Math that makes worlds

From the May 18 New Yorker article World Without End, by Raffi Khatchadourian:

The design allows for extraordinary economy in computer processing: the terrain for eighteen quintillion unique planets flows out of only fourteen hundred lines of code. Because all the necessary visual information in the game is described by formulas, nothing needs to be rendered graphically until a player encounters it. Murray compared the process to a sine curve: one simple equation can define a limitless contour of hills and valleys—with every point on that contour generated independently of every other. “This is a lovely thing,” he said. “It means I don’t need to calculate anything before or after that point.” In the same way, the game continuously identifies a player’s location, and then renders only what is visible. Turn away from a mountain, an antelope, a star system, and it will vanish just as quickly as it appeared. “You can get philosophical about it,” Murray once said. “Does that planet exist before you visit it? Sort of not—until the maths create it.”

This article gets to something fundamental about the mathematical experience for me: even when you’re making the rules, the rules talk back, and give you worlds to explore that you couldn’t even have conceived of. The description of the programmers being drawn into exploring their own, unfathomable creation resonates; that’s the story of mathematics from the beginning. They found a way to make it visual and more broadly experiential.

Hopefully the game, No Man’s Sky, will be fun for everyone who plays it!


Quick Physical Games for the Math Classroom

We hold these truths to be self-evident, that kids need to move around, and creating opportunities to move during math class can pay off in spades.

Therefore, we have a collection of some of our favorite math/movement quick activities to share. These are especially good for K-4, though they’re adaptable to older and younger grades too. They provide a dose of movement, fun, and mathematical practice in an abbreviated time frame–perfect for station breaks and transitions.

If we’re missing any of your favorites, let us know! A PDF with Common Core tagging is available at our Lessons page.

Teacher-led Games

  • Groups (2-5 minutes)
    The teacher calls out a number (3), and the students have 10 seconds to get themselves into groups of that size. It might be impossible for everyone to get in a group every time, but each new number gives everyone another chance.In the basic game, just call out single numbers. Once students get the gist, you can call out addition or subtraction problems (i.e., “get into groups of 7-4.”)

    Don’t forget to call out a group of 1 and a group of however many students are in the entire class at some point in the game.

  • Stand Up/Sit Down (2-5 minutes)
    The rules are simple: if the teacher gives the number 10, students stand up. Any other number, they sit down. The trick is, the teacher will say things like “7+3” and “14 -5” (pick appropriate sums and differences for your students to solve mentally). This is a great game to try to “trick” the students by standing up or sitting down on when they should be doing the opposite.There are endless variations. For example:
    -stand when the number is larger than 5; sit if it is 5 or below
    -stand when the number is even; sit when it is odd
    -stand if the digit 1 appears on the number; sit otherwise.
  • Bigger/Smaller/Equal (2-5 minutes)
    If the teacher says a number greater than 10, students expand their bodies to take up as much space as they can (while keeping their feet firmly planted on the ground—no running around). If the teacher says a number less than 10, students shrink their bodies to take up the least space they can. If the teacher gives the number 10 exactly, students hold their body neutrally and make an equals sign with their arms.As before, the teacher moves to sums and differences once students get the rules.
  • Rhythmic Clapping/Counting (2-5 minutes)
    The teacher claps/counts out a rhythm. Students imitate the rhythm of the clap and the count.
  • Skip Counting with Movement (2-5 minutes)
    Make up a movement that comes in 2, 3, or more parts. Whisper the first parts, and call out the final move loudly.
    Example: Windmills. Whisper “1” and touch your right hand to your left foot. Whisper “2” and touch your left hand to your right foot. Call out “3” and do a jumping jack! Continue counting like this up to 30, calling out the multiples of 3 and whispering the numbers in between.Example: http://mathandmovement.com/pdfs/skipcountingguide.pdf
  • Circle Count (2-5 minutes)
    Stand in a circle and try to count off as quickly as possible all the way around the circle. Start with 1, then the student on your right says “2,” and the student on their right says “3,” and so on until the count comes back to you. Challenge the kids to go as quickly and seamlessly as possible.When everyone can do this proficiently, count by twos, fives, tens, or threes. You can also start at numbers greater than 1, or try counting backward.

