# Facts, Factors, and a Countdown to 2023

### Countdown to 2023:

The last 10 seconds of the year, we like to countdown from 10 to the new year. I like a mathematical way of counting down so I try to make an equation with the numbers from 10 to 1 that equals the coming year. This year I could have based my countdown  on last year’s countdown and said
(10-9+8×7×6)(5-4)(3)(2)+1 = 2023,
but this is a blog about factoring so I want a countdown that takes you to the prime factors of 2023 first. Here’s my countdown: (Note: Even though I used 1 as a factor twice in the countdown, I am very much aware that 1 is not a prime factor of any number.)

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Here’s the final shot from the Countdown:

### A Puzzle for 2023:

2023 has something in common with 2022. When either number and their reverses are squared, something interesting happens…it’s almost like looking in a mirror!

Only 50 numbers less than 10000 can make a similar claim to fame:

### Factors of 2023:

• 2023 is a composite number.
• Prime factorization: 2023 = 7 × 17 × 17, which can be written 2023 = 7 × 17².
• 2023 has at least one exponent greater than 1 in its prime factorization so √2023 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √2023 = (√289)(√7) = 17√7.
• The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 2023 has exactly 6 factors.
• The factors of 2023 are outlined with their factor pair partners in the graphic below.

### More About the Number 2023:

What do 2023 tiny squares look like?

2023 is the sum of consecutive numbers in five different ways:

And it is the sum of consecutive odd numbers in two ways:

2023 is a palindrome in base 16 because
7(16²) + 14(16) + 7(1) = 2023.

This tweet demonstrates that the prime factors of 2023 have a relationship with the digits of 2023.

I saw this tweet on Twitter and immediately recognized something else cool about 2023.

That might seem like a lot of mathematical mumble jumble, but with a little bit of explanation, it can be understood. And even though I made the problem look scarier because I substituted 2+0+2+3 for 7, some older elementary students who already understand powers, factorials, and/or remainders, will get it. I’m confident you can too.

You could also give the following list of facts to older elementary students and ask them to use it to find the remainder when they divide 823,543 by 5040.

1×5040 =    5040
2×5040 = 10080
3×5040 = 15120
4×5040 = 20160
5×5040 = 25200
6×5040 = 30240
7×5040 = 35280
8×5040 = 40320
9×5040 = 45360

Because its factors, 17 and 289, are hypotenuses of Pythagorean triples, 2023 is also the hypotenuse of some Pythagorean triples:
952-1785-2023 which is 119(8-15-17) , and
1127-1680-2023 which is 7(161-240-289).

Ureczky József also pointed out in the comments of this post, that 2023 is the short leg in SIX Pythagorean triples, and thus
2023² = 2046265² – 2046264²
2023²= 292327² – 292320²
2023² = 120377² – 120360²
2023² = 17255² – 17136²
2023² = 41785² – 41736²
2023² = 7225² – 6936²

One of those triples is a primitive triple. Can you determine which one?

Ureczky József shared one more amazing fact in the comments that I’m replicating here:

2023=9³+8³+7³+6³+5³+4³+3³+2³-1³

### More Mathematical Tweets About 2023:

These tweets are more or less in the order I saw them, not in order of mathematical difficulty. I will add more as I see them.

v

### Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution.

Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

### Factors of 1622:

• 1622 is a composite number.
• Prime factorization: 1622 = 2 × 811.
• 1622 has no exponents greater than 1 in its prime factorization, so √1622 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 1622 has exactly 4 factors.
• The factors of 1622 are outlined with their factor pair partners in the graphic below.

### More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

# 131 Playful Math Carnival

Welcome to the Playful Math Education Blog Carnival featuring the amazing prime number 131, whose digits can mutate into other prime numbers right before your eyes!

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Yessiree, 131 is prime, and so is 113 and 331. Do I need to mention that 3, 11, 13, and 31 are also prime numbers?
131’s next trick happens when you add up all the 2-digit PRIMES that begin with a 4:
41 + 43 + 47 = 131.
Because 131is a palindrome, it reads the same forwards and backward. Here’s another trick: 131 is 65 in BASE 21 and 56 in BASE 25.

