how you can do some mathematical research of your own. After that, you make a response video explaining what you’ve come up with. That’s Collaborative Mathematics.

His first video was about ERMER numbers, like 12312 or 94794. Core Challenge: How many ERMER numbers are even? To learn all about it and get involved, check out this video.

On his site, Jason says, “when possible, students should work with a team of problem solving peers. Our ideas are formed and refined as we communicate our thoughts to others and as we hear a diversity of ways of interpreting the same concepts.” So don’t feel like you have to do it all alone. It is collaborative after all!

And don’t worry if things are tough! “Struggling in mathematics is not a bad thing! We expect sore muscles when we exercise and try to improve at, say, basketball. Why would we expect mathematical growth to be painless? We must exert ourselves to grow. There is glory in the struggle! “

And if you liked that. Here’s the second video challenge.

* * *

Up next is a simple little site I found called the Petals Challenge.

The secret of the game is in the name of the game: Petals around a roseHow many petals?

The secret of the game is in the name of the game: Petals around a rose
How many petals?

It’s a kind of riddle, because there aren’t really instructions. The only way to make sense of it is to give it a try. Good luck, and never tell anyone the secret of the game!

* * *

theseusLastly, here’s a great game called Theseus and the Minoataur. You’re Theseus, and you must exit a labyrinth while a minotaur chases you. The Minotaur is faster than you are, though, so you’ll have to be clever!

Unfortunately, this is a java game, which some computers won’t be able to play, so as a bonus, watch this beautiful animation from Numberphile showing the creation of the Dragon Fractal.

Bon appetit!