Math Circles at Home 3: Domino Variations–Get it Covered

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Here are two classic questions about dominoes, with variations for each.

1. The classic: Imagine a standard 8×8 chess board. Remove the opposite corners of the board; now there are 62 spaces. Can you cover the board with 31 dominoes, if each domino covers exactly two spaces?


  • What is the removed corners are adjacent?
  • What if you remove 4 spaces at each corner of the board?
  • Go back to a full chess board; it can be easily covered with 32 dominoes. When I did this, I found that 16 were placed vertically, and 16 horizontally. Can you place them in any other combination?

2. The classic: Imagine a strip 8 squares long, covered with dominoes and monominoes (monominos are also called ‘squares’).

In this figure, the strip is covered by two brown dominoes and four green monominoes. You could, of course, just use four dominoes, or 1 domino and 6 monominoes, etc. In how many ways can this you cover the strip using only these figures?


  • What if the strip were 100 squares long?
  • What if you had brown and blue dominoes, and yellow and green monominoes?
  • What if you had monominoes, dominoes, and triominoes (formed by connecting three squares in row)?
  • What if these figures were made by linking together equilateral triangles rather than squares?

The variations are endless!

PS. Online math circles start next week. Learn more here!

And in case you missed it:

Math Circles at Home Take 1: 1d Tic-Tac-Toe

Math Circles at Home Take 2: Knight Moves


Math Circles at Home: Knight Moves

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Idea #3: Knights of all Kinds

The initial puzzle: The most interesting chess piece is the knight because of how it moves. A knight moves along an L-shape: two spaces up and one space over. Or one space up, and two spaces over. For visual learners and those who like overkill–here’s a two minute and 14 second video of someone moving a knight on a chessboard.

If a single knight is on an infinite chessboard, can it reach every space (eventually!)? How do you know? Can you use the rotational symmetry of the grid to make a concise argument that all of the spaces can be reached? [Suppose you can show you can reach the space directly to the left of the starting square, and the space above that; I claim that’s all you need to show. Why?!]


After you explore this initial question, extend!

Chess has a problem, and it’s a problem that’s 100’s or 1,000’s of years old–the rules never change (I imagine serious chess players would disagree. If that’s you, write your own blog). Thankfully, this is a (virtual) math circle, so we will immediately change the rules.

  • Think of the traditional chess knight as a (1,2) knight. A (1,3) knight is able to move one up and three over (or vice versa). Can a (1,3) knight reach every space on an infinite chess board? If not, which spaces can it reach?
  • What about the (1,4), (1,5), and (1,6) knights?
  • What about (2,5)? (4,8)? (3,9)? (7,11)? Explore.

Generalize and Extend More

  • Try to answer the question in general–how can you tell whether an (m,n) knight can reach every square on an infinite board, just by look at m and n?
  • What other chess pieces that don’t exist can you imagine, and what questions can you ask about them?


Idea #4: Math and Music

The connections between mathematics and music are at the heart of the newly released WFMT Explainer Series. The series features Eugenia Cheng, Scientist in Residence at the Art Institute, and leader of the Math Circles of Chicago’s summer experience for high school students. These videos are short, informative, and of interest to anyone who enjoys math or music. Highly recommended!

For more ideas, see Math Circles at Home: Take 1.



Math Circles at Home: Take 1

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I know parents and caregivers are struggling to find activities for their children to engage in as we wait to see when school will relaunch. I thought MC2 could provide some ‘curated’ suggestions for giving your child a math circle like activity that you can try at home. Rather than providing a laundry list of links, I feel concise and specific suggestions might be most helpful. So here’s a list…of two ideas:

Idea #1 1d Tic-tac-toe

The words in italics describe the progression of a typical math circle. Initially, engage in a game that is accessible. Next, try to ‘solve’ the game, describing a strategy that works for the initial version of the game. Then generalize and create!

The initial rules of the game: Draw seven blank spaces in a row. Take turns putting x’s (and only x’s; no o’s!) in open spaces. The person who gets three or more x’s in a row wins. It’s important to understand that no one ‘owns’ an x–what matters is who was the person who added that last x that led to at least three x’s being consecutive.

Play a few times and start to ask questions:

  • Are there particular moves for player 1 or player 2 that are ‘bad’?
  • Can player 1 or player 2 always win?
  • Can you describe what moves will ensure that victory? Is there more than one choice of moves that will lead to victory no matter what?