We have many different attractions this month. You can go to any category quickly here:

### Carnival Attractions:

Arithmetic

You’ve heard of the three R’s, reading, and writing and ‘rithmetic, but what is arithmetic? Joseph Nebus shares a few comics about basic arithmetic and explains what they mean:

Arithmetic is also television’s Lisa Simpson’s favorite subject in school and she will miss it greatly as she recovers from the mumps. In this blog post, Safi explains Dr. Hibbert’s comforting words to her about polygons, hypotenuses, and Euclidean algorithms.

### Art

You can always count on Robert Loves Pi to produce a beautiful and complex geometric design. This one he calls Two Rhombic Polyhedra with Tessellated Faces. Here’s another one:

Paula Beardell Krieg helped students create big, beautiful geometric artwork and origami in Summer Projects with Teens.

Also, check out Paula’s Paper, Books, and Math Workshop for many more ways to learn math through art.

### Big Prize, Little Chance of Winning

Several years ago Mental Floss wrote about carnival games that offer big prizes but have very little chance of being won. This carnival has a couple of those as well. They are called unsolved math problems. Even if winning probably isn’t going to happen, that doesn’t mean the games and activities aren’t fun. Explaining Science updates us on a very famous unsolved problem, The Goldbach’s Conjecture. Supercomputers have worked on it, but we are no closer to a solution.

In A Neat Unsolved Problem in Number Theory That Kids Can Explore, Mike’s Math Page explores the new-to-me Collatz conjecture that for every positive n, the sum 3 + 8n will equal a perfect square plus an even number. It’s a simple enough conjecture for kids to understand and it is fascinating, yet mathematicians have not been able to prove or disprove it yet!

### Creative Writing

Subha laxmi Moharana (Angel Subu) writes creatively about some tough topics in high school mathematics in Math Poem. I think her words could be turned into a rap.

Poetrywithmathematics shares Doug Norton’s lovely mathematical poem Take a Chance on Me.

What if graphs were self-conscious about their looks? High School aged students can consider that thought as they read the imaginative blog post, To Infinity and Beyond.

### Displays

There’s a cozy classroom place that promotes mathematics in Our New Math Space. It was designed for older students by Continuous Everywhere But Differentiable Nowhere and includes many pictures.

Have you considered displaying a weekly math joke? MathEqualsLove shares a fun joke and a puzzle for kids to gather around and enjoy.

Super Safi uses another episode from the Simpsons to teach about the quadratic formula.

### Food for Thought

Anybody can cook or do math. Really? What does that even mean? Math4Love explains both in What We Mean When We Say, “Anyone Can Do Math.”
Math with Bad Drawings makes a similar point in The Adventures of Captain Math.

### Games

Joyful Parenting made a simple kindergarten-age counting game and called it Snack Math, but even older kids might enjoy figuring out exactly how many crackers are required to play the game.

How many are in the jar. What is a good estimate? Add Steve Wyborney’s clues one by one to get an even better estimate. He has 51 New Esti-Mysteries that also happen to teach several different math concepts.

For older students, Kent Haines a free game he calls Last Factor Loses. I played it a few times with a student. Making prime factorization a game really did make it more fun.

### Geometry

Bn11nb enjoys the geometry of architecture. The pictures in this post are worth a look and could be an inspiration to your students.

### House of Mirrors (Reflecting on Mathematics Teaching)

We often reflect on the effectiveness of our teaching methods. Sometimes we are advised to require students to use more strategies. We might ask them to notice or wonder about a concept. These two thoughtful posts will certainly give you cause for reflection:

“The More Strategies, the Better?

Noticing and Wondering: A powerful tool for assessment

Robert Kaplinsky shares ten things he’s embarrassed to tell you. Has he been reading your mind and mine?

### Money

What is your favorite part of a cupcake? What if you could buy just that part? What if you wanted to put a whole cupcake together? How much would that cost? Your child can learn about money and decimals exploring those answers with Mathgeekmama’s  Money Math Problems.

### Museum of Mathematics

Beads can be a fun manipulative when learning mathematics. Joseph Nebus has begun his 2019 Mathematics A-Z series by writing about the Japanese abacus. He compares it to a slide rule and the Chinese abacus. He also describes how to use it to add, subtract, and multiply numbers. Students could have some fun using it to understand place value, too.

