Author Archives: Paul Salomon

The Dice Lab, Sum of Cubes, and Double Polyhedra

Welcome to this week’s Math Munch!

It’s the final Thursday of September, so it’s time again for a recap of the month’s best from our Facebook page. This month we have a new sort of dice, a beautiful illustration of a numerical fact, and some wonderful new sculpture work from Rinus Roelofs. Let’s dig in.

First, check out this wonderful image. Meditate on it, and see if you can figure out what’s going on, even if you can’t understand the notation.


It’s showing us a simple way to compute a sum of cubes. They can be broken down and reconstructed as a square! Consider the sum of the first 3 cube numbers, for example: 1+8+27=36, and 36 is the square of 6. One step further, 6 is the sum of the first 3 numbers.

So in the picture above, the sum of the first 5 cubes is equal to the square whose side length is the sum of 1 through 5.  AMAZING, and a beautiful illustration. Can you see why it always works, not just for 1 through 5? That’s key! And now test your understanding: What is the sum of the first 100 cube numbers?

dicelablogoUp next, we’ve met Henry Segerman plenty of times on Math Munch, including a look at the project he shares with Robert Fathauer, called The Dice Lab. They make mathematically interesting dice that have, in most cases, never been produced before. There newest creation (also last? see the video to see what I mean) is a 48-sided dice. Very cool. Can you think of a use for a 48-sided die?  It sure looks cool. Reminds me of a rhombic dodecahedron. Do you see the connection?

Finally, another familiar face – the incredible mathematical artist, Rinus Roelofs – has been making incredible things. We met Roelofs in July, but his facebook page has been full of activity since then. His recent work has focused on double-covered polyhedra.  You’ll have to click over and browse to see what I mean. Recently he posted a project I might want to take on. These are fold-up models for his creations. Check out the gallery below.


I’m not 100% sure how that cube one works, but I think I can figure it out, and I bet some of you can too. Of course, I’m sure we’ll make mistakes, but if we keep on learning, I bet we can get this figured out. If anyone ends up making a template of their own, email it to us and we’ll share it on the site.

Until next time, bon appetit!


Maria Chudnovsky, Puzzlebomb, and Some Futility

Welcome to this week’s Math Munch!

This week we meet an incredible mathematician, take on a tough number puzzle, check out a wonderful mathematical card trick, and much more.

Maria Chudnovsky

A while ago we shared an interview with mathematician Fan Chung Graham.  The interview was posted by Anthony Bonato, The Intrepid Mathematician. Well, this week we share another of his interviews, this time with Maria Chudnovsky, graph theorist and star of not one, but two television commercials. (A rare feat for a mathematician.) Maria is also a winner of the extraordinary MacArthur “Genius” Grant. You can check out the video below or click here for the full interview.

Up next, our friends over at The Aperiodical do a lot of great things for the math world. One contribution is the monthly Puzzlebomb put on by Katie Steckles.

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This month’s puzzle is MODOKU, a sort of sudoku style puzzle where columns and rows span the possible remainders mod 7 and mod 5. Check it out! Thanks to Katie for such a lovely puzzle! You can click below for an interactive version with complete instructions.

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Finally this week, it’s time again to look at a Futility Closet, a phenomenal blog containing the odd mathematical tidbit. We’ll take a look at three of them.

Screen Shot 2016-07-21 at 12.51.33 AMHere’s a weird arithmetic fact I found there. Do you see what’s going on there? I have absolutely no idea how often this kind of thing is true, if ever again, but it gets me thinking.

2016-07-14-a-square-triangleHere’s another incredible one. We’ve posted about Pascal’s (Yang-Hui’s) Triangle lots of times (1 2), and I’ve come across a lot of fascinating stuff about it, but this is new to me. Apparently, “the product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square.” Can you prove it?

Now on to the biggie…  This is such a cool card trick! Here’s the trick as explained by Futility Closet:

“I hand you an ordinary deck of 52 cards. You inspect and shuffle it, then choose five cards from the deck and hand them to my assistant. She looks at them and passes four of them to me. I name the fifth card.”         !!!!!!!!!!


The key to the magic is this chart:

{low, middle, high} = 1
{low, high, middle} = 2
{middle, low, high} = 3
{middle, high, low} = 4
{high, low, middle} = 5
{high, middle, low} = 6

Can you figure out how it works from the chart alone? You’ll need a good assistant to get on board, and it wouldn’t hurt to practice a bit. Then get ready to impress. Oh, and if you can’t figure out the trick from the chart alone, then just head over to Futility Closet and read the full explanation.

Well that’s it for this week. Hope you found something delicious. Bon appetit!

SliceForm, Rinus Roelofs, and krazydad

Welcome to this week’s Math Munch!

For the 5th and final Thursday of June we will once again take a look at some of the goodness over on our facebook page, and oh my goodness what a huge load of goodness we have indeed! For an appetizer, how about this little visual problem posted by ThinkFun Games? (If you remind me in the comments, I’ll tell you the neat way I thought about solving it.)

