Welcome to the 178th edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.
Bookmark this post, so you can take your time browsing.
There’s so much playful math to enjoy!
By tradition, we start the carnival with a puzzle/activity in honor of our 178th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.
Activity: Nicomachus’s Theorem
Welcome to 2025, a perfectly square year — and the only one this century!
2025 = (20 + 25)2
- When is the next time we’ll have a perfect-square year?
- Can you find the only perfect square less than 2025 that works by this pattern? When you split the number’s digits into two smaller numbers and square their sum, you get back to that number.
2025 = the sum of all the numbers in the multiplication table, from 1×1 to 9×9
2025 = the sum of the first 9 perfect cubes
- When is the next time this will happen, that the year is the sum of the first n perfect cubes?
And by Nicomachus’s theorem:
2025 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93
so it must also = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2
Try it for yourself with small numbers: Get some blocks, and build the first few perfect cubes. Then see if you can rearrange the block to form the sum of those numbers squared.
Can you show that…
- 13 = 12
- 13 + 23 = (1 + 2)2
- 13 + 23 + 33 = (1 + 2 + 3)2
- 13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2
- 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2
Older Students: Can you see that the pattern would continue as long as you want? How might you prove that?
Here’s the formula for triangular numbers, to get you started:
(1 + 2 + 3 + … + n) = n(n + 1)/2
Contents
And now, on to the main attraction: the blog posts. Some articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.
- Talking Math with Kids
- Exploring Elementary Arithmetic
- Adventuring into Algebra and Geometry
- Scaling the Slopes of High School Math
- Enjoying Recreational Puzzles and Math Art
- Teaching with Wisdom and Grace
- Bloggers: Will You Help Keep the Carnival Going?
Talking Math with Kids
“Kids have a lot of math ideas, even kids who don’t think they do, so the more we can make math about the ideas kids have, the more kids will like math.”
—Dan Meyer
- Dan Meyer explains the value of talking math in the classroom: Kids Like Learning About Themselves More Than Math.
- Christopher Danielson ponders the challenges of talking math with kids: Questioning Piaget.
- Simon Gregg’s students are Making Numberblocks.
- For my entry to the carnival, a post on some of my favorite ways to talk math with kids: Real Math for Early Learners.
- Pam Harris’s weekly mental-math puzzle is a great discussion prompt: MathStratChat.
- Marilyn Burns talks fractions with kids as they try to understand: How Was Amber Reasoning?
- Claire and her kids talk percents in medieval England: Hands-On Homeschool Percentage Maths {Domesday Book}.
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Exploring Elementary Arithmetic
“When I was typing up these puzzles, my five-year-old became very interested in what I was working on. We did the first couple of puzzles together, and then he really surprised me by solving a few of the puzzles entirely on his own.”
—Sarah Carter
- Sarah Carter posts 30 Square Logic Puzzles, a fun way to practice lots of addition.
- John Golden (host of last month’s Playful Math Carnival 177) develops a new math game, and asks for input on improving it: Make a Difference.
- Sam Blatherwick has fun with arithmetic puzzles: Differences.
- David Marain poses a fraction challenge: 1/2 + 1/3 + 1/6 = 1 and Beyond.
- 1st Class Maths shares a nice puzzle on the Properties of Numbers.
- NCTM launches my favorite annual mathematics challenge: Play the 2025 Year Game.
- Pat Ballew plays with big multiplication problems in A problem with carries, and a solution.
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Adventuring into Algebra and Geometry
“We as math teachers often teach about rules and ‘mathematical truths’ that work all the time, so it is logical to us that certain math things won’t change… but students may not expect invariants to show up.”
—Karen Campe
- Karen Campe launches a series of posts on using technology to build understanding: Go For Geometry!, and Go For Geometry! 2.
- Randall Munroe’s word graphs are a fun way to think about how coordinates work: Features of Adulthood.
- Paul Rowlandson makes art with geometry: Pleasing Patterns and Playful Programming with Logo.
- Elissa Miller’s students celebrate math in Unit Circle Art IX.
- And Randall Munroe reports on a new discovery that may revolutionize high school geometry: Unit Circle.