Life Through a Mathematician’s Eyes is giving museum tours in A History of Mathematics-August. K-12 students could be fascinated by the mathematical relics from the Smithsonian founded in August 1846 as well as the Seven Bridges of Königsberg solved by Euler in August 1735.

### Pumpkin Patch

Erin of Sixth Bloom’s Pumpkin Math-Preschool Activity will engage your little ones as they learn to count and sort pumpkin-shaped macaroni or candies.

They will also love decomposing numbers using pumpkin seeds and  Mathgeekmama’s cute Pumpkin cards.

### Posters

Digital Educators Alliance offers free posters of admirable women in math and related fields:

While Sara Van Derwerf set of 112 New Math Fail Posters will delight students as they notice and wonder about and LEARN from grown-ups’ computing mistakes.

### Puzzles

7Puzzle gives some clues about a 3-digit number. Can you figure out what it is?

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Alan Parr writes about a newspaper puzzle called Evens Puzzles. He suggests that students can make their own and hints that he has thought up several variations of it. I look forward to reading about those!

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American Calendars for September had more than a week’s worth of palindromes. Would palindromes make a good puzzle? Yes! Print off a 100 chart and try Denise Gaskins’s A Puzzle for Palindromes. Also, check out her new Morning Coffee feature each week for more math teaching tips.

### Next Month’s Carnival

That’s it for this month’s Math Education Blog Carnival. The 132nd Carnival will be next month at Arithmophobia No More. Would you like to share a post or host the carnival? Go to Let’s Play Math for details!

# 127th Playful Math Education Blog Carnival

Ladies and gentlemen welcome to the Playful Math Education Blog Carnival featuring the incredible number 127 of the famous Mersenne Prime family! Let’s give a big hand to. . . . .

2²-1 = 3, the smallest single Mersenne Prime;
2³-1 = 7, the smallest double Mersenne Prime;
2⁷-1 = 127, the smallest triple Mersenne Prime; and finally
39-digit 2¹²⁷-1, the smallest quadruple Mersenne Prime!

It took NINETEEN YEARS (1857 to 1876) for Édouard Lucas to test Mersenne Prime Number 2¹²⁷-1 BY HAND to successfully verify that it is indeed a prime number. It is the largest Mersenne Prime that has ever been verified by hand calculations!

Ladies and gentlemen, Prime number 127 has one more amazing feat up its sleeve that you will have to see to believe:

In this month’s blog carnival posts, the amazing Desmos will delight and astound young and old alike. There will even be an Easter egg hunt! The blog post links (in turquoise) are joined by several links from Twitter (in blue-violet) and a few from other places such as Youtube (in red). Stay as long as you like and ENJOY what the many carnival participates have to offer in 20 different amusement areas!

Art and Mathematics

1. Stephanie showed off her colorful and impressive Tessellation Math Art Wall on twitter.
2. David Petro used all 84 pieces of a 21st-century pattern block set to create a lovely symmetrical design.
3. After this year’s very long winter, I’m especially glad Colleen Young collected some lovely and amazing springtime Desmos drawings and gifs in It’s Springtime. . . .
4. Do you see mathematics everywhere? Continuous-Everywhere-but-Differentiable-Nowhere can and does, even on a student’s shirt. Read the story and see the t-shirt design replication in Desmos in Going Off the Beaten Path.
5. Wanting to have your students create an art project in Desmos? 1ntegration-by-Parts has given that assignment many times and has links for student directions to help them meet your expectations in Desmos Art Project (Update).
6. You must click on the turtle face link in Desmos Art! With just ONE equation, a magnificently detailed turtle face was produced in the Desmos calculator. I was so impressed. I tweaked that single equation by changing the number 16 to 7.29, added some color restrictions as well as equations to make a mouth and some eyes to produce my own Desmos Art piece I call Blue-eyed Beauty.

Classwork/Homework that is enjoyable

1. Elementary-school-age students will enjoy that Matt Vaudrey shared.
2. I-Speak-Math has a mathematics homework solution students LOVE. Read about it in Meaningful Homework and CPM.
3. Jennifer Michaelailis has a pro tip on how to keep students who need a little extra help in class from feeling defeated.
4. If you want a free math education gathering in your area, here’s how to get one started. Also, check out Denise Gaskins’s resources to keep the group going.