  • How can you change the rules of the game? The first, perhaps most obvious generalization is to change the number of spaces. I encourage you to explore that first.
  • What’s the fewest number of spaces that a game can have? [3] Who wins in that case?
  • Who can always win with games of 4, 5, and 6 spaces? Organize your thinking in a table, and continue to explore bigger numbers. Is there a fundamental difference between games with an even and an odd number of spaces?
  • At some point, jump to a big number, like 171. Can you give a winning strategy for player 1 or player 2 there?
  • What big mathematical ideas are at play? How does symmetry support your thinking?

Invent & QED

Now it’s time to make up your own variation and explore. Here’s an old post that can help support your brainstorming if you need it, but I believe there is something particularly valuable about kids generating and exploring their own ideas. That’s what QED is all about.

Idea #2 Tangrams

My initial love of mathematics was seeded by toys like soma cubes, rubik’s cubes, and, plain old cubes. I also have early memories of playing with tangrams.

Tangrams are a collection of seven shapes–5 isosceles triangles, a square, and a parallelogram. Play with tangrams generally involves arranging these shapes into a figure shown as a silhouette–the number of possible tangram puzzles is limitless.

  • You can find instructions here for creating your own tangrams if you don’t have any on hand.
  • There are any number of websites that give tangram challenges–here’s one that has links to a few 100 at a variety of  difficulty levels.

I believe that many children will be given school work to do at home in the upcoming weeks to continue their academic studies. I hope you’ll help them continue their enrichment studies too–we’ll keep the posts coming. 🙂



Camps: Ours and Others

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Registration for MC2 Summer 2020 is open! We’ve also launched a new webpage that lists our summer sites (much like our Locations page does for the Academic Year). This new site also explains our summer fees–please know that we have a sliding scale, and we expect that many of our campers will attend for free.

To enter the summer lottery, login to your account on our registration page and sign up as usual–the summer lottery will run on March 31st.


While we hope you’ll join us this summer, what’s most important is that you do fun math somewhere! Some other suggestions:

Summer Plans 2020

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Our summer plans have been approved! We will share finalized details in early March, but here some initial details:

1. Locations, Times, Dates, and Grade Levels

A. Back of the Yards College Prep: Rising 6th and rising 7th graders, July 13th to July 17th, 9AM-4PM (the meeting time may shift, but this will be an ‘all day’ camp)

B. Payton Prep: Rising 6th, rising 7th, and rising 8th graders, July 13th to July 17th and July 20th-July 24th, 9AM-noon.

C. Jones Prep: Rising 9th to 13th graders, July 13th to July 24th, Noon-3PM

2. Sliding Scale Tuition

Full tuition in any of these programs is $300 per week.

Students who have their school counselor confirm that they should receive reduced tuition will pay no tuition for camp.

Other students may make a financial needs statement and pay $150/half-tuition.

3. Signing Up

We expect to open summer registration in early March, with a lottery to be held by the end of the month. Many more details soon!

MC2 Friends in the News

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Every once and a while Po-Shen Loh opens the year with a talk for MC2 students. So it was exciting to see last week that the New York Times featured an entire article on one of Dr. Loh’s recent discoveries. This Professor’s ‘Amazing Trick Makes Quadratics Easier–highly recommended!

I also wanted to make a plug for Five Fabulous Math Activities for Your Math Circle. MC2 supporter Japeth Wood and friends have nearly met their stretch goal for their Kickstarter Campaign–please support them, as this book will be a great resource for math circles like ours across the country!


Reading & Writing & Arithmetic

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1. Reading

Roots of Unity, the math opinion page on Scientific American’s webpage comes highly recommended. I’d like to point you to a recent post from December 27th: The Math Reading Challenge 2020. The post encourages you to read math related books in 12 categories, like:

  • A math-related book from the year you were born
  • A math book that helps you make something
  • A nonfiction math book written by a woman [Our colleague Eugenia Cheng’s Beyond Infinity is listed-yay!]
  • A graphic novel about math or mathematicians


Read a book and join the conversation at #MathReadingChallenge2020 on twitter (it’s a community, not a contest). I’m in!

2. Writing

This one is a contest, for 15-18 year olds. Apply for the Steven H. Stogatz Prize for Math Communication. Cash prizes. Submissions are due April 22nd, and, can be a song, a play, a video, etc., etc., celebrating the universality of math.

The contest is being run by MoMath aka the National Museum of Mathematics in NYC. One must hope that a trip to New York is part of the award.

3. My Writing

Today the American Mathematical Society posted my essay, “The Future of Math Enrichment: Math Contests or Math Circles?“. Please read!


QED 2019: Horsing Around or How to Win $1,000,000/N

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Congratulations to all students who participated in QED 2019! You’ll find glamorous photographs in the blogpost following this one, and we hope you are still coveting the ‘fabulous prizes’ you took home Saturday.

Silly contests and serious art fairs

We try to keep QED from being too serious. Lawrence Tanzmann runs our annual Guesstimathon, an internet-free challenge with prompts like, “The number of times Horse appears in the song Old Town Road” or “The number of pages in the first edition of the Harry Potter series” (you can find all the prompts here). The Lowest Positive Integer game asks you to pick the smallest number that no one else does–this year’s winner chose 19 (here’s the distribution of entries). Most popular was our million dollar lottery. You could enter as many times as you like (someone entered 1 billion times), with the unfortunate side effect that the grand prize was $1,000,000 divided by the number of entries. Generously we rounded up the five hundredths of a cent grand prize to a penny.

The highlights, of course, were the QED entries themselves. While some of the project names were also silly (A Can of Worms, Frogs on a Log, Peppermint Experi-MINT), the mathematical content was anything but. Like an Art Fair, there was no competition between entries, just the opportunity to share what you created with the public. Every judge I spoke to after the event was deeply impressed by the level of work they saw.

Judges from all over

We’re proud that we draw judges from many walks of mathematical life, from professors to post-docs to graduate and undergraduate students, teachers from primary to middle to high school, along with some combinations of parent-engineer-financial industry wiz-data scientist. Thanks to you all!  Stu Abram, Maneesha Pradeep, Paul DeRonne, Nailea Curiel, Rutha Dixon, Sam Dodds, Aimee Hart, Jaime McLauglin, Marco Mendez-Duarte, Cornelia Mihailia, Jeremy Judge, Sarah Reitzes, Youjin Lee, Graham Rosby, Minh-Tam Trinh, Matt Rosenberg, Hannah Butler, Melissa Ramirez, Serg Cvetkovic, Abhinav Gandhi, John Marchetta, Faith Medlock, Maxime Bergeron, Kara Fischer, Nate Harman, Edward Huh, Peyton Morgan, Todd Pytel, Kristen Schreck, Eric Xu, and Kevin Zhou.

A word of thanks for our sponsors and friends

Peter Tingley went all out as this year’s QED speaker. Dr. Tingley, a professor at Loyola, obtained 600 hackey sacks with QED & Loyola logos to give away (see the photo), for teaching both the mathematics and the practice of juggling. Lawrence Tanzmann ran the Guesstimathon for the nth year running, where I know n to be somewhere between 4 and 7. Steve Starr hosted our Visiting Students who we expect to return in 2020 with new projects!

Julienne Au managed the judges and Mike Caines managed QED’s logistics; that they did so while also managing the logistics of having their newborn onsite was all the more impressive. Lead judges Paul DeRonne, PJ Karafiol, Dan Kang, and Matt Moran made everything run smoothly. Esperanza Baeza, Anabelle Mendez, Grace Shin, Ralph Banasiak, Paul Buckner, and Dzevida Duheric took care of everything else. Go team QED!

The American Mathematical Association again gave every QED participant a recreational math book, and Wolfram gave them each Mathematica, Wolfram Programing Lab, and Wolfram Alpha, the most advanced mathematics software in the world. Chicago’s own Citadel provided food and shirts. Thanks to you all!

QED Updates, Juggling News

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QED, our annual math symposium is just around the corner. We have important announcements!

  • The pre-registration deadline for QED has been extended due to the recent school closure. Please pre-register by Tuesday, November 19th. (The previous deadline was the 12th). High School students, be reminded that your project papers must be submitted by that date! Middle schoolers do not need to submit papers in advance. Pre-register here!
  • Are you still looking for a project idea? MC2 teachers just conducted a problem brainstorming session–email if you’d like to see one!
  • If you aren’t ready to participate in QED this year, we take visitors. Sign up here to get an intro to QED, visit projects, try our Guesstimathon, brainstorm an idea for next year (while eating pizza), and attend the QED Talk.
  • Speaking of our QED Talk, this year’s speaker is Peter Tingley of Loyola University. Dr. Tingley is a leader of the Chicago Math Teacher’s Circle, and, I claim, he will teach every QED participant and guest how to juggle. We will provide the juggling balls–just come ready to learn.

So, QED. Imagine a science fair. But with math and computer science projects. Where you do things like guess the number of calories in a single serving of Chocolate Chip Cookie-Doug Cheesecake from the Cheesecake Factory. And you learn how to juggle.

You have to come.