1679 and Level 6

Today’s Puzzle:

Can you complete this multiplication table puzzle? It’s a level 6, so some of the clues might be a little tricky. Use logic every step of the way, and everything will work out for you!

Factors of 1679:

  • 1679 is a composite number.
  • Prime factorization: 1679 = 23 × 73.
  • 1679 has no exponents greater than 1 in its prime factorization, so √1679 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1679 has exactly 4 factors.
  • The factors of 1679 are outlined with their factor pair partners in the graphic below.

More About the Number 1679:

From OEIS.org we learn that
1 + 6 + 7 + 9 = 23, AND 1679 is divisible by 23.
This is cool: 1679 is the smallest multiple of 23 that can make that claim!

1679 is the hypotenuse of a Pythagorean triple:
1104-1265 -1679, which is 23 times (48-55-73).

1677 and Level 4

Today’s Puzzle:

Using logic, write the numbers from 1 to 10 in both the first column and the top row so that this puzzle forms a multiplication table.

Factors of 1677:

  • 1677 is a composite number.
  • Prime factorization: 1677 = 3 × 13 × 43.
  • 1677 has no exponents greater than 1 in its prime factorization, so √1677 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1677 has exactly 8 factors.
  • The factors of 1677 are outlined with their factor pair partners in the graphic below.

More About the Number 1677:

The last digit of every prime factor of 1677 is 3.

1677 is the hypotenuse of a Pythagorean triple:
645-1548-1677, which is (5-12-13) times 129.

1665 Why I Am Rooting for England over Denmark Today

Today’s Puzzle:

England is playing Denmark in the Euro 2020 Semi-Finals today. Why am I rooting for England when I don’t really follow soccer, as we call it in America? It ISN’T because 5 of my great-grandparents claim English ancestry, and just only one of my great-grandparents was born in Denmark. England wins 5 to 1. No, that isn’t the reason why I’m excited today and created this soccer/football puzzle by freehand in paint:

How many pentagon and hexagon transformations are there on the game ball?

The reason I am rooting for England is that an ingenious mathematics teacher, AC@eymaths, created and shared an exciting and even better transformation puzzle on Twitter: pdf of a transformation puzzle worksheet.

The transformation looked like a wonderful idea, still, I wasn’t sure what all the fuss was about or the meaning of “Miss, is it coming home?” The next day I asked about it:

I googled and found this music video from Three Lions.

I love the enthusiasm shown even while expressing these lyrics:

“Everyone seems to know the score, they’ve seen it all before
They just know, they’re so sure
That England’s gonna throw it away, gonna blow it away
But I know they can play.”

I know that exact feeling! I’ve had high hopes for a team that only disappointed me. I love how everyone in England is in the moment and feeling enthusiastic no matter what! I have watched the music video over and over again. It’s also wonderful that so many mathematics teachers at several different levels are embracing their students’ excitement:

Factors of 1665:

Since this is my 1665th post, I’ll share factoring information about the number 1665.

Obviously, 1665 ends with a 5, so it is also divisible by 5.
6, 6, and 1 + 5 use up all the digits and give us three 6’s (three of the same multiple of 3), so 1665 is divisible by 9 and, of course, by 3.

The prime factors of 1665 work together to give us several repdigits as factors, too: 111, 333, and 555.

  • 1665 is a composite number.
  • Prime factorization: 1665 = 3 × 3 × 5 × 37, which can be written 1665 = 3² × 5 × 37.
  • 1665 has at least one exponent greater than 1 in its prime factorization so √1665 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1665 = (√9)(√185) = 3√185.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1665 has exactly 12 factors.
  • The factors of 1665 are outlined with their factor pair partners in the graphic below.

More About the Number 1665:

1665 is the sum of two squares in TWO different ways:
39² + 12² = 1665, and
33² + 24² = 1665.

1665 is the hypotenuse of FOUR Pythagorean triples:
513-1584-1665, calculated from 33² – 24², 2(33)(24), 33² + 24²,
but is also 9 times (57-176-185),
540-1575-1665, which is (12-35-37) times 45.
936-1377-1665, calculated from 2(39)(12), 39² – 12², 39² + 12²,
but is also 9 times (104-153-185), and
999-1332-1665, which is (3-4-5) times 333.

1661 A Little About Palindromes

Today’s Puzzle:

A number is a palindrome if it reads the same backward as it does forward. For example, 1661 is a 4-digit palindrome. Today’s puzzle asks you to explore number palindromes and their factors.

Half of the time when a 4-digit palindrome is divided by 11, we get a 3-digit palindrome. Why does that happen?

Are there 3-digit palindromes that were NOT included in the Divided by Eleven column in the table above?

All 2-digit palindromes are divisible by 11. They are 11, 22, 33, 44, 55, 66, 77, 88, 99.