Creative Writing

1. Many people have a personal story to tell that explains why they love mathematics. Through a Mathematician’s Eyes opens up and shares her experiences in My Story. What obstacles did she face? How did she feel about enjoying a subject so many others hated?
2. Philip Jose Pacis played with some math vocabulary and wrote a poem he titled Fractions about a fractured relationship.
3. Calendars and time are mathematical topics. How many other mathematical terms do you see in Maggie C.’s poem On Time? Do any of those words have more than one meaning?

Decimals

1. Math Geek Mama has come up with Decimals on a Number Line Game and includes everything you need to teach this concept with complete confidence.
2. Would you like to play Decimal Pickle? On Twitter, Mrs. Unger explained how to play it as well as a few of its variations.

Desmos First Aid Station?

1. There’s no need to call the paramedics when it’s time to learn about parametric functions. Suzanne Van Oy has come to the rescue! She sees a lot of value in parametric functions and their graphs. Why is she so excited about them? What’s all the fuss? Suzanne answers those questions and more in Why Parametrics?.
2. Suzanne Van Oy also recently put an incredible spinning Desmos Birthday Cake on twitter. How does she make Desmos gifs that don’t look like they need to be put on life support? Six months ago she blogged about how to do it in Making a Great Desmos Gif. She certainly knows what she’s doing!
3. Sometimes Desmos doesn’t do what you expect. Your work might need some first aid. DesmosGraph (Unofficial)’s post: Desmos Traps: Why Is It Not Working may have the diagnosis and cure you seek.
4. Although this post from Gold & Ratios is two years old, I still wanted it included on this list. It teaches how to add color and texture to Desmos.

Eggstraordinary Mathematical Easter Egg Hunt from Twitter

1. Cliff Pickover shared Don M. Jacobs equation for an egg laying on its side.
2. Tamás Görbe shared an equation for an egg that is standing up on its wide end.
3. Get out your markers and start coloring Paula Beardell Krieg’s and Three eggs! Paula also shows how
4. Mathigon shared a gif of a colorful, nine-piece, tangram-like egg puzzle that can also transform into a rabbit and a goose.
5. Robert Bosch shared that have a few mathematical traits in common.
6. Ha! Doesn’t this always happen? I found more fabulous eggs the day after the hunt: Daniel Mentrard’s eggs made in Geogebra,
7. And these Eggsponenential eggs created by Traci Jackson!

Exponents and Exponential Functions

1. Christopher Danielson on Twitter that generated a lot of thinking from adults. I’m sure it would do the same for kids who understand a little bit about exponents.
2. Jongarland6 was able to get ELL students conversing with each other in English about exponential functions. How it was done is described in Desmos Sparking Academic Conversations.

Fractions

1. Math Play Day gives ideas from 20SomethingKids and 1KookyTeacher about playful ways first-grade kids learned about fractions.
2. Mathgeek Mama published some adorable free Equivalent Fraction Robot Puzzles.
3. A recently released YouTube video has a little girl teaching about fractions in Maths 4 Kids’ Fractions of Shapes and Fraction Vocabulary.
4. Here’s another one featuring the same little girl: Fractions of Amounts Using the Bar Model
5. Every carnival has food concessions. You can have a lesson at breakfast on fractions based on CTSPEDMATHDUDE’s post Sausage Fractions: Real Life Example.
6. When teaching fraction division, should you start with rules or diagrams? Filling the Pail speaks from experience in Fraction Division and explains the advantages and disadvantages of both approaches.