Student-pair Games

  • Finger Speed-Sums (1-5 minutes)
    Students meet in pairs with one hand behind their back. On the count of three, they each put forward some number of fingers. Whoever says the sum first wins. Then the pair breaks up and each person finds a new person to play with. Advanced players can use two hands instead of just one.
  • Finger Speed-Differences (1-5 minutes)
    Same as speed-sums, except whoever find the difference between the two numbers first wins.
  • Five High Fives (1 – 2 minutes, or longer with the exploration)
    Students try to give a high-five to five different classmates. When they’ve gotten their five high-fives done, they sit down. This game is part mystery: sometimes it will be possible for everyone to get a high-five; sometimes not. The difference (which the teacher knows but the students don’t) is that it is only possible if there are an even number of people giving high-fives. Try this game at different times and let students guess whether they think everyone will get a high-five or not. Why does it only work sometimes, not always?If you make it four or six high-fives instead of five, then everyone will be able to get their high-fives every time.

Why we love these games

Getting kids moving is a win-win. Movement refreshes your students while giving them another take on math concepts. These games are super quick and super fun for everyone.

Tips for the classroom

  1. Make sure kids never feel ashamed if they don’t already know the right answer. You can also tweak competitive games to make them collaborative.
  2. You enthusiasm is critical in these games. Figure out your favorites, and expand on them, or get the students to come up with their own variations. If you’re into them and having a good time, the kids will have a good time too.

Counting Collections and Dots and Boxes, fractional version

I just rewrote our write up of Counting Collections, and reclassified it on our Lessons page as a Foundational Activity. The reclassification was motivated in part by a conversation with a Kindergarten teacher, who mentioned that she had been having the kids in her room count collections every Friday. “I help kids who need it,” she said, “but honestly, they’re so focused and engaged, I could probably disappear and the class would still run all right without me.” When lessons work this fluidly, it is easy to feel that you aren’t doing enough as a teacher. In fact, it’s the opposite: the necessity of heroics, theatrics, and great feats of charismatic teaching can be a sign that the activity you’re pushing at the kids might not be a good fit. The easy path can sometimes be the best one, especially when the teacher is underworked because the students are doing the heavy thinking.

Speaking of student inspiration, I was playing Dots and Boxes with a student last week, and he made a play I hadn’t seen before: instead of connecting two dots with one edge, he made two half edges. Suddenly, it was a new game. I made four quarter edges; he played three third-edges. Then we started connecting using multiple fractions: 1/2, 1/3, and 1/6. This fractional version of Dots and Boxes goes like this. On your turn, you have to add a total length of 1 unit to the board. If you complete a box, you get a point, and an extra unit length to add before your turn is over. Whoever completes the most boxes wins.

So after 8 turns from Blue and 7 from Red, a game might look like this.

Dots and Boxes 1


It’s Red’s turn. What move can Red make? Any full edge will give Blue a box to complete. But what about adding half-edges?

Dots and Boxes 2

Now where can Red go? Adding halves will give Blue a box, but Red can add three thirds!

Dots and Boxes 3

Does Blue have a move that won’t cost a box? I’ll let you figure it out. This game is totally new to me, but it’s clear that it will end (every turn adds a unit length of line segment to the board, after all), and this particular game won’t end in a tie either. I think it’s more likely to end in a blowout for the winning player.

Try out the game and let me know how it goes!

Cheryl’s Birthday Party, Meta-logic, and the known unknown

I almost missed the Cheryl’s Birthday Party internet phenomenon this week. An awkwardly written logic problem went viral, and the internet was abuzz with attempts to solve it. Check out the NYTimes treatment of the origial pnroblem, and the afterparty.

The problem is an example of a metalogic puzzle. Logic puzzles usually consist of organizing what you know and don’t know to suss conclusions out of ambiguous seeming clues. Metalogic puzzles are a different sort of beast: they require you to imagine what characters know and don’t know to solve problems. In a metalogic puzzle, a character not knowing something can be critical information.

Here’s some examples of my favorites.

The Three Hats.

Three princesses are taken prisoner by an ogre. For sport, the ogre puts them in a row from eldest to youngest, so that each sister can only see her younger sisters. Then he blindfolds them and says, “I have three white hats and two black hats. I will put one hat on each of your head. If any one of you can tell me the color of your own hat when I take off your blindfold, I’ll let all three of you go.”

The ogre takes the blindfold off the eldest princess. She looks ahead at the hats on her two younger sisters, thinks, and finally says, “I don’t know what color my hat is.”

