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?

1

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.

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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?

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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?

 

 

3

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

3

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.

1

Revisiting Internal Motivation

There is a tension between intrinsic and extrinsic motivation in teaching mathematics. Our answer to the classic student questions Why do I need to learn this? is a good measure of where we look for motivation. You can appeal to the extrinsic, or instrumental, rewards: you need math to succeed in get a good grade, to succeed in middle school, high school, college math, to get a good job, and so on. And of course, that’s what a lot of people do.

On the other hand, you can take the tougher route of appealing to the intrinsic rewards. You need to learn math because it is beautiful, challenging, elegant, amazing. The reward of math is that it is engaging right now, in the present moment, and you should learn it because something in you needs to know it.

Any reader of this blog knows where we stand. Our name is Math for Love, after all. (Early motto: the only reason to do math is for love.)

_________________

Sidenote: Creating the conditions that encourage the growth of intrinsic motivation is nontrivial; it defies the casual effort. It is one of the central jobs of a teacher, and the reason that teaching is a serious profession.

In a recent talk at Los Alamos, Bill Gates described the difficulty of reforming education as greater than the difficulty in curing malaria.

New technology to engage students holds some promise, but Gates says it tends to only benefit those who are motivated.”And the one thing we have a lot of in the United States is unmotivated students,” Gates said.

If we could automate what it takes to instill curiosity, passion, and love for a subject in a group of kids, then there wouldn’t be much of a reason to respect the work of teaching. But the nut of creating student motivation from tech solutions has barely begun to be cracked. In fact, it’s precisely because motivating is so deviously hard that teaching ranks as one of the most interesting and respectable professions (in my eyes at least. And in the eyes of those nations that tend to have more well-educated populations.) Not surprisingly, those who don’t understand how difficult creating motivation is are the same ones who malign teachers.

Sub-sidenote: Gates might have been in the camp that thought education reform was easy and teaching was rote before. If so, it sounds to me like he’s coming around. Nothing like working in education to see how hard it is to change it. There’s a possibly apocryphal story about some founding father—I forget which one—who tried and failed to reform schools, so went on to found the country… an easier job.)

_________________

A natural reaction when considering intrinsic vs. instrumental motivation would be to conclude that trying to motivate students using both internal and external rewards would be the best way to go. But new research hints that two motives may not be better than one. In the study described by its authors Amy Wrzesniewski and Barry Schwartz, the researchers surveyed over 11,320 West Point cadets and found that among those with strong internal motivations, those with powerful extrinsic motives actually did worse in every capacity—graduation rates, performance in the military, etc.—than those without them.

In other words, extrinsic motivations may perhaps weaken the long term power of love the work. If this is really true and applies more broadly, it suggests that wanting the money, fame, renown, etc. from doing great work actually gets in the way of achieving it. We get external prizes when we take our eyes off them, and focus on our passion for the real work.

In other words: the only reason to do math is for love.

There’s more to say about motivation, and the depth and complexity of it will always make teaching a fascinating profession. But for now, I’ll leave you with this wonderful RSAnimate video of Dan Pink on the counterintuitive nature of motivation.

 

Playing with Math and more

Early summer is a great time of year in Seattle and Eastern WA, where we’re splitting our time right now. We’ve been pouring energy into a few projects: launching our new game, Prime Climb; supporting Seattle Summer School with curriculum and professional development support; piloting a Math for Love summer camp, which just wrapped up after an excellent week; and setting up for fall. But in the midst of this, there has been some moments to down-regulate, and these times to breathe have been vital. I’ve gotten to read and take walks and relax, and I can feel my mind expanding as it gets a chance to rest.

I thought I’d give an update of some interesting mathematical things coming out right now: a board game (ours), a book, a video game, and a video series. All of them cost money, but depending on your interests, needs, and financial status, I think they all might be worth it.

Prime Climb.

Our new board game, Prime Climb, had a great showing on Kickstarter last month, and we’re working on the next step of getting the design finalized and the game put out into the world—not to mention the fascinating and challenging work of figuring out how domestic and international distribution works (we hope to be big in Japan). It’s very hard to predict what the future of Prime Climb will be, but you can ensure you get a game from the first printing if you pre-order this summer. Just click the “games” link above, or click here. You can get the game shipped to you in the US for $35.

Playing with Math. 