Games and Educational Toys

1. On the spur of the moment, I came up with a very quick Yahtzee variation that I played with one of my students. We counted the number of rolls it took us to get a Yahtzee. Lowest score wins. He beat me badly every round as the graphic above this category attests, but on a different day, Lady Luck was with me more than him.
2. Denise Gaskin also has a tried and true Yahtzee game variation that she calls Six Hundred. You only need to print her directions and scoring sheets, provide six 6-sided die and pen or pencils, and you’ll be ready to make memories in more ways than one.
3. Autism Awareness Week was earlier in April. In this post, Special Educational Resources Blog reviewed three games made by Orchard Toys: Money Match Cafe (teaches about money), Look and Find Jigsaws (teaches number and letter recognition), and Bus Stop (uses processes like 3-2+4-1 to figure out how many kids are on the bus when the bus arrives at the bus station.)
4. The Mathematical Tourist shared how to play a game called Clobber. The game has been around since 2001, but the best strategy to win the game is still a mystery. I’m sure getting clobbered will be just as much fun as winning.

Geometry

1. What did an insightful five-year-old tell Paula Beardell Krieg about triangles?
2. Simon Gregg showed some pictures demonstrating how students had fun exploring squares in several different ways.
3. Similar geometric shapes line up and beg you to compare their dilations in Paula Beardell Krieg’s post About Halfway There.
4. Robert Loves Pi creates wonderful 3-D rotating digital geometric models. This one he calls the Twelfth Stellation of the Triakis Tetrahedron.
5. Desmos is a great tool, but sometimes I need to actually see how people use it to teach. Bearsemath.com does exactly that by sharing some pictures of Desmos Geometry being presented to a class of 10th graders.

Giving Back

1. When Women Inspire gives Three Worthy Reasons to Teach Charity to Your Kids. One of those reasons is that they will naturally learn the mathematics of money management.
2. Read the impressive CBS News account of how a STEM Robotics team made a toddler wheelchair for a  two-year-old whose family couldn’t afford one.
3. LMS Life Skills was practically speechless! Her class designed quilts blocks by using linear equations. Then the class made two quilts and donated them!

Linear Equations

1. Wheeler’s Thoughts on Teaching used a bank balance problem to teach about solving a system of linear equations. The students were able to think about the problem and work on it with much fewer hints from their teacher. That makes teaching math much more fun for the students as well as the teacher.
2. Jeff Lay created an Easter egg hunt activity to review linear equations, and he is happy to share the google docs he made with you.
3. Ms. Wheeler exclaimed that sometimes crayons and paper do the trick while her glass created stain class art.
4. Coincidentally, Ian Maclellan also had his class produce some stain glass art with linear equations.
5. Alicia Phillips shared one of her student’s projects that used only linear equations and was made on Desmos.

Literature and Mathematics

1. Imagine this carnival ride: a catapult that will send you flying through the air! Lana Pavlova and Meredith Wilkes have assisted Math Book Magic in creating the perfect design of an unforgettable carnival ride in Play with Your Math with Little Pea. How far will this catapult take you?
2. Erikson Institute writes how Anno’s Flea Market by Mitsumasa Anno, Which Would You Rather Be? By Willaim Steig, and Whose Shoes? By Stephen R. Swinburne are Three Books That Encourage Simple Graph Explorations with Young Ones.
3. Life Through a Mathematicians Eyes loves to curl up with a good book that features mathematics. Find out which books she has gathered and plans to read in her MathReadathon.
4. Kelly Anne Garner received several must-have mathematics in literature book suggestions from Twitter to build a fabulous math library. Check out the whole thread.

Museum of Mathematics

1. Chirag Mittal took charge of April’s birthday celebration of Leonard Euler. Did you know that Euler is credited with being the first to use letters from our alphabet and the Greek’s alphabet to represent some very important functions and numbers: Σ, f(x), e, i,  and π?
2. Alan Paar of Established 1962 has put together a tour of Wendover School and the way teachers taught and students there learned several mathematical topics and other subjects from 1868 to 1930.
3. Jo Morgan retweeted a tweet that caught my eye and was, therefore, the catalyst for bringing  MathigonOrg’s expansive and interactiveTimeline of Mathematics to this month’s carnival.

Number Theory

1. The number 127 is a centered hexagonal number as demonstrated by the graphic above.
2. Ramblings of a Writer recently marveled about how many things come in fours in her post Exploring the Number Four.
3. Dr. Helen J. Williams has pictures from a very playful session on “Fiveness”.
4. Science Switch had a few things to write about Belphegor’s Prime number, 1000000000000066600000000000001, in The Most Evil Number.