The ogre takes the blindfold off the middle princess. She looks ahead at the hat on her younger sister, thinks, and says, “I don’t know what color my hat is.”

Then the ogre takes the blindfold off the youngest princess. She looks ahead into the woods–she can’t see anyone else’s hat. Then she thinks, and says, “I know what color my hat is.”

What color is her hat? How did she know?

I love this puzzle, because it doesn’t seem like there’s enough there to answer it. The secret lies in imagining yourself in the head of each princess as they imagine themselves in the head of their sisters. It requires a fantastic leap of imagination. I’m not going to solve it for you, but consider:
What does the eldest sister know about her sisters’ hats?
What does the middle sister know from the fact that the eldest sister doesn’t know her own hat? What does it tell us that even with this information she doesn’t know the color of her own hat?

I’ve recently come up with my own series of metalogic puzzles. I don’t think these are too hard if you can get past the fact that they seem so devoid of usable information to start.

The Known Unknown

Abby and Bill play a game, where they each think of a whole number and try to guess each other’s number by asking each other questions which must be answered truthfully.

Game 1

Abby and Bill each pick a number in the 1 to 30 range.
Abby: Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number half my number?
Bill: I don’t know. Is your number half my number?
Abby: I don’t know.
Bill: I know your number.
What is Abby’s number?

Game 2

Abby and Bill each pick a number in the 1 to 40 range.
Abby: Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number twice my number?
Bill: I don’t know. Is your number twice my number?
Abby: I don’t know. Is your number twice my number?
Bill: I don’t know. Did you think we chose the same number?
Abby: We didn’t.
What are their numbers?

Finally, here’s a classic that I’d say is a level more difficult, and seems even more impossible to solve than the previous ones. It’s definitely doable though–you just need to remember who knows what, and write it all down.

The Surveyor’s Dilemma

A surveyor stops at a mathematician’s house to take a census.

Surveyor: “How many children do you have, and how old are they?”
Mathematician: “I have 3 children. The product of their ages is 36.
S: “That’s not enough information.”
M: “The sum of their ages is the same as my house number.”
S: “That’s still not enough information.”
M: “My eldest child is learning the violin.”
S: “Now I have enough information.”

What are the ages of the mathematician’s children?

Play is where love begins

I recently wrote a piece for the New York Times Numberplay blog on what we do to help people fall in love with math. I thought I’d include it here.

No matter whom we work with, our initial goal is for them to have an authentic, mathematical experience; that is the first step to helping anyone — teachers, students, parents, kids, you name it — develop a love for math. The question then becomes, what does authentic mathematics look like, and how can you get people to experience it, especially if they’ve been turned off from the subject?

The answer is play. Play is the engine of learning, and our minds are wired to play with mathematics. When people are able to play with mathematics, they learn more profoundly and expertly than they would otherwise. The enemy of play is fear, and insistence that others do things your way, the “right” way. If you share mathematics as something you love, a gift you and everyone else gets to play with, and have the discipline to reserve judgment, what you end up seeing is that people of all ages will quickly take up the invitation to play with you, and soon they are doing math with a level of creativity, interest and rigor that you might have thought would be impossible.

All we need to do to open people up to mathematics then, is give them permission, and the right circumstances, to take ownership of their mathematical experience and begin playing. There are critical elements to provide:

A worthy mathematical question or problem, with a low barrier to entry and a high ceiling,
A safe atmosphere, free of judgment,
Time, and whatever encouragement and support is necessary to coax people into playing.

I think the Broken Calculator problem qualifies as the right kind of mathematical question, potentially, though working with teachers or students I usually change the question to: What numbers are possible to get on the calculator and what numbers are impossible? That way, people can mess around and get positive feedback. You made 12 on the calculator? Great! That tells us something new — what else is possible? That’s what I mean by a low barrier to entry. But the problem is designed to resist easy answers, too (unless you’ve worked on this kind of question before), so it is unlikely that anyone without training would be able to say, “Here’s the answer. Now what?”

I recently demoed a pared-down variation of the Broken Calculator problem in a third grade classroom (without square roots) as part of a teacher professional development partnership. I was worried that I’d have to sell the problem to the students, but the opposite was true: The moment the kids saw that they were in control of how they approached the question, they couldn’t get enough. Kids know how to play.

In the end, play really is the magic ingredient. Einstein called play the highest form of research; happily, play is also the best way to win anyone over to math. Play is where love begins.

