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…

24 responses »

  1. I can’t find the question and answer session. That link takes me to the Amazon page. I have his book and love the game! Thanks for the post on this. Nancy Rooker

    • Hi Nancy! Thanks for coming by Math Munch! My copy of Elwyn’s book is on order. I can’t wait to dig into it.
      Thanks for catching the mis-link. It’s all fixed now. I’ve also just added the interview to our Q&A page. Hope you enjoy the interview!

  2. Pingback: Sam Loyd, Weight Problems, and Exercises | Math Munch

      • Hi Noah,
        I did play in a Dots-and-Boxes tournament on Little Golem, just to try out their system. I won three games and lost two, but the wins came in short games where my opponent gave up quickly, and the losses came in games that were played to the end. Still, it was fun to try out.
        Of course, to get a really high score in a game of Dots-and-Boxes, you need to play on a very big board with a very helpful opponent! 😉
        I hope you have the chance to enjoy playing some Dot-and-Boxes.

  3. the game dots and boxes seems fun I might try this game I played it some few times but I really didn’t knew who made the game and the post really help me I guess I never thought about who made the game before and when did Elwyn Berlekampt retire from UC Berkley because one of my cousins graduated from UC Berkley May 18 2013 and I was wondering if my cousin got to see him soo thanks for the post im going to play the game lets see if I win.-Israel Rivera

  4. the Game was really fun it was hard nut fun I won two to eight but I learned some moves but it was really fun thanks for the game.-Israel Rivera

  5. I played the game before I saw the video. So your saying the best strategy is to not give away any positions at the beginning to get more boxes but also possibly loose some? How long would you say it takes to play someone with the same or close competition skill?

  6. That dots and boxes game is fun. I like how you can chose how big or how small it can be, but i don’t like how it’s restrained.

  7. It was cool and I thought this game was like fun. I thought this game was annoying because I always lost, except for a few times.

  8. This game is fun. It has more strategies than tic tac toe. I always wanted something more difficult to play and this is perfect.

  9. Dots and Boxes was really challenging I played with 4 rows and won 4 of 10 times. At the end I figured out how to win. It is a great game and you must pick your move carefully.

  10. I learned after watching the video on how to play dots and boxes that the player that encloses the box last gets that square as a reward! I’ve played this game before, but never really knew how to play the game correctly, that’s probably why I would lose most of the time!! I played the dots and boxes game with a friend and made the grid a 3×3 and won five out of nine boxes. I enjoyed this game…. thanks for posting it!

  11. I have played this game several times and i have noticed that the most of the time the winner is the person to make the last move. Can you play this game with different shapes?

    • Hi Michelle,
      I like how you’re paying attention to strategy, and I love your question. You can definitely play Dots-and-Boxes without using a square or rectangular board—any shape you could cut out of grid paper would be fine. I’ve never played D&B on a non-square grid before, though. It would be fun to try to play it on a triangular grid!
      Hope you try it out—let me know if you do. And I hope you’ll drop by Math Munch again!

  12. Pingback: Fullerenes, Fibonacci Walks, and a Fourier Toy | Math Munch

  13. I always saw my friends playing this game in elementary school, but I never knew the story behind it or how to play it! Loved to see the math behind it!

  14. I think I have played this game before. I think this game is really fun to play because you really have to think of what you have to put next. It is hard. I usually win though. In my opinion 8 am good at this game.

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