Optical Illusions

1. When there was a day off from school in the middle of the week, BMore Energy found plenty of kid’s activities in Manhattan’s Museum of Illusions.
2. Love Travelling takes us on a trip to see the fun-filled Vilnius Museum of Illusions. There is so much to see there!
3. While Matematickcom shows how to make a paper optical illusion that you can make yourself in very little time.

Puzzles

1. Math with P. Nik gives instructions and several examples of his Three Elastic Bands puzzles. He said they were easy to make, so I made the one at the top of this category. Follow P. Nik’s instructions and you probably won’t need to click on the tiny answer key under the puzzle.
2. When Simona Prilogan of Fiat Lux writes a number puzzle, it is much more than it appears.  You have to study patterns inside the puzzles to figure out what the relationship really is. Give her Wednesday Math Puzzle a try!
3. Simona included a bonus, information about Bolsover Castle, in her Monday Math Puzzle. You will find two different Math Puzzles in the middle of reading about the Castle!
4. This clever tie matching exercise from Math with P. Nik feels more like a puzzle than a worksheet. Can you match the graph families with the correct equation families?

Statistics and Probability

1. Yes, you can do statistics in Desmos! You can make Normal, Poisson and Binomial distributions and even graph box and whisker plots in Desmos! Colleen Young shows you what that looks like in her post Statistics with Desmos.
2. Does El Niño play much of a part in rising global temperatures? In New Kid in Town, Open Mind answers that question and includes line graphs to help us visualize global temperature data collected since 1979.
3. This year Easter occurred on April 21st. That seemed rather late to me, but it isn’t the lastest it could be. In Joseph Nebus’s post, What Dates Are Most Likely for Easter?, he’ll direct you to a post he wrote two years ago where all the data is lined up to figure out the probability.

Telling Time

1. What time is it? There is more than one valid way to give the correct time, and one way should not be labeled as a smarter way to give the time than the others. That’s the message given in Dan Meyer’s Don’t Teach Math “the Smart Way”. He even suggests a lovely game from Desmos to get kids talking about telling time.
2. After a long winter with snow causing several school days to be replaced with “e-learning days,” Educational Technology in Action wrote about using that same Desmos talking time activity in Desmos for meaningful e-learning days.

Vocabulary

1. On Twitter, Jo Morgan shared a photo that truly enhanced the 1679 definition of a Rhombus.
2. Joseph Nebus of NebusResearch regularly writes about mathematics-themed comics. Here is a comic about the difference in definitions of vertex and apex. It also has a graph theory puzzle and three other comics about story problems involving addition and subtraction, slope intercept form, and paradoxes.
3. What does the word Asymptotic mean? Hazel Clementine shared a catchy musical definition.

Thanks for coming to this month’s carnival! I hope you enjoyed it. I had a wonderful time hunting for goodies to put in the carnival and organizing it. I felt like I was on an Easter egg hunt looking for the best eggs!

Math Misery? will host May’s Playful Math Education Blog Carnival. Perhaps YOU will consider contacting Denise Gaskins and volunteering to host a future carnival! There are two open dates in the summer still available this year.

You can also visit The 126th Playful Math Education Blog carnival hosted by Math Mama Writes. . . or the 157th Carnival of Mathematics hosted by Lines Curves Spirals for more mathematical adventures!

# 1371 Today is a Good Day to Review Proof by Induction

0² = 0
1²  = 1

Does that pattern hold for all natural numbers? Could we claim that n²  = n?

Yes, we can, and I’ve written a proof to prove it! The proof uses a valuable concept in mathematics called induction. I remember being introduced to proofs by induction when I was in Junior High. Nowadays, if it is not part of Common Core, it wouldn’t be taught much anymore. Nevertheless, I will use it here to prove that n² = n.

Using a similar proof, we can also prove that n³ = n, n⁴ = n, n⁵ = n, n⁶ = n, and so forth!

Today is the perfect day to review how to use proof by induction so try your hand at proving at least one of those mathematical statements on your own. Use the same steps in my example: prove true for n=1, assume true for n = k, prove true for k + 1, write your conclusion. then have a very Happy April Fools’ Day, Everyone!

Today is also a very good day to review that (x + y)² = x² +2xy + y²  and NOT x²  + y², a very common error students make. