$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.

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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.
1
+ 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:
630-1512-1638.

Solution:
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!)

Is GCF < LCM?
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?

Puzzles

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.

 

1627 Color-Coded Prime Numbers

Today’s Puzzle:

Study this color-coded chart of prime numbers. 1627 is the smallest prime number that begins something special. Can you figure out what that is?

Also, why do you think I’ve underlined some of the other prime numbers on the list?

Memorizing which numbers are prime can be a big time-saver in mathematics. How many in a row can you recite without looking?

Factors of 1627:

  • 1627 is a prime number.
  • Prime factorization: 1627 is prime.
  • 1627 has no exponents greater than 1 in its prime factorization, so √1627 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1627 has exactly 2 factors.
  • The factors of 1627 are outlined with their factor pair partners in the graphic below.

How do we know that 1627 is a prime number? If 1627 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1627. Since 1627 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1627 is a prime number.

More about the Number 1627:

1627 is the sum of two consecutive numbers:
813 + 814 = 1627.

1627 is also the difference of two consecutive squares:
814² – 813² = 1627.

What do you think of that?

1626 is a Centered Pentagonal Number

Today’s Puzzle:

1626 is the 26th centered pentagonal number and it is also one more than 5 times the 25th triangular number.

Today’s puzzle is for you to figure out why the following two expressions are equivalent. The first expression is the formula for the nth centered pentagonal number, and the second expression is one more than 5 times the formula for the nth triangular number.
(5n²-5n+2)/2, when n = 26, and
1 + 5n(n+1)/2, when n = 25.

Factors of 1626:

  • 1626 is a composite number.
  • Prime factorization: 1626 = 2 × 3 × 271.
  • 1626 has no exponents greater than 1 in its prime factorization, so √1626 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 1626 has exactly 8 factors.
  • The factors of 1626 are outlined with their factor pair partners in the graphic below.

More about the Number 1626:

2(813)(1) = 1626 means that 2(813)(1), 813² – 1², 813² + 1² is a Pythagorean triple, and
2(271)(3) = 1626 means that 2(271)(3), 271² – 3², 271² + 3² is as well.

Calculate those expressions and you will have found the only two Pythagorean triples containing the number 1626.

1625 is a Centered Square Number

Today’s Puzzle:

Because 1625 is the 29th centered square number, it is one more than four times the 28th triangular number. Can you draw lines on the graphic below separating out one tiny square and dividing the rest of the graphic into four equal triangles each with a base of 28 tiny squares?

Factors of 1625:

  • 1625 is a composite number.
  • Prime factorization: 1625 = 5 × 5 × 5 × 13, which can be written 1625 = 5³ × 13
  • 1625 has at least one exponent greater than 1 in its prime factorization so √1625 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1625 = (√25)(√65) = 5√
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1625 has exactly 8 factors.
  • The factors of 1625 are outlined with their factor pairs in the graphic below.

More about the Number 1625:

1625 is the sum of two squares FOUR different ways:
40² + 5² = 1625,
37² + 16² = 1625,
35² + 20² = 1625, and
29² + 28² = 1625.

1625 is the hypotenuse of TEN Pythagorean triples:
57-1624-1625, calculated from 29² – 28², 2(29)(28), 29² + 28²,
180-1615-1625, which is 5 times (36-323-325),
400-1575-1625, calculated from 2(40)(5), 40² – 5², 40² + 5²,
455-1560-1625, which is (7-24-25) times 65,
572-1521-1625, which is 13 times (44-117-125),
625-1500-1625, which is (5-12-13) times 125,
825-1400-1625, calculated from 35² – 20², 2(35)(20), 35² + 20²,
975-1300-1625, which is (3-4-5) times 325,
1020-1265-1625, which is 5 times (204-253-325), and
1113-1184-1625, calculated from 37² – 16², 2(37)(16), 37² + 16².

 

1624 Applying Twelve Divisibility Rules to Permutations of 1234567890

Today’s Puzzle:

Can you use divisibility rules to find a number that uses all ten digits exactly once and is divisible by all the numbers from 1 to 10?”

EVERY such number will be divisible by 1, of course. That’s the divisibility rule for 1. That one was super easy.

But guess what! No matter how you arrange those ten digits it will be divisible by 3 and by 9. Why? Because the sum of the digits of 1234567890 is 45, a number divisible by both 3 and 9. That’s the divisibility rule for 3 and for 9.

If the number you create ends with a zero, it will also be divisible by 5 and 10. That’s the divisibility rule for 5 and the divisibility rule for 10. AND it will also be divisible by 2 and by 6 because those are the divisibility rules for even numbers and for even numbers divisible by 3.

That leaves only three divisors to worry about: 4, 7, and 8. This post will talk about an easy and fun way to deal with those divisibility rules!

Did you know that no matter how many digits a number has, if the last digit is 2 or 6 and is preceded by an odd number, then that number will be divisible by 4? Also if the last digit of that number is 0, 4, or 8 and is preceded by an even number, then that number will be divisible by 4 as well? That’s a divisibility rule for 4.

If that number happens to have nine digits and we multiply it by 10, then the new product will also be divisible by 8. That’s because of the divisibility rule for 4 and the number getting multiplied by 2 × 5.

So 7 is the only divisor that is making any real trouble for us! Here’s how we will deal with division by 7: We will separate our number into smaller parts, all of which are divisible by 7. It would be easiest to separate it into three 3-digit numbers and then have those 9-digits be followed by a zero.

3-Digit Multiples of 7:

Now in the chart below we have all of the 3-digit multiples of 7. I have colored in all the numbers with a zero, and all numbers that repeat any of their own digits. We won’t be using any of the numbers that have been shaded in. However, we want to use at least one of the numbers printed in green. In fact, a green number should be your first choice because it is divisible by 4, while your second and third choices will be numbers that have not been shaded in.

And just like that, you can take ANY two numbers from the chart followed by a green number and zero, and you will have a number that is divisible by all the numbers from 1 to 10.

If you take care that none of the digits repeat, which is very likely with so many choices, it will also be a number that has each of the ten digits exactly one time!

For example, if I choose 812 as my green number, and 357 as my second number, my leftover digits are 9-4-6. I can check all permutations of 9-4-6 to see if any of them are on the chart, and I notice that 469 is there so I form the number 3574698120. Here’s proof that it is divisible by every number from 1 to 10:

Now you pick three of your own 3-digit numbers, attach a zero, and see how it does!

Let’s Explore the 11-and 12-Divisibility Rules:

My chosen number is also divisible by 12 because it satisfies the 3- and the 4- divisibility rules. Is it divisible by 11?

3574698120 is NOT divisible by 11. I know that because the sum of the red numbers, 26, minus the sum of the blue numbers, 19, is 7 which is not a multiple of 11.

However, if I switch the 357 with the 469, my number still satisfies all the previous divisibility rules, but now 4693578120 has an additional factor! The sum of its red numbers, 28, minus the sum of its blue numbers, 17, is 11, clearly a multiple of 11.

The secret to having the bonus that your 10-digit number is divisible by 11, is having every other number add up to either 17 or 28. Those are the sums we should look for because 28 + 17 = 45, the total of all the digits, and 28 – 17 = 11, the difference satisfying the 11 divisibility rule.

Thus, 4693578120 is divisible by all the numbers from 1 to 12. Here’s a table showing all those divisions:

Now you give it a try! Can you also use the 3-digit multiples of 7 Chart to find your own 10-digit number that is divisible by all the numbers from 1 to 12? Try it! I think you will have fun solving this puzzle, even if you have to try several different sets of 3-digit numbers to find one that works.

Factors of 1624:

  • 1624 is a composite number.
  • Prime factorization: 1624 = 2 × 2 × 2 × 7 × 29, which can be written 1624 = 2³ × 7 × 29
  • 1624 has at least one exponent greater than 1 in its prime factorization so √1624 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1624 = (√4)(√406) = 2√406.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1624 has exactly 16 factors.
  • The factors of 1624 are outlined with their factor pair partners in the graphic below.

Factor Tree for 1624:

There are several possible factor trees for 1624. Here’s one of them.
That one used the factor pair from 28 × 58.

Wait a Minute!

1624 has all different digits and is divisible by 4 and by 7… Does that mean we can use it to form a number with 10 different digits that will be divisible by all the numbers from 1 to 10??!!!

Yes, we can. In fact, in this case, I was able to determine the needed multiples of 7 in my head because they were all 1- or 2-digit multiples!

Here’s what I did: 1624, is divisible by both 4 and 7, so 16240 is divisible by 4, 7, and 8. The missing digits are 3, 5, 7, 8, and 9. What multiples of 7 use just those digits? How about 7, 35, and 98?! Easy Peasy!

Those ten digits can be arranged in six different ways all of which are divisible by all the numbers from 1 to 10:
7359816240,
7983516240,
3579816240,
3598716240,
9873516240, and
9835716240.

Are any of those permutations of 1234567890 also divisible by 11? Yes!!! I used pencil and paper to make sure every other digit of my chosen10-digit number added up to 28 or 17. Here are all those divisions:

It is also divisible by 13, 14, 15, and 16, but I didn’t use a divisibility rule for 13, so I didn’t include any of those divisors in the chart.

4-Digit Multiples of 28:

Here is a chart showing all the 4-digit multiples of 28. Numbers containing zero or a repeated digit have been shaded. Use the chart to pick a 4-digit number that can be placed before the zero in your 10-digit number. Then find 1-, 2-, or 3-digit multiples of 7 to make your own 10-digit number that uses all the digits and is divisible by all the numbers from 1 to 10. You might even be able to satisfy the 11-divisibility rule so that your number is divisible by every number from 1 to 12. Make sure you are having fun working on this problem!

After solving the puzzle in several different ways, you might want to explore this puzzle that Andy Parkinson posted on Twitter:

Since 4967 is a relatively large prime number, 480 Factors likely can be beaten. Maybe YOU will be the one who finds a 10-different-10-digit number with more factors!

More about the Number 1624:

1624 is the hypotenuse of a Pythagorean triple:
1120-1176-1624, which is 56 times (20-21-29).

2(28)(29) = 1624. That makes 1624 four times the 28th triangular number.

1621 Give This Purple Egg a Crack!

Today’s Puzzle:

Here’s a level-5 purple Easter egg for you to try. All you need to do is write the numbers from 1 to 12 in both the first column and in the top row so that those numbers and the given clues function like a multiplication table. Go ahead. Give it a crack!

Factors of 1621:

  • 1621 is a prime number.
  • Prime factorization: 1621 is prime.
  • 1621 has no exponents greater than 1 in its prime factorization, so √1621 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1621 has exactly 2 factors.
  • The factors of 1621 are outlined with their factor pair partners in the graphic below.

How do we know that 1621 is a prime number? If 1621 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1621. Since 1621 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1621 is a prime number.

More about the Number 1621:

OEIS.org informs us that 1621 is in an interesting group of prime numbers.

I have verified it. They really are all prime!

1621 is the sum of two squares:
39² + 10² = 1621.

1621 is the hypotenuse of a Pythagorean triple:
780-1421-1621, calculated from 2(39)(10), 39² – 10², 39² + 10².

Here’s another way we know that 1621 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 10² = 1621 with 39 and 10 having no common prime factors, 1621 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1621. Since 1621 is not divisible by 5, 13, 17, 29, or 37, we know that 1621 is a prime number.