There are only eight 3-digit palindromes that are divisible by 11. They are 121, 242, 363, 484, 616, 737, 858, 979. Some 3-digit palindromes are prime numbers. Others are divisible by 101 or 111. Still, there are plenty of 3-digit palindromes that are composite numbers but not divisible by 11, 101, or 111.

Most 5-digit palindromes are NOT divisible by 11. I was able to construct one that is, 76967, because the red digits minus the blue digits are 2312 = 11, a number divisible by 11.

Here’s another: 81818. It works because 242 = 22, a number divisible by 11.

What 5-digit palindrome can you construct that is divisible by 11?

Will an N-digit palindrome be divisible by 11? What difference does it make if N is an even number or if N is an odd number?

Factors of 1661:

1661 is divisible by 11 because the red digits minus the blue digits equal 0, a number divisible by 11.

  • 1661 is a composite number.
  • Prime factorization: 1661 = 11 × 151.
  • 1661 has no exponents greater than 1 in its prime factorization, so √1661 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1661 has exactly 4 factors.
  • The factors of 1661 are outlined with their factor pair partners in the graphic below.

More about the Number 1661:

1661 is also a palindrome in base 18:
1661₁₀ = 525₁₈ because
5(18²) + 2(18¹) + 5(18º) =
5(324) + 2(18) + 5(1) =
1620 + 36 + 5 =

Prime Number 1657 is the 24th Centered Hexagonal Number

Today’s Puzzle:

Draw six triangles on the graphic below to show that 1657 is one more than 6 times the 23rd triangular number.

Factors of 1657:

  • 1657 is a prime number.
  • Prime factorization: 1657 is prime.
  • 1657 has no exponents greater than 1 in its prime factorization, so √1657 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1657 has exactly 2 factors.
  • The factors of 1657 are outlined with their factor pair partners in the graphic below.

How do we know that 1657 is a prime number? If 1657 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1657. Since 1657 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1657 is a prime number.

More About the Number 1657:

1657 is the sum of two squares:
36² + 19² = 1657.

1657 is the hypotenuse of a primitive Pythagorean triple:
935-1368-1657, calculated from 36² – 19², 2(36)(19), 36² + 19².

Here’s another way we know that 1657 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 19² = 1657 with 36 and 19 having no common prime factors, 1657 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1657. Since 1657 is not divisible by 5, 13, 17, 29, or 37, we know that 1657 is a prime number.

Do you notice anything else special about the number 1657 in this color-coded chart?

$1653, Fake Primes, and Taxman’s “Rigged” Counting

Today’s Puzzle:

If you’ve ever played Taxman or watched someone else play it, you know that the Taxman gets all available factors of each card you take. You can only take a card if the Taxman can also take at least one card on that turn. When the game is over, the Taxman gets ALL the leftover cards.

1653 is the 57th triangular number so if all the cards in the puzzle were envelopes containing dollar amounts indicated on the outside of the envelope, there would be $1653 at stake.

Many people start Taxman by taking the largest prime number followed by the largest prime number squared. What if a person claimed long before the game started that the only way he or she could lose is if the game is rigged? Most people have never played this game and might believe that claim especially if they perceive that the person making that claim is pretty good at math. Besides given the opportunity, won’t the Taxman take far more than his fair share just so he can spend it on frivolous projects? The fact that all remaining cards at the end of the game go to the Taxman will make the rigged claim seem even more plausible. Furthermore, what if “our math whiz” confidently called out his or her first two number choices, fake prime number 57 followed by perfect square 49? (57 is a composite number, but it often fools people into thinking it’s prime. You could call it a fake prime because it looks like a prime number but isn’t actually prime. Other fake primes are 51, 87, and 91.)

For today’s puzzle, I would like you to play this Taxman game with the mistaken assumption that 57 and 51 are prime numbers. Of course, the Taxman will know better. It will still be possible to win, but it will be much more difficult.

You can print the cards to play Taxman from this file: 10 Factors 1650-1660 with Taxman Scoring Calculator. You might choose to have someone else be the Taxman while you stand far enough away not to be able to see the factors listed on the top of the cards. Whether you are close to the cards or far away, don’t allow yourself any do-overs.

I’ve included a taxman scoring calculator in that excel file. Only enter numbers under “My Cards” and “Taxman Cards”. The rest of the data will auto-populate. You win if your tax rate is less than 50%. I would be very interested to know if you win or if you lose.

Factors of 1653:

  • 1653 is a composite number.
  • Prime factorization: 1653 = 3 × 19 × 29.
  • 1653 has no exponents greater than 1 in its prime factorization, so √1653 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1653 has exactly 8 factors.
  • The factors of 1653 are outlined with their factor pair partners in the graphic below.

More About the Number 1653:

1653 is the hypotenuse of a Pythagorean triple:
1140-1197-1653, which is (20-21-29) times 57.

1653 = 29 × 57.
1653 is the 29th hexagonal number, and
1653 is the 57th triangular number.

All hexagonal numbers are also triangular numbers. Can you look at the graphic above and see why that’s true? The broken line that I drew might be helpful. It separates the odd numbers from the even ones.

1653 is the 57th triangular number because (57)(58)/2 = 1653.
It is the 29th hexagonal number because 2(29²) – 29 = 1653.

Yes, YOU Can Host a Playful Math Education Blog Carnival

I had so much fun hosting the 146th Playful Math Education Blog Carnival. Kelly Darke of Math Book Magic will host the 148th Carnival. Probably neither of us should host the 147th Carnival, but YOU most certainly can! By YOU I mean anyone who has ever blogged even just a little bit about math. For example, if you normally blog about art, you could create a carnival that mostly focuses on mathematical art. The same could be said for photography, games, puzzles, storybooks, and so forth.

If now isn’t a good time for you to take on an extra project, remember there are plenty of other months open for you to volunteer!

But how do you host the Playful Math Education Blog Carnival, you ask? First of all, let Denise Gaskins know you would like to host the carnival.

You can also contact her through Twitter:

After you get assigned a month and a carnival number, you should pick a day in the last full week of the month as your goal to publish your carnival.

You might be interested in knowing how I approach creating a carnival:

Number Facts or a Puzzle:

Traditionally you start with some facts or a puzzle about the current carnival number. You can find several facts about your number at Pat Ballew’s Math Day of the Year Facts, Wikipedia, or OEIS.org. Also, check The Carnival of Mathematics which is about 48 numbers ahead of the Playful Math Education Carnival. What interesting facts were written about your number around four years ago in that carnival? You don’t have to be fancy; you can simply state a fact or two about your carnival number.

I, on the other hand, am obsessive. If I were hosting the 147th Carnival I would find as many facts about the number 147 as I could. I would think about all those facts and try to come up with a way to marry my number with something about a carnival, a fair, or even a circus. After a couple of weeks of imagining, I would finally be able to tell you about the great contortionist, Hexahex. Perhaps you’ve heard of his mother, Polly Hex. Hexahex can contort himself into 82 different “free” positions. He wants to stretch himself a little bit and add 65 more “one-sided” positions for a total of 147 “one-sided” positions in his repertoire. He is allowed to count positions that are reflections of the first 82 positions, but only if they aren’t exactly the same or merely a rotation of any of those first 82. Below is a graphic showing those first 82 positions as well as their reflections. Put an X above the 17 positions in the bottom three rows that don’t qualify as different, then count up the rest. You will then see that Hexahex can indeed contort himself 147 ways!

See, I told you I am obsessive! If you host the 147th carnival, you can use my graphic and story about Hexahex if you like. If you don’t want to use it, that’s okay, too!

As Denise Gaskins advised,

You decide how much effort you want to put in. Writing the carnival can take a couple of hours for a simple post, or you can spend several days searching out and polishing playful math gems to share.

I try to start writing a draft of my blog carnival post long before my deadline. I collect pictures (good advice on finding pictures here) and quotations whenever I find something I like, and enter them into my post ahead of time. If I have the framework in place, then all I have to add at the last minute are the blog post links, and the job doesn’t seem overwhelming.

Make sure you have the right to use any image you post. Either create a graphic yourself or find something marked “Creative Commons” — and then follow the CC rules and give credit to the artist/photographer.

I typically use graphics I’ve made or embed tweets from Twitter that just seem to have the perfect picture or quote.

Finding Blog Posts for Your Carnival Through Your Blog’s Reader:

Second, you look for blog posts. I found some blog posts because I subscribe to them, but you can also find blog posts by searching your reader. You may think blogging is dead, but it most certainly isn’t. I blog on WordPress, and its reader is easy to search. The search terms I used included math art, math poetry, math games, math puzzles, math geometry, and math algebra. Here are blog posts I found recently, most of which were written after the last carnival was published. Others were written before my carnival, but somehow I missed finding them before. If you hosted the 147th carnival, you will want to organize the posts into different categories or age groups and write a brief introduction to each post, but you could include as many or as few of these posts as your heart desires as well as other posts that you find. Here’s a bonus: if you also blog on WordPress, as soon as you hit the submit button, then WordPress will let the authors know that their post was included in your carnival! I have not organized these blog posts, but click on any of them that look interesting to you and consider including them in your carnival. If they don’t look interesting, a good introduction written by you might make them appeal to more people.

Finding Blog Posts on Twitter:

Twitter has SO many wonderful, playful ideas about mathematics. Most of them do not come from blog posts, but some of them do. Often when I see a tweet that refers to a math blog post or something else I like, I hit the like button. You can check my Twitter profile to see what appeals to me. Twitter also has a search feature. I’ve searched for individuals that I know who blog. I’ve also used words like “math blog” in my Twitter search to find blog posts I haven’t seen before. Be aware that you may find posts that are old or have no date on them, but plenty of recent blog posts are just waiting for you to find! Also, Denise Gaskins will retweet some blog posts that she’s found. Here are the blog posts I found on Twitter AFTER my carnival was published. Again, if you were hosting the carnival, the posts to include in your carnival would be up to you. You would organize them into different categories or age groups and write a brief introduction for each post.