I’ve read about a third of Playing with Math, but I wanted to get this review out now since a crowd-funding campaign for the book is underway. Sue Van Hattum set out years ago to put together the book she wanted to read, and Playing with Math is the result. The organization is simple: Sue contacted dozens of the people she saw doing cutting edge work in mathematics teaching and asked them to write a brief piece about their work. Among those contributing to the book are leaders of math circles, homeschooling parents, innovative teachers, bloggers and writers, and more, each telling stories that encapsulate their perspective and the results of their experience. There are math puzzles, games, and projects peppered throughout the pages as well.

Playing with Math is an exciting book. There’s a thrill of people really experimenting here, working out the possibilities of how math can be taught. The authors share their missteps and their successes; through it all, you start to see a coherent philosophy taking shape. Learning math requires struggle and joy… it’s serious play. But over and over, the right challenges at the right time empower the student, and build momentum for their continuing journey in mathematics. It’s easy to say it–the details are what’s tricky. This book is a story of those details.

Not surprisingly, the pieces can be a bit uneven, but it’s easy to turn the page if one of the stories doesn’t speak to you. And there are some gems in this book. The Kaplans’ story of leading a prison math circle is a laugh-out-loud pleasure to read, and Colleen King’s description of turning math subjects into student-designed games is a vivid picture of a teacher discovering a new way to teach as her students discover a new way to think. This book is a snapshot of the work of some of the trailblazers of math education working now, and worth reading.

Support the Playing with Math campaign here, and get yourself a copy of the book. ($25 in the US.)

Mathbreakers. 

Even the best math video games tend to be about skill building. Add in good graphics, first person game play, etc., and you still tend to have a textbook approach to math underneath the play. But there’s a game out that looks like it might be different. Mathbreakers, on Kickstarter till Saturday, seems to be about creative play with mathematics in a way that other games aren’t. It looks like a Legend of Zelda with mathematics underneath it; you don’t rise in the game by answering a specific math problem, but by finding creative ways to make the necessary numbers in any way you can. It looks like a leap forward in math gaming, and I want to play it.

This campaign has three days to make its last 15%. You can pledge here and get the game for $25.

James Tanton’s Geometry in the Great Courses

I’ve never bought a Great Courses DVD–they strike me as overly expensive and not necessarily better than what you can get for free online. That was before I saw that James Tanton has a Great Courses course on geometry. James is one of the best math teachers and curriculum designers in the country, and one of the few people who always teaches me something new. James has tons of free videos on his website, and a great monthly newsletter, but this new course looks like it’s a step up in terms of production values and pacing. If you’re looking for a video geometry program and planning to spend some money on it, you should check out James Tanton’s new geometry in the Great Courses.

Expensive at $320 – 375. But if you go for Great Courses, then this is one to get.

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Primo – the beautiful, colorful, mathematical board game

Primo

A Totally Unique Mathematical Board Game from Math for Love

More than a year ago we had an idea to build a game around what felt like one of the unsung ideas in math: prime numbers make multiplication easy. (Why does no one learn this in school?) We realized that with the right color-coding, it would be possible to see multiplication and division as combining or removing colors.

Primo BoardA year later, Primo is ready. The game plays beautifully in play test after play test. It’s one of the most mathematically rich games we have ever seen, and at the same time avoids that icky “educational game” feel. Primo is a real game and it’s worth playing because it’s fun. Really fun.

The game is a race. Arithmetic is the engine, but not the end. Players add, subtract, multiply, and divide their way to the center of the board, knocking each other back to start and collecting Primo cards as they land on primes (the red circles) along the way. It’s a very easy game to learn, and infinitely replayable. Kids (and adults) practice their arithmetic without even noticing that they’re learning, and the game makes prime numbers intuitive and multiplication legible.

We decided to launch Primo via Kickstarter, and the campaign is up and running right now. This means that you can get a first run copy of Primo by donating to our campaign right now.

But if you really want to help us get Primo into the world, tell people about it. The beauty and terror of Kickstarter is that if we don’t make our goal, we don’t get to print up the first run of 1000 copies, and Primo gets relegated to some dusty closet. Email your friends and colleagues, post on Facebook, tweet, and let others know that they can get the game here:
https://www.kickstarter.com/projects/343941773/primo-the-beautiful-colorful-mathematical-board-ga

We hope you love this game. Thanks for your support.

Dan & Katherine
Math for Love