Spring Classes, Summer Classes, and Julia Robinson Festival

Recently, in an art gallery in Ballard, I saw the amazing painting above. (Check out the artist’s website here.) I love this kind of mathematical art–the tessellation in the background a kind of blanket that subsumes the floor and clothes of the people in the picture. It’s a meeting of mathematical structure and organic complexity.

We’re going to be exploring all kinds of beautiful geometric structure this spring and summer! Registration is open for Math for Love‘s upcoming:

These can fill up fast, so sign up now if you would like your student to join us!

Read on to learn more (or skip to the bottom for the Problem of the Moment).


Spring Sunday Class Registration is Open 

Math for Love classes are a chance to learn beautiful, powerful mathematical ideas from mathematicians wholove to teach.

Sign up your K-8th grader now for a spot in our classes at the Phinney Neighborhood Association, starting 
Sunday, April 19.

Our theme this spring is Making and Breaking Conjectures.
Find out more here.


Fourth Annual Julia Robinson Festival!

The mission of the Julia Robinson Mathematics Festival is to inspire students to explore the richness and beauty ofmathematics through activities that encourage collaborative, creative problem-solving.

Join us Saturday, April 4th for this noncompetitive celebration of great ideas and problems in mathematics. Held at the HUB on UW’s campus, and open to all students grades 4 – 10.
To learn more and sign up, click here.
To volunteer, click here.

Price: $10 – 15. Free and reduced registration is available. Use the discount code “scholarship” to get an additional 50% discount.
Made possible with financial support from the UW Math Department and the Puget Sound Council of Math Teachers.


Math for Love Summer Math Camp!

We are expanding our popular Summer Math Camp from last year, with four 1-week sessions in Seattle and Bellevue.Registration is open now. Sign up your 8-10 or 11-15 year old for one or more sessions, and see what will unfold in these playful mathematical explorations of shape and structure.
Learn more here.


Problem of the Moment

A Fault-Free Rectangle is a rectangle made of dominoes that contains no horizontal or vertical “faults,” that is, lines that would allow you to pull the rectangle apart into two rectangles.
A student recently constructed the fault free rectangle on the right. Is this the smallest fault-free rectangle possible (not counting the single domino)? If so, what is the next largest fault-free rectangle you can build?


Three Square Problem and Variations

I just saw an absolutely charming problem on Numberphile that I was shocked never to have seen before. They call it the Three Square Problem (featuring Professor Zvezdelina Stankova).

 Three Square Problem

Three Square

  • Prove that \alpha + \beta + \gamma = 90^\circ

It feels, as Prof. S says in the video, like a beautiful conjecture.

I highly recommend trying to come up with a proof. There are many (54!), and I’ve come up with about seven since I saw the video. More intriguing, though, is the question of whether this type of thing happens in any other cases. Let’s look at a picture that summarizes the most surprising part of this problem.

Three Square View 2

It is not particularly surprising that \alpha = 45^\circ. The shock is that \gamma + latex = 45^\circ as well. Are there other right triangles we can draw on a grid that have this property? The answer, it turns out, is yes.

five by three

  • Prove that \gamma + \beta = 45^\circ.

Let’s go into even greater generality. Suppose we have two rectangles with integer side lengths. (Everything that comes later will refer to the picture below.)General case

  • Big Question 1: For what collections A, B, C, and D will the x + y = 45^\circ?
  • Big Question 2: Given any A and B, does there always exist a C and D so that x + y = 45^\circ?

Here are some of the answers I’ve discovered so far:

A = 2, B = 3, C = 5, D = 1.
A = 3, B = 4, C = 7, D = 1.
A = 4, B = 5, C = 9, D = 1.

  • Prove the examples above all satisfy x + y = 45^\circ.
  • Define a pattern in the numbers above. Will always work?

Here’s another observation about the list of numbers above. A^2 + B^2 = \frac{C^2 + D^2}{2}! What could that have to do with things?

Wild Conjecture: If A^2 + B^2 = \frac{C^2 + D^2}{2}, then x + y = 45^\circ.

  • Prove or disprove the wild conjecture.

Here’s another sequence of solutions where x + y = 45^\circ:

A = 3, B = 5, C = 4, D = 1
A = 5, B = 7, C = 6, D = 1

  • Prove that these sets of A,B,C,D satisfy x + y = 45^\circ.