That blog post doesn’t appear to be recent, but it did lead me to this one published in May 2021: Accelerate Vs. Remediate.

Embedding tweets on your blog can make the post seem VERY long. I would select a few of my favorite tweets with pictures to embed in the carnival and just use the blog links to take my readers directly to most of the posts.

You can also post a link to your carnival on Twitter with a thank you acknowledging the Twitter handles of people whose blog posts you used. That isn’t a required step, but it will help to get the word out to more people to visit your carnival.

Some Final Steps:

After you’ve organized all the blog posts into different categories or age groups and written briefly about them, stop looking for more blog posts, because there will always be more, and if you don’t stop looking, you will never be finished! It is a good idea to make sure the links you’ve included really do take your readers where you think you are sending them. I admit that I’ve messed up on that detail before.

To finish up, you will want to include a link to the previous playful math carnival and a link to the website of the next carnival, if known. You can find that information here. You will want to include an invitation for others to host future carnivals. It is also courteous to direct your readers to the current edition of the Carnival of Mathematics. Lastly, proofread and publish! Good luck and have fun!


1638 Factors and Multiples

Mathematical Musings:

I like the way this tweet shows familiar relationships of several unfamiliar math terms.

Recalling that MANY people confuse factors with multiples, I was inspired to make something similar that will hopefully help people to know which is which:

Factors of 1638:

1638 is even, so it is divisible by 2.
+ 8 = 9 and 6 + 3 = 9, so 1638 is divisible by 9.

  • 1638 is a composite number.
  • Prime factorization: 1638 = 2 × 3 × 3 × 7 × 13, which can be written 1638 = 2 × 3² × 7 × 13.
  • 1638 has at least one exponent greater than 1 in its prime factorization so √1638 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1638 = (√9)(√182) = 3√182.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1638 has exactly 24 factors.
  • The factors of 1638 are outlined with their factor pair partners in the graphic below.

More about the Number 1638 and Today’s Puzzle:

1638₁₀ = 666₁₆ because 6(16² + 16¹ + 16º) = 6(256 + 16 + 1) = 6(273) = 1638.

1638 is the hypotenuse of a Pythagorean triple:

It may be a little confusing because of the words greatest and least, but remember factor ≤ multiple so
Greatest Common Factor < Least Common Multiple.
Or simply, GCF < LCM.

Since these numbers have several digits, skip counting to find common multiples isn’t practical. An easy way to solve the puzzle is to pay attention to the exponents in the prime factorizations:
630 = 2 × 3² × 5 × 7,
1512 = 2³ × 3³ × 7, and
1638 = 2 × 3² × 7 × 13.

Rewrite the prime factorizations to contain all the bases used in any of the prime factorizations with the appropriate exponents:
630 = 2¹ × 3² × 5¹ × 7¹ × 13º,
1512 = 2³ × 3³ × 5º × 7¹ × 13º, and
1638 = 2¹ × 3² × 5º × 7¹ × 13¹.

Write the bases using the SMALLEST exponents for the Greatest Common Factor:
GCF = 2¹ × 3² × 5º × 7¹ × 13º = 2 × 3² × 7 = 126.

Write the bases using the LARGEST exponents for Least Common Multiple:
LCM = 2³ × 3³ × 5¹ × 7¹ × 13¹ = 98280.
(Aren’t you glad we didn’t skip count to find it!)

126 < 98280. Most certainly!

As you might expect, 630-1512-1638 is 126 times (5-12-13).


The 146th Playful Math Education Blog Carnival

Welcome to the146th Playful Math Education Blog Carnival!

What kind of math does the number 146 make?

146 is the 6th octahedral number because 6(2·6² + 1)/2 = 146.
That means that 1² + 2² + 3² + 4² + 5² + 6² + 5² + 4² + 3² + 2² + 1² = 146.

Base ten number 146 looks interesting when it is written in some other bases:
146₁₀ = 123₁₁ because 1(11²) + 2(11¹) + 3(11º) = 146, and
146₁₀ = 222₈ because 2(8² + 8¹ + 8º) = 146.

The factors of 146 are 1, 2, 73, and 146. Coincidentally, 1 + 2 + 73 + 146 = 222.

You can read other ways 146 and the numbers from 121 to 150 make math at Pat Ballew’s Math Day of the Year Facts.

Here are the attractions at this month’s carnival. Click on one to be transported right there!

Notice Patterns, Wonder, Create Math!

Graphics that let us notice patterns and wonder about them are fun, but students don’t have to wait for some teacher somewhere to make them. Denise Gaskins, the original playful math carnival creator, reminds us that students can be Math-Makers, and she invites them to have their creations published! Check out some student creations that have already been published.

