# 1641 and Level 3

### Today’s Puzzle:

This is a level 3 puzzle so the clues have been placed so that you can know what order you should use the clues. Place the factors of 90 and 30 in the appropriate cells, then work your way down the puzzle cell by cell filling in the factors of the clues as you go. ### Factors of 1641:

1 + 4 + 1 = 6, so 1641 is divisible by 3. (It isn’t necessary to include multiples of 3 in the sum to determine divisibility by 3.)

• 1641 is a composite number.
• Prime factorization: 1641 = 3 × 547.
• 1641 has no exponents greater than 1 in its prime factorization, so √1641 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 1641 has exactly 4 factors.
• The factors of 1641 are outlined with their factor pair partners in the graphic below. ### More About the Number 1641:

1641 is the difference of two squares in two different ways:
821² – 820² = 1641, and
275² – 272² = 1641.

From OEIS.org we learn that the number formed from 1²6²4²1² is a perfect square:
Recall that 1² = 1; 6² = 36; 4² = 16; and 1² = 1. Those squares form the number, 136161.
Sure enough, 136161 = 369².

# 1640 A Level 2 Flower

### Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table. ### Factors of 1640:

• 1640 is a composite number.
• Prime factorization: 1640 = 2 × 2 × 2 × 5 × 41, which can be written 1640 = 2³ × 5 × 41.
• 1640 has at least one exponent greater than 1 in its prime factorization so √1640 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1640 = (√4)(√410) = 2√410.
• 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 1640 has exactly 16 factors.
• The factors of 1640 are outlined with their factor pair partners in the graphic below. ### More about the Number 1640:

Since 1640 = 40 x 41, we can be sure that 1640 is the sum of the first 40 even numbers.

1640 is the sum of two squares in two different ways:
38² + 14² = 1640, and
34² + 22² = 1640.

1640 is the hypotenuse of a Pythagorean triple in FOUR different ways:
360-1600-1640, which is 40 times (9-40-41),
672-1496-1640, calculated from 34² – 22², 2(34)(22), 34² + 22²,
but it is also 8 times (84-187-205),
984-1312-1640, which is (3-4-5) times 328, and
1064-1248-1640, calculated from 2(38)(14), 38² – 14², 38² + 14²,
but it is also 8 times (133-156-205).

1640₁₀ = 2222₉ because 2(9³ + 9² + 9¹ + 9⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.
1640₁₀ = 2020202₃ because 2(3⁶ + 3⁴ + 3² + 3⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.

# 1639 and Level 1

### Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will make this puzzle function like a multiplication table. ### Factors of 1639:

1 – 6 + 3 – 9 = -11 so 1639 is divisible by 11.

• 1639 is a composite number.
• Prime factorization: 1639 = 11 × 149.
• 1639 has no exponents greater than 1 in its prime factorization, so √1639 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 1639 has exactly 4 factors.
• The factors of 1639 are outlined with their factor pair partners in the graphic below. ### More about the Number 1639:

1639 is the hypotenuse of a Pythagorean triple:
561 1540 1639, which is 11 times (51-140-149).

1639 is the 22nd nonagonal number because
22(7·22 – 5)/2 =
22(154 – 5)/2=
22(149)/2 =
11(149) = 1639.
Mathworld.Wolfram has illustrations of the first 5 nonagonal numbers.

# 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).

# 1637 Flower Challenge

### Today’s Puzzle:

Mother’s Day in the United States is this Sunday so I made this challenging flower puzzle for the occasion.

Use logic to write the numbers from 1 to 10 in each of the four boldly outlined areas so that those numbers and the given clues work together to make four multiplication tables. ### Factors of 1637:

• 1637 is a prime number.
• Prime factorization: 1637 is prime.
• 1637 has no exponents greater than 1 in its prime factorization, so √1637 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1637 has exactly 2 factors.
• The factors of 1637 are outlined with their factor pair partners in the graphic below.

How do we know that 1637 is a prime number? If 1637 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1637. Since 1637 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1637 is a prime number. ### More about the Number 1637:

1637 is the sum of two squares:
31² + 26² = 1637.