What happens if we sum the squares? Once again, 3^2 + 5^2 = 9 + 25 = 34 is double 4^2 + 1^2 = 17. This is evidence in favor of the Wild Conjecture. Not nearly a proof.

Notice what’s happened here. We began with an isolated question about a cool relationship between specific angles. By asking the next natural questions, we have very surprising variations of that original relationship, and an entirely different pattern emerging in the squares of the sides of the rectangles. True understanding comes not from solving one problem, but in solving families of problems in multiple ways, and following the natural questions as far as we can.

I’ll leave you with a suggestive diagram for a proof of the original three square problem.AnswerHere we have the original 1 by 2 right triangle (AEB) next to a scaled up 1 by 3 right triangle (CDB). Can  you see why their two small angles (the ones at B) sum to 45 degrees? Do you see how this picture could generalize?




what’s awesome about algebra

As students are enrolling in pre-algebra and algebra in droves, I thought I would post this video, which is part 1 of a magic trick illustrating the mind-boggling power of algebra. It is very common to learn how to do algebra without learning why to do it. To me, this magic trick gives a sense of how useful and powerful the technique of algebra can be.

Part 1 ends with a mystery, including a $100 bounty for anyone who can find a counterexample to the trick. If you know a student (or are a student) who thinks you can find a way to break this trick, let me know if you do.

Fall Registration Open

If you are in Seattle or Bellevue, we have a host of great classes and circles to offer this fall, all focusing on the beauty, power, and elegance of mathematics.

Saturday Classes at the PNA in Phinney Ridge

This Saturday session runs for six sessions, from October 18 – November 22. This session’s topic: Games, Logic and Arithmetic.

Kindergarten & 1st grade
Section 1 
12:05pm – 12:55pm Sign up now!
Section 2 1:05pm – 1:55pm     Sign up now!

2nd & 3rd grade
12:05pm – 12:55 Sign up now!

4th & 5th grade
1:05pm – 1:55pm Sign up now!

6th, 7th & 8th grade
2:05 – 2:55pm Sign up now!

NEW! – Sunday Classes at ROMP in Bellevue

This new Sunday session runs for eight sessions, from October 5 – November 23. Games, Logic and Arithmetic will be the topic for these new Bellevue classes too.

2nd & 3rd grade
10:00 – 10:55 Sign up now!

4th & 5th grade
11:00 – 11:55 Sign up now!

6th, 7th & 8th grade
12:00 – 12:55 Sign up now!

Math Circles

Math Circles meet before or after school, and highlight our favorite games, puzzles, and mathematical ideas. Dates and times are variable. Click the signup link to learn more, or check with your school.

APP at Lincoln — Sign up now!
Catharine Blaine K-8 – Sign up now!
McGilvra Elementary School – Sign up through the school
Queen Anne Elementary – Sign up now!
Thurgood Marshall – Sign up through the school.
View Ridge Elementary 2nd & 3rd grades – Sign up now! 
View Ridge Elementary 4th & 5th grades- Sign up now!
West Woodland Elementary – sign up through the school


From 21 to 500: Game & Math Salon

Here’s a fun, very simple classroom game you can play for multiplication.

You may know the game 21, aka blackjack. In classrooms, I like to play with a deck that only includes numbers from 1 to 10.

Twenty-one. Each player gets two cards (face up). They can “hit” to take another card, or “stay” to stick with what they have. Whoever gets as close to 21 without going over wins. (Traditionally this game is played against the dealer in casinos. It’s fine to play it that way as well.)

Here’s how 21 becomes 500:

Five Hundred. Each player gets two cards. As in 21, they can “hit” or “stay.” The difference in 500 is that you multiply the numbers on your cards together. The goal is to get as close to 500 as possible without going over.

So in the image above, the 19 in Twenty-One would be a 240 in 500. Worth sticking in either case.

I only just made this game up last week, and haven’t played too much, so please experiment. Is 500 the best number to have as the bust point? Still, the game makes kids estimate, make a single strategic decision, and multiply one digit numbers and two digit numbers. And it takes almost no time to teach it.

Announcement: Math Salon on August 16!

For Seattleites: we’re happy to announce that we have a Math Salon on the calendar. Supported and hosted at the Greenwood Library, this event is a great opportunity for you and your kids to spend a Saturday afternoon playing with math. If you’re interested in joining us, please rsvp here.

Want to volunteer? Email dan [at] mathforlove.com to join us.