Carrot Ranch noticed that Maths Is Everywhere: Clocks, Numbers and place value, patterns and algebra, measurement and geometry, probability and statistics, and much MORE.

Anna noticed something cool about the multiplication table. Can you notice it, too?

Mathematical Art

Nisha-designs decorated mugs with some lovely Abstract Geometric Circle Triangle Art.

K’s Dreamscape has a tutorial for you to make Simple Geometric Art! using cardboard, paint, paintbrushes, and painter’s tape.

Kreativekavya of Fremont forum uses circles, lines, and rectangles in Geometric Art!

Dianna Kolawole shares bright geometry art by Maranda Russel in Wordless Wednesday Geometric Art.

RobertLovesPi made a beautiful Pentadecagon and Its Diagonals.

FracTad’s Ractopia describes how to Create a Geometric Eye Using Desmos.

Sarah Carter of MathEqualsLove has created a gorgeous 3-colored origami Harlequin Cube and shows pictures of the steps taken in her post.

Karmen of Gallery K has made math digitally in some stunning Geometric Art.

Tessellation Art

Tessellation Art is the subject of Bumbastories’ What Four?

RobertLovesPi regularly publishes tessellations like Two Versions of a Tessellation Featuring Regular Hexagons, Regular Pentagons, and Tetraconcave, Equilateral Octagons.

I especially liked how his A Tessellation of Regular Hexagons, Golden Triangles, and Rhombi turned out. It seems to change depending on where you focus your eyes.

Mathematical Photography

You can make math using a camera! Marlene Frankel, A Photo’s Worth searched for and captured lots of geometry in Lens-Artists Photo Challenge #141 Geometry.

Tina Schell of Travels and Trifles photographed some geometric examples of the Fibonacci sequence.

Oh, the Places We See found geometry everywhere but carefully selected some geometric photographs from around the world.

Jazzersten photographed More Greek Geometric Art at the Museum of Fine Arts.

Estimation Booth

Steve Wyborney has engaging Esti-Mystery puzzles ready for every day for the rest of the school year!

Chasing Unicorns humorously blogs about Organizing Jelly Beans. How many jelly beans can you eat each day to keep yourself below the estimate of refined sugar consumed per American per day?

Within 1%, how long is the hypotenuse of this right triangle? If certain criteria are met, John D. Cook’s blog post, Hypotenuse Approximation, can help you be the first to find the correct answer and win the prize.

Fractions, Ratios, and Decimals

Henri Picciotto of Henri’s Math Education Blog updates us on how to use fraction rectangles to help students make sense of adding, subtracting, or comparing fractions with different denominators.

1001 Math Problems shares an engaging and delicious Chocolate Problem involving fractions.

Third-grade teacher, Ms Victor, couldn’t help but see fractions while eating lunch in When Your Teacher Brain Is on Overdrive.

Jillian Starr shares how to transition from unit fractions to more complex fractions in Teaching Fractions Through One Whole.

Do you bake using ratios instead of measuring cups? Kat from the Lily Cafe does and will show you how to use ratios and a scale to Make Flatbread. What a tasty way to make math!

Duane Habecker of The Other Math, More Than What’s in the Textbook invites you to solve ratio problems using Tape Diagrams.

Read how much laughter can be had learning long division involving decimals in FiveHundredaDay’s post It’s not them, it’s me.

Carnival Games

Ajitadeshmukh shares the game, The Number Detective [Spying the number]. This is a game that uses an ordinary deck of playing cards and reinforces the concepts of adding, subtracting, multiplying, and/or dividing. It can be played by children in early elementary grades and up.

Primary Ideas shared how well a game of Noggle (Number Boggle) went when it was played in a Google session remotely.

Anna of one+epsilon designed a logic game called Dot, Dot Poof! Here’s a bonus: Kids 6 and up might inadvertently learn a little linear algebra playing it, too!

A Game of Linear Equations by Bethany of MathGeekMama will help students find solutions to their problems!

MakeMathNotSuck blogged about Theresa Wills’s Playing Cards.io Interactive Math Games for middle grades. It is really exciting that these games can be played in real-time with a partner.

Wendi Bernau made an Easter Egg Hunt Escape Room for her 15- and 17-year-old kids. The escape room included puzzles based on their current schoolwork. The 17-year-old had to solve a puzzle that required calculations, graphing, and trigonometry. The kids liked the escape room so much that they are already talking about doing it again next year.

Harsh Sharma writes about How Math Games and Puzzles Improves Brain Activity. It turns out that Failing/Losing is as important in brain development as Succeeding/Winning is!

Hands-on Math

The Pi Project lets you listen in on the delightful conversation about knitting and fairies and the place value police in The Beauty of Base Ten Blocks.

Melissa Packwood of The Florida Reading Coach blogged about some Affordable Math Manipulatives that can assist students in learning mathematics.

Inclusiveteach.com shares some ideas to Make Your Own Maths Manipulatives.

House of Mirrors

Reflections are important topics in geometry and coordinate math. Our House of Mirrors is full of fascinating reflections.

Ted Jennings, shared a beautiful picture of an alligator and a turtle in Reflections.

Hannah Michaela of CoC-GetFit gives a geometric definition of a mirror image, shares a few examples in pictures and a thoughtful poem about mirrors and reflections in Mirror Image.

Beth of I didn’t have my glasses on made math by photographing a reflection that is happening at the front and the back of a pond in Argo Park.

Ritva Sillanmäki wrote a poem and made math by photographing a reflection that happens on the left and the right side of a river.

Bushboys World has several amazing pictures of birds in See My Reflection.

I shared a couple of puzzles where the squares of two numbers look like they are looking in a mirror.

Museum of Mathematics

All over the world math is being made on this day, April 28. Pat’s Blog shares some famous ways math has been made in the past On This Day in Math, April 28th.

David Campbell of Culturico writes about the beloved Louis Carroll in Portrait of a mathematician in love with the art of writing.

Indrajit RoyChoudhury tells us about Bhaskaracharya, a 12th century Indian mathematician and astronomer in Arjuna’s Arrows and Algebra. Bhaskaracharya discovered differential calculus 500 years before the births of either Newton or Leibniz.

Papannasons  has written an essential biography of 20th Century Indian mathematician Srinivasa Ramanujan. Knowcusp reviews the movie about Ramanujan in The Man Who Knew Infinity: A tale of one of the Greatest Mathematicians of all times. While (Roughly) Daily mentions him and several other great mathematics in “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

LA of Waking up on the Wrong Side of 50 is featured in the controversial current events area of the Math Museum in Anything Can Happen Friday: Math. LA includes the actual newsletter in which Oregon instructs its math teachers to allow for more than one correct answer. LA is upset thinking that now Oregon math teachers must accept incorrect math like 2 + 2 = 79. Perhaps Oregon is just welcoming some of Denise Gaskins’ math rebels who might say that 2 + 2 = 79 – 75, or some other of the infinite number of possible non-simplified yet still very much correct answers.

Likewise, the College Fix reported that Oregon math teachers have been instructed to let their students show their work by making TikToks, silent videos, or cartoons about the math they are learning, in other words, let students make their own math. I think about Ramanujan who taught himself math from an old textbook and then created his own mathematical symbols and terminology when he dreamed up more advanced mathematics. Later when he was told he needed to prove his ingenious mathematical formulas with rigorous proofs, did it help him or restrict him?

Esther Brunat has “curated a collection of Math TikToks” that now belong in a modern Museum of Math.

Adding, Subtracting, Multiplying, Dividing, Etc.

Have you ever experience joy when skiing? Bill McCallum of Illustrative Math compares that feeling to being fluent adding and subtracting numbers.

Laura of Riddle From the Middle describes why third-grade students often struggle with determining which operation to use in SoCs – the right teacher makes a world of difference.

Tess M Perko of River to Humility has written a sweet short story: The Imagination Grandpa Story 3: The Multiplication Staircase.

With doses of frustration and humor, Joseph Nebus of NebusResearch explains why No, You Can’t Say What 6/2(1+2) Equals.

Bethany of MathGeekMama shares her game that makes learning order of operations fun and not impossible!

Math Story Time and Other Books

1 + 1 + 1 = 3. Any number greater than one can be partitioned in a similar fashion. Patricia Nozell reviews a perfect picture book, I Am One: A Book of Action by Susan Verde. A little math can be learned while one person works with another and another to make the world a better place.

Writing this post has introduced me to Perfect Picture Book Fridays. Susanna Leonard Hill reviewed Little Ewe: The Story of One Lost Sheep, by Laura Sassi. Your 3- to 5-year-old will love counting logs, frogs, and other rhyming nouns as you read this book together.

Sue Heavenrich of Sally’s Bookshelf blogged about Bracelets for Bina’s Brothers, a picture book about estimation for 3-6-year-olds, and concluded that Math + Art > Numbers. Activities to make the math in the book more meaningful are also included in the blog post.

Wrenbeth22 of Miss Beth has a Book reviewed The Boy Who Loved Math by Deborah Heligman and LeUyen Pham. This is the story of Paul Erdös, a famous twentieth-century mathematician who made friends all over the world by sharing the math he loved.

Darlene Beck-Jacobson reviewed three biographical storybooks: Queen of Physics by Teresa Robeson, Code Breaker, Spy Hunter by Laurie Wallmark, and Counting on Katherine by Helaine Becker in Celebrate Girls and Women in STEM Day with Some Great Books.

Patricia Tilton of Children’s Books Heal reviewed Wonder Women of Science by Tiera Fletcher and Ginger Rue as part of Women’s History Month. The book is perfect for 9 to 12-year-olds. She also made me aware that Nerdy Book Club reviewed the same book. From that review, I learned the delightful true story of a human calculator named Tiera Fletcher that I am anxious for you to read as well!

MikesMathPage tells us that James Tanton’s Solve This book is full of incredible math projects to do with kids. In this post, he and his son explore a little topology in Going back to James Tantons’ amazing Möbius Strip cutting project.

In Monday’s Math Madness, Willow Croft thoroughly enjoys a 15th-century maritime manuscript called The Book of Michael of Rhodes. There is a lot of math in the book, but even if the reader doesn’t like math much, it won’t take away from the thrilling adventure. It is suitable for high school students and older.

Kelly Darke of MathBookMagic and FairyMathMother would like you to know about Math Book Wisdom: An Early Math Resource Book. It isn’t a book to read to kids, but it is filled with math wisdom for the parents and teachers who teach children.

Crow Intelligence reviewed a book that interests me a lot:  Playing with Infinity – Mathematical Explorations and Excursions by Rózsa Péter.  I only need to decide if I will read it in English or try to get through it with the little bit of  Hungarian I know!

The Enchanted Tweeting Room

Jo Morgan blogs about some wonderful ideas for teaching Place Value Tool, Powers, Simple Linear Graphs and more that she’s found on Twitter and elsewhere in 5 Maths Gems #143.

On Mondays, MathEqualsLove blogs about many must-read tweets she finds on Twitter. You will want to check out Volume 80, and Volume 81.

The Whispering Spot

Imagine someone whispering at a spot inside a building and someone else clear across the room being able to listen to them clearly! Such a whispering spot exists at this carnival! See what happens when three math teachers teach by listening to their students:

When a student didn’t understand a mathematical concept, he broke a rule by leaving the classroom. Kaneka Turner of BlackWomenRockMath details how she listened to the student with her ears, her eyes, and her heart in The Art of Listening. By so doing, she successfully helped him make the connections needed to understand the lesson while simultaneously letting him know he was truly understood. What trajectory would his life be on now, if she had not listened as she did?

The Heinemann Blog features an interview between Marilyn Burns and Lucy Calkins on Listening to Learn. By listening to the interview or reading its transcript, you can learn how Marilyn Burns interviews individual students and listens to them to advance their understanding of mathematics.

In the second half of Bill Davidson’s podcast interview with Robin Ramos, she describes how she scripts questions and listens to not just individual students but to a classroom of students at the same time!

Listening is key anytime we talk with a math maker. You can read Life Through a Mathematician’s Eyes’ interview of an up-and-coming mathematician: Akshay Thakur for the Inspirational Corner.

Of course, teachers need to be listened to as well. See Research Minutes’ Teacher Stress and Burnout in the Wake of Covid 19.

Poetry Corner and Some Trigonometry

In Math Makers Write a Poem, Denise Gaskins gives us some ideas and examples of student-written mathematical poetry.

I also have found some examples of people making math by writing poetry. Even if a poem speaks negatively about math, it gives us all an opportunity to LISTEN to students and meet them where they are.

Trigonometry for Dogs is a short, sweet poem by Lyna Galliara.

My heart broke when I read Looking at Love Lost, by murisopsis of A Different Perspective. It is a poem about falling out of love with mathematics in high school beginning with trigonometry. Simply saying Trig is Easy doesn’t help and only makes a person not feel heard. Perhaps Wyrd Smythe’s Explanation of Trig Basics might have been helpful?


Craftgossip.com shares an easy Easter Egg Sudoku Puzzle that even preschoolers can do.

Puzzle a Day challenges us to solve A Mathematical Multiplication Puzzle with a six-digit product without using a calculator. I can attest that it can be done!

De Graw Publishing’s blog gives us Number Problems and Easy Sudoku Puzzles for Kids: Math and Logic Games Problems for Children.

Sarah Carter of MathEqualsLove shares a new puzzle in Number Ball Puzzles by Naoki Inaba. She translates the rules from Japanese to English so that you can have some idea where to put the missing numbers in the puzzle. Be warned, for the bigger puzzles, you might need to use your eraser a lot.

Sara also shared a sequence puzzle. Her students have enjoyed predicting the next letter in the sequence.

Maggie Heffernam suggested to Brian Marks of Yummy Math that he write a math activity when a real-life man was paid in greasy pennies.

Bedtime Math has a musical mathematical puzzle for you in Mile-Long Xylophone.

Math Teaching Strategies

Some teachers have half of their students in class and half remote over zoom. Keeping the at-home kids engaged can be difficult. Libo Valencia of Fresh Ideas for Teaching has six proven strategies to engage students in these hybrid classes.

You or your students can easily make Original Which One Doesn’t Belong puzzles!

Dan Draper of Opinions Nobody Asked For explores Area Models and Grid Method.