1637 is the hypotenuse of a Pythagorean triple:
285-1612-1637, calculated from 31² – 26², 2(31)(26), 31² + 26².

Here’s another way we know that 1637 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 26² = 1637 with 31 and 26 having no common prime factors, 1637 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1637. Since 1637 is not divisible by 5, 13, 17, 29, or 37, we know that 1637 is a prime number.

Do you notice anything else special about the number 1637 in this color-coded chart? # 1636 A Tricky Mystery Puzzle

### Today’s Puzzle:

It took me a while to figure out the logic of this puzzle so don’t get discouraged if you find it a bit tricky, too. If its logic eludes you, this short video I posted on Twitter should help.

### Factors of 1636:

• 1636 is a composite number.
• Prime factorization: 1636 = 2 × 2 × 409, which can be written 1636 = 2² × 409.
• 1636 has at least one exponent greater than 1 in its prime factorization so √1636 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1636 = (√4)(√409) = 2√409.
• The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1636 has exactly 6 factors.
• The factors of 1636 are outlined with their factor pair partners in the graphic below. ### More about the Number 1636:

1636 is the sum of two squares:
40² + 6² = 1636.

1636 is the hypotenuse of a Pythagorean triple:
480-1564-1636, calculated from 2(40)(6), 40² – 6², 40² + 6².
It is also 4 times (120-391-409).

1636⁴ = 7,163,630,838,016. Thank you OEIS.org for that fun fact!

# 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.

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.

# 1635 The Logic to This Puzzle Is a Real Mystery

### Today’s Puzzle:

You can easily find a way to start this puzzle, but just a few factors later, it’s a mystery what to do next. Give it a shot, and see what I mean. Writing all the factor pairs for the clues often is helpful, but not as helpful as we might hope this time. Here’s a video explaining what to do to find a few more factors of this puzzle using logic:

### Factors of 1635:

• 1635 is a composite number.
• Prime factorization: 1635 = 3 × 5 × 109.
• 1635 has no exponents greater than 1 in its prime factorization, so √1635 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 1635 has exactly 8 factors.
• The factors of 1635 are outlined with their factor pair partners in the graphic below. ### More about the Number 1635:

1635 is the hypotenuse of FOUR Pythagorean triples:
99-1632-1635, which is 3 times (33-544-545),
552-1539-1635, which is 3 times (184-513-545),
900-1365-1635, which is 15 times (60-91-109), and
981-1308-1635, which is (3-4-5) times 327.

OEIS.org informs us that there’s something special about the first nine decimals places of the fifth root of 1635.
Its fifth root is 4.392416875…
Can you figure out what is so special about that?

# 1634 Be Prepared for April Showers

### Today’s Puzzle:

If you learn the multiplication and division facts in a standard multiplication table, you will be prepared to solve this somewhat tricky April Shower puzzle. You will also be able to solve MANY other mathematical challenges. Use logic to solve it, not guess and check, and it will be much less challenging to find the missing factors. ### Factors of 1634:

• 1634 is a composite number.
• Prime factorization: 1634 = 2 × 19 × 43.
• 1634 has no exponents greater than 1 in its prime factorization, so √1634 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 1634 has exactly 8 factors.
• The factors of 1634 are outlined with their factor pair partners in the graphic below. ### More about the Number 1634:

1634 is part of exactly two Pythagorean triples. Here are the formulas you can use to calculate those two triples:
2(817)(1), 817² – 1², 817² + 1, and
2(43)(19), 43² – 19², 43² + 19².

Do you see the factors of 1634 prominently displayed in those formulas?

# 1633 and Level 5

### Today’s Puzzle:

It might be tricky in a few places, but use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues behave like a multiplication table. ### Factors of 1633:

• 1633 is a composite number.
• Prime factorization: 1633 = 23 × 71.
• 1633 has no exponents greater than 1 in its prime factorization, so √1633 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 1633 has exactly 4 factors.
• The factors of 1633 are outlined with their factor pair partners in the graphic below. ### More about the Number 1633:

1633 is the difference of two squares in two different ways:
817² – 816² = 1633, and
47² – 24² = 1633.