Circle areasThe shape consists of overlapped color circles.  Which two colors have their total visible areas equal? (click to enlarge)

Now onto the main course. I have to show you this incredible new math art tool called SLICEFORM STUDIO. Click over and check out their gallery to begin with. Just gorgeous.

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My first creation on

There’s a tutorial page as well, but the best thing to do is probably just to start playing with the app itself.  DIG IN! The site is sort of made for people who can use laser cutters to do the paper and stuff, but you can also just click “trace and export strips” and then color it in and export the image. On the right, you can see my first creation. Email yours to and we’ll stick it in our readers’ gallery.

Alright, up next is an amazing mathematical artist by the name of Rinus Roelofs. (You might remember the paper project of his that we shared at new year.) Well, Rinus is just an unblievable and prolific maker of incredible and beautiful things. Check out his website. (He has two, I think)

I follow Rinus on facebook, and he’s always posting pictures of his works in progress, and they are stunning. First, check out this gallery of Interwoven Ring Patterns he recently posted. Then take a look at his timeline photos. Lots of overlapping patterns and Möbius shapes.

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A completed galaxy puzzle.  Each colored area has rotational symmetry

Finally, have you ever heard of Galaxy Puzzles?  I hadn’t either, but you can find lots of them over on the wonderful puzzling site, krazydad. The puzzle begins with lots of dots, and your goal is to separate the dots by making enclosures that have 180 degree rotational symmetry. Print and play galaxy puzzles are available as well as an interactive online version. There are lots of other puzzles available as well, but I think Battleships is a pretty cool. You might give that a try too.


But wait, there’s more. With 5 Thursdays in a month, there’s just lots to share, so you also get some bonus stuff!

That’s it for June. See you next time. Bon appetit!

Temari, Function Families, and Clapping Music

We have a rare 5 Thursdays this month, so we get an extra rerun post. This one features a Q&A with mathematical artist Carolyn Yackel and much more beautiful stuff. Enjoy!

Math Munch

Welcome to this week’s Math Munch!

Carolyn Yackel Carolyn Yackel

As Justin mentioned last week, the Math Munch team had a blast at the MOVES conference last week.  I met so many lovely mathematicians and learned a whole lot of cool math. Let me introduce you to Carolyn Yackel. She’s a math professor at Mercer University in Georgia, and she’s also a mathematical fiber artist who specializes in the beautiful Temari balls you can see below or by clicking the link. Carolyn has exhibited at the Bridges conference, naturally, and her 2012 Bridges page contains an artist statement and some explanation of her art.

temari15 temari3 temari16

Icosidodecahedron Icosidodecahedron

Truncated Dodecahedron Truncated Dodecahedron

Cuboctahedron Cuboctahedron

Temari is an ancient form of japanese folk art. These embroidered balls feature various spherical symmetries, and part of Carolyn’s work has been figure out how to create and exploit these symmetries on the sphere.  I mean how do you actually make it…

View original post 339 more words

Wordless Videos, isthisprime, and Fan Chung Graham

Welcome to this week’s Math Munch! For the final Thursday of May, we’ll be looking back at some of this month’s posts from our Facebook page. We’ll see some wordless videos by The Global Math Project, look at a prime number quiz game, and meet Fan Chung Graham, one of the world’s leading mathematicians.

I don’t know much about The Global Math Project, but I know James Tanton is involved, and that is always a good thing. (Remember his Exploding Dots?) Well, they’ve posted a couple of wonderful videos featuring Tanton’s “math without words.” Need I say more? See for yourself.

If you like those, here are some more math without words from Tanton’s website.

Up next is a neat little thing by Christian Lawson-Perfect from The Aperiodical. Christian bought and set up a little quiz game. Click over and see for yourself how it goes… I’ll wait… click below…

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It’s good practice for divisibility tests and getting your prime recognition up, but I suppose it’s not all that mathematical, is it? But Christian did something interesting. He recorded data from all the games played, and he wrote a summary of the results. I love all the charts and graphs in there. The one below shows how likely a number is to be missed by players.

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Finally, I hadn’t heard of Fan Chung-Graham until I found an interview of her posted on Facebook. She is one of the world’s leading mathematicians in several fields, and though she recently retired, she still conducts some research. The interview is a little academic, but it’s still nice to get to know such a talented mathematician.


Well that wraps up the week and month. I hope you’ve found some tasty math.  Have a great week and bon appetite!

Near Miss, Curiosa Mathematica, and Poincaré

Welcome to this week’s Math Munch!

For this last Thursday of April, we’ll be taking a look at some recent posts from our facebook page. Craig Kaplan writes about “near miss” polyhedra, a Pythagorean gif takes us to an curious math blog, and we find a beautiful portrait of a great mathematician.  Let’s go!