Probability and Statistics

Joseph Nebus of Another Blog, Meanwhile posts humorous statistics every Saturday like this cumulative bar graph showing Star Wars Movies versus Star Trek Movies. His vertical axis is a hoot.

Mr. Rowlandson of Pondering Planning in Mathematics has been Thinking About Probability Trees. Do you add or multiply the fractional probabilities? His blog post spells out what to do.

Athletes are constantly making math. Greg Pattridge of Always in the Middle writes about the statistics produced with every play in It’s a Numbers Game! Baseball.

Lunatic Laboratories uses alliteration to tell a tale of tails in One-tailed vs. two-tailed tests in statistics.

Did you know that if you get 11,000 steps a day, you will walk a million steps every quarter and just over 4 million steps a year? LisaFeatherstone had a daily goal of walking 10,000 steps and still made the 4-million steps goal. She used a spreadsheet to track the data her fitbit gave her and wrote a formula to predict when she would meet her goal.

Lvonlanken of The Shy Genealogist analyses the data she’s collected to determine which John Smith is her ancestor in Sorting the Land Records. Some genealogical programs will provide you with all kinds of statistics from your family tree. See the stats the Chiddicks Family found in My Family Tree in Numbers. I was pleased that they didn’t simply accept every statistic. They made predictions of the results and compared their predictions with the statistics the program produced.

MSCNM uses probability and statistics to answer the question Should You Buy a Lottery Ticket?

Blue Ribbons

Jo Morgan of Resourceaholic recently celebrated seven years of blogging by reviewing the very best teaching ideas and resources from the previous year and naming the winners of her (Maths) Gem Awards. Check it out!

The pandemic hasn’t stopped some people from doing good. Leila Zerai writes for LondonNewsOnline about a Student Winning the Prestigious Lewisham Mayor’s Award for Offering Free Online Maths Tuition.

A short story, Advanced Word Problems in Portal Math, is a finalist in the Nebula Best Short Story Contest. The reviewer didn’t care for the story because the math references were hard to understand. Let me tell you a little secret: I think that’s the way it was meant to be because I didn’t get the math references either! The story was just a fun way to make math. Another example of purposeful over-our-heads math was in a Barnaby comic. I know how to find the determinant of a two by two matrix and how to multiply binomials, but I look forward to Joseph Nebus explaining that comic sometime soon. It is still a funny comic even if I don’t fully understand it yet.

Math Memes and Comics

Joseph Nebus of Nebusresearch explains the mathematics of a comic in Where Else Is a Tetrahedron’s Centroid.

Design a Carnival

I hope you had a wonderful time at this month’s carnival! This month the Carnival of Mathematics #192 was hosted at Eddie’s Math & Calculator Blog. Perhaps you would like to design your own carnival.

Simran M Karkera of MSCNM tells the story of a girl who loved math that used trigonometry and calculus to design a roller coaster that thrilled her previously-mocking friends in A Mathematical Ride!

Last month the 145th Playful Math Carnival was hosted by Mathhombre. Perhaps you would like to host the next carnival or one later in the year. You don’t have to go overboard like I probably did. I was having so much fun, I couldn’t stop myself! To volunteer to host a carnival go to Denise Gaskins’ Carnival Volunteer Page.

Math Happens When Two of 1632’s Factors Look in a Mirror!

Today’s Puzzle:

Both 12 and 102 are factors of 1632. Something special happens when either one squares itself and looks in a mirror. Solving this puzzle from Math Happens will show you what happens to 12 and 12².

You can see that puzzle on page 33 of this e-edition or this pdf of the Austin Chronicle. You can find other Math Happens Puzzles here.

This next puzzle will help you discover what happens when 102 and 102² look in a mirror!

Why do you suppose the squares of (12, 21) and (102, 201) have that mirror-like property?

Factor Trees for 1632:

There are many possible factor trees for 1632, but today I will focus on two trees that use factor pairs containing either 12 or 102:

Factors of 1632:

  • 1632 is a composite number.
  • Prime factorization: 1632 = 2 × 2 × 2 × 2 × 2 × 3 × 17, which can be written 1632 = 2⁵ × 3 × 17.
  • 1632 has at least one exponent greater than 1 in its prime factorization so √1632 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1632 = (√16)(√102) = 4√102.
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1632 has exactly 24 factors.
  • The factors of 1632 are outlined with their factor pair partners in the graphic below.

More about the Number 1632:

1632 is the hypotenuse of a Pythagorean triple:
768-1440-1632, which is (8-15-17) times 96.

1632 is the difference of two squares in EIGHT different ways:
409² – 407² = 1632,
206² – 202² = 1632,
139² – 133² = 1632,
106² – 98² = 1632,
74² – 62² = 1632,
59² – 43² = 1632,
46² – 22² = 1632, and
41² – 7² = 1632.

That last difference of two squares means 1632 is only 49 numbers away from the next perfect square, 1681.