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Scaling the Slopes of High School Math
“When it comes to data visualization, the stakes can be high — especially when the data we’re sharing could save lives or change behaviors. A simple graph showing cold temperatures may not convey the urgency of the situation, and worse, it may even mislead.”
—Mike Cisneros
- Mike Cisneros demonstrates how to make data tell a story: Cold weather context: graphing lessons from the polar vortex.
- And Druin shares New (to me) Tools for Graphing Categorical Data.
- Karen Campe posts the January Calendar Problems, but the puzzles can be taken off the calendar and used for fun anytime. Or watch her blog for next month’s calendar.
- Sarah Carter challenges her precalculus students with an Arithmetic Sequence Puzzle.
- Pat Ballew brings up a probability puzzler: Pick Two and Get One of Each?
- The Girls’ Angle Bulletin is available to download or read online.
- John D. Cook hosts the 235th Carnival of Mathematics.
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Enjoying Recreational Puzzles and Math Art
“The key is in the image, and in the careful way Dudeney worded his story. If you have a child like mine, you know to look for a trick. But are you crafty enough to figure it out? (I had to peek, but then my kids usually managed to stump me, too!)”
—Denise Gaskins
- I post a problem from Henry Ernest Dudeney’s Amusements in Mathematics that reminds me of my trickster son: Memories: The Oral Story Problem Game. And in case you missed it last month, check out Henry Dudeney’s Pebble Game.
- Clara Moskowitz highlights her pick for The 7 Coolest Mathematical Discoveries of 2024.
- Eric Elter shares a printable poster of 2025 in all its forms, from school to university.
- Thaddeus Wert reviews Lewis Carroll in Numberland. Meanwhile, Pat Ballew posts a wide-ranging compilation: Lewis Carroll, Happy Birthday, and RIP.
- JoAnne Growney collects math-related poetry, including Geometry in New York City and From Fibs to Fractals.
- Jo Morgan links to a fun daily Knight Puzzle in a recent MathsGems post.
- Erick Lee and students create mathematical art using Knot Tiles (inspired by Annie Perkins’ #MathArtChallenge).
- Jonathan Halabi poses A Math Question: the Crazed Carpenter.
- Colin Beveridge looks at a new variation on the Monty Hall problem: The Goat In The Machine.
- Tanya Khovanova shares A Puzzle from the Möbius Tournament.
- Matthew Scroggs plays with the Friendly squares puzzle. And Colin Beveridge digs into another of Scroggs’ puzzles: A Scroggsvent treat.
- Ben Orlin illustrates Puzzles of strange coinage (increasing in difficulty).
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Teaching with Wisdom and Grace
“Anyone wanting to know how to create lovely maths tasks might like to know Don’s secret. He would sit with a coffee and play. Just play about with the mathematics until something lovely happened. And that is what his tasks offer pupils — an opportunity to play with mathematics and for something worth talking about to be revealed.”
—Tom Francome
- Tom Francome has been working to digitize Don Steward’s old notes (stored in over 140 ring-binders!), and shares some favorites: Don Steward’s blog ‘Median’: a source of high-quality teaching and practice materials.
- Dylan Kane warns that Conceptual Understanding Doesn’t Happen All At Once.
- Can you plan a whole lesson around a single problem? Marilyn Burns taught A Lesson Designed Around One Problem: 100 ÷ 3.
- Ben Orlin illustrates The illusory consensus of math reform.
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Bloggrs: Will You Help Keep the Carnival Going?
And that rounds up this edition of the Playful Math Blog Carnival. I hope you enjoyed the ride.
The next installment of our carnival should open sometime during mid- to late-February, but we don’t have a host scheduled.
We need volunteers! Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Playful Math Blog Carnival, please speak up!
You can leave a comment here below, or send me an email. Visit our blog carnival information page for more details.
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“Playful Math 178: Nicomachus’s Carnival” copyright © 2025 by Denise Gaskins. Image at the top of the post copyright © NewAfrica / Depositphotos. “Nicomachus theorem 3D” illustration by Cmglee via Wikimedia Commons (CC BY-SA 3.0).
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