Professor Povey’s Perplexing Problems

Posted on by Denise Gaskins

Check out this new puzzle book for upper-level high school students & adults:

Professor Povey picture

Thomas Povey is a Professor of Engineering Science at the University of Oxford, where he researches jet-engine and rocket technology. In his new book Professor Povey’s Perplexing Problems, he shares his favorite idiosyncratic stumpers from pre-university maths and physics.

These problems “should test your ability to grapple with the unfamiliar,” Povey writes. “You will learn to tease new problems apart, and apply things you already know in ways you had never considered. You have all the tools you need, but you should see what amazing things you can do with them.”

Can You Solve This?

Alex Bellos shared one of Professor Povey’s puzzles in The Guardian. Can you figure it out?

Professor Povey cover

The book starts off with geometry, but most of the chapters focus on various topics from physics. Some of the puzzles are accessible through applied common sense, but for many of them, it helps to have taken an algebra-based (high school level) physics course.

Kitten is just finishing up her physics textbook, and she still has one more year of homeschooling. I’m hoping to work several of these puzzles into our schedule this year. It should be great fun!

Spoiler

If like me you’re a bit rusty on your physics, don’t worry. Each answer is thoroughly explained—‌in fact, it takes a bit of discipline to close the book and try your hand at each problem before reading on. I wish they’d put the solutions in the back rather than in the main text, to make it easier to browse the problems without reading spoilers.

Speaking of which, here’s the answer to the video puzzle above…

Playful Math Snacks for August: Logic Puzzles

Posted on by Denise Gaskins

The August “Let’s Play Math” newsletter went out last week to everyone who signed up for Tabletop Academy Press math updates. This month’s issue focuses on logic puzzles for all ages, including a newly-discovered deleted scene from Harry Potter and the Sorcerer’s Stone. What fun!

If you missed this month’s edition, no worries—‌here are some great puzzles from the Let’s Play Math blog archive:

The Centauri Logic Challenge

Sign Up Today

If you missed this month’s edition, no worries — there will be more playful math snacks coming soon. Click the link below to sign up today!

And remember: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

Math with Many Right Answers

Posted on by Denise Gaskins
The discussion matters more than the final answer.
The discussion matters more than the final answer.

One of the most persistent math myths in popular culture is the idea that mathematics is primarily about getting right answers.

The truth is, the answer doesn’t matter that much in math. What really matters is how you explain that answer. An answer is “right” if the explanation makes sense.

And if you don’t give an explanation, then you really aren’t doing mathematics at all.

Try This Number Puzzle

Here is a short sequence of numbers. Can you figure out the rule and fill in the next three blanks?

2, 3, 5, 7, ___, ___, ___, …

Remember, what’s important is not which numbers you pick, but rather how you explain your answer.

Possibility 1

Perhaps the sequence is the prime numbers?

2, 3, 5, 7, 11, 13, 17, …

The prime numbers make a wonderful sequence, though it isn’t the one I was thinking of.

Continue reading Math with Many Right Answers

Infinite Cake: Don Cohen’s Infinite Series for Kids

Posted on by Denise Gaskins

Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits
Prerequisite: able to identify fractions as part of a whole

This is how I tell the story:

  • We have a cake to share, just the two of us. It’s not TOO big a cake, ‘cuz we don’t want to get sick. An 8 × 8 or 16 × 16 square on the graph paper should be just right. Can you cut the cake so we each get a fair share? Color in your part.

Bobby Flay German Chocolate Cake

  • How big is your piece compared to the whole, original cake?
  • But you know, I’m on a diet, and I just don’t think I can eat my whole piece. Half the cake is too much for me. Is it okay if I share my piece with you? How can we divide it evenly, so we each get a fair share? How big is your new piece? Color it in.
  • How much of the whole, original cake do you have now? How can you tell?
  • I keep thinking of my diet, and I really don’t want all my piece of cake. It looks good, but it’s still just a bit too big for me. Will you take half of it? How big is that piece?
  • Now how much of the whole, original cake do you have? How could we figure it out?
    [Teaching tip: Don’t make kids do the calculation on paper. In the early stages, they can visualize and count up the fourths or maybe the eighths. As the pieces get smaller, the easiest way to find the sum is what Cohen does in the video below‌—‌identify how much of the cake is left out.]
  • Even for being on a diet, I still don’t feel very hungry…

Continue reading Infinite Cake: Don Cohen’s Infinite Series for Kids

Puzzle: Crystal Ball Connection Patterns

Posted on by Denise Gaskins

K4 matchings

In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.

Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.

On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.

Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.

For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.

TheWizardBySeanMcGrath-small

  • Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?
    T(0) = ?
    T(1) = ?
    T(2) = ?
    T(3) = ?
    T(4) = 10 connection patterns (as above)
    T(5) = ?
    T(6) = ?
    and so on.

Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.

Feature photo at top of post by Christian Schnettelker (web designer) and wizard photo by Sean McGrath via Flickr (CC BY 2.0). This puzzle was originally featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #76.

April 2015 Math Calendar

Posted on by Denise Gaskins

AprilMathCalendar

Six years ago, my homeschool co-op classes had fun creating this April calendar to hand out at our end-of-semester party. Looking at my regular calendar today, I noticed that April this year starts on Wednesday, just like it did back then. I wonder when’s the next time that will happen?

A math calendar is not as easy to read as a traditional calendar — it is more like a puzzle. The expression in each square simplifies to that day’s date, so your family can treat each day like a mini-review quiz: “Do you remember how to calculate this?”

The calendar my students made is appropriate for middle school and beyond, but you can make a math calendar with puzzles for any age or skill level. Better yet, encourage the kids to make puzzles of their own.

How to Use the Math Calendar

At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

As a puzzle:
Cut the calendar squares apart, then challenge your students to arrange them in ascending (or descending) order.

Help Us Make the Next Math Calendar

If you like, you may use the following worksheet:

Submission details here: Kids’ Project — More Math Calendars?

2015 Mathematics Game

Posted on by Denise Gaskins

[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both via Flickr (CC BY 2.0, text added).]

2015YearGame

Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

— Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2015 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

  • You must use all four digits. You may not use any other numbers.
  • Solutions that keep the year digits in 2-0-1-5 order are preferred, but not required.
  • You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
  • You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
  • You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

My Special Variations on the Rules

  • You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
  • You MAY NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.

Click here to continue reading.

Math Storytelling Day: The Hospital Floor

Posted on by Denise Gaskins

[Feature photo above by Christiaan Triebert via flickr (CC BY 2.0).]

Have you ever heard of Math Storytelling Day? On September 25, people around the world celebrate mathematics by telling stories together. The stories can be real — like my story below — or fictional like the tale of Wizard Mathys from Fantasia and his crystal ball communication system.

Check out these posts for more information:

My Math Story

tiles2

My story begins with an unexpected adventure in pain. Appendicitis sidewhacked my life last week, but that’s not the story. It’s just the setting. During my recovery, I spent a lot of time in the smaller room of my hospital suite. I noticed this semi-random pattern in the floor tile, which made me wonder:

  • Did they choose the pattern to keep their customers from getting bored while they were … occupied?
  • Is the randomness real? Or can I find a line of symmetry or a set of tiles that repeat?
  • If I take pictures from enough different angles, could I transfer the whole floor to graph paper for further study?
  • And if the nurse finds me doing this, will she send me to a different ward of the hospital? Do hospitals have psychiatric wards, or is that only in the movies?
  • What is the biggest chunk of squares I could “break out” from this pattern that would create the illusion of a regular, repeating tessellation?

I gave up on the graph paper idea (for now) and printed the pictures to play with. By my definition, “broken” pattern chunks need to be contiguous along the sides of the tiles, like pentominoes. Also, the edge of the chunk must be a clean break along the mortar lines. The piece can zigzag all over the place, but it isn’t allowed to come back and touch itself anywhere, even at a corner. No holes allowed.

I’m counting the plain squares as the unit and each of the smaller rectangles as a half square. So far, the biggest chunk of repeating tiles I’ve managed to break out is 283 squares.

tiles1

What Math Stories Will You Tell?

Have you and your children created any mathematical stories this year? I’d love to hear them! Please share your links in the comments section below.

 
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If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.

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Which I am going to say right now. Thank you!

“Math Storytelling Day: The Hospital Floor” copyright © 2014 by Denise Gaskins.

Fractions: 1/5 = 1/10 = 1/80 = 1?

Posted on by Denise Gaskins

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading Fractions: 1/5 = 1/10 = 1/80 = 1?

Reblog: Patty Paper Trisection

Posted on by Denise Gaskins

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

trisection2

I hear so many people say they hated geometry because of the proofs, but I’ve always loved a challenging puzzle. I found the following puzzle at a blog carnival during my first year of blogging. Don’t worry about the arbitrary two-column format you learned in high school — just think about what is true and how you know it must be so.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

trisection

One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.

One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why …

[Click here to go read Puzzle: Patty Paper Trisection.]