Craig Kaplan

Craig Kaplan

First is an article from a wonderful mathematician and mathematical artist by the name of Craig Kaplan. His name has popped up on Math Munch before (1, 2 ,3), in case it sounds familiar. You can check out Craig’s stuff on his website, Isohedral, or download his really great game, “Good Fences,” which I have on my iPhone.

near missWhat I really wanted to share, however was Craig’s writing on “A New Near Miss.” This is a polyhedron that almost is… but just isn’t. It looks pretty good, but it can’t be. You’ll have to read to see what I mean.

PythagorasPerigalP.gifUp next, I found this little gif on our facebook page, and I absolutely loved it. It demonstrates the Pythagorean Theorem which says that as long as that’s a right triangle there, the big square on bottom is exactly as big as the two smaller squares combined. The animation shows you how to chop up the middle-sized square and recombine it with the small one to make the big one. I knew there were demonstrations/proofs like this, but this one opened my eyes to something I didn’t quite know before.

This gif sent me off on a journey through the internet to track down the source, and it led me to a site called Curiosa Mathematica. It’s a math blog featuring lots of random math goodies. There’s lots to see and get into (much like Math Munch). Here’s a quote I found there.  I hope you find something you like too.

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Finally, I was really taken by this piece of art (below). It’s a portrait of French mathematician Henri Poincaré, and it was drawn by Bill Sanderson. I can’t find much info on Bill, but WOW the piece is so cool. I love how he’s surrounded by his mathematical creations. I was hoping he had done more, and I did find a couple more (below), but not all I had hoped for.


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French Mathematician Henri Poincaré

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Alan Turing

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Isaac Newton

Have some illustrative talent? I’d love to see your mathematician’s portrait. Feel free to send us something… anything.

I hope you enjoy your weekend and find something tasty out there in the mathematical interwebs. Bon appetit!

2016, ScienTile, and a New Algorithm

Welcome to this week’s Math Munch!

In this week’s post we check out a tile designing contest from 2010, learn about some breakthrough news in computational algorithms, and get a DIY project to ring in the new year.

Speaking of the new year… welcome to the new year!! 2016 is 11111100000 in binary, by the way. Pretty cool right!? The five 0’s at the end tell you that 2016 has five 2’s in its prime factorization. That is, you can divide 2016 by 2 five times and still get a whole number. The big bunch of 1’s at the start means its also divisible by a number that is one less than a power of 2. 63, basically.

That is to say, 2016 = (26–1)(25). I think that signals a promising year. Bring it on.

2016 card project

DIY Möbius strip project to ring in the new year

cord project template

template here

Here’s a great way to start your year off. How about a paper folding project from mathematical artist Rinus Roelofs? I found the project posted by on their facebook page. According to the post, “this card is the representation of a Möbius strip.”

Click here for the downloadable, printable, cuttable, foldable, template and make your own!


A tile pattern I designed. Anyone else thinking bee hive?

Up next, I’ve been playing a lot of board games that use square or hexagon tiles, and I’ve been thinking about what other tiles might make for a cool game. First off, here’s a little tile I came up with that always leaves hexagons in the places where they meet. Might make a neat game  where you build a bee colony. Who knows.  But in my searching for groovy tiles, I found ScienTile.

ScienTile was an “open tile design competition” initiated by Dániel Erdély, a Hungarian mathematician and mathematical artist featured previously on MM for his spidrons.  In fact, ScienTile was meant to commemorate the 2010 Bridges Conference, which was in Hungary. Sadly, I don’t think the ScienTile competition was repeated in later years, but the results from 2010 are quite beautiful. I was most struck by the picture below, a tile designed by Gabor Gondos. I also really liked this one by the wonderful Craig Kaplan (featured previously here), but all the submissions can be found here.

A gorgeous and flexible tile design by Gabor Gandos

A gorgeous and flexible tile design by Gabor Gondos

All these graphs are isomorphic, and the new algorithm could tell you that... really fast!

All these graphs are isomorphic, and the new algorithm could tell you that… really fast!

Finally, some breakthrough math news from the computational world. Computer scientists have develop a new fast algorithm for solving” the graph isomorphism” problem, which simply checks whether or not two graphs (think connect-the-dots pictures) are really the same. All the graphs in the gif on the right are isomorphic, because they can be morphed into each other without changing the connectivity of the dots.


The 5,2 Johnson Graph

The new algorithm breaks a computational record that was unbroken for the last 30 years, which is a crazy long time in computer terms. Congratulations László Babai, who can be seen below presenting his breakthrough paper at the University of Chicago. His algorithm actually doesn’t cover all types of graphs, but Babai was able to show that the only type of graph not covered were the highly symmetric Johnson Graphs. You can see one of these on the right.

You can find more info on the record-breaking algorithm in this article from or this write up from Quanta Magazine.


László Babai presenting his record-breaking algorithm

Have a great week, and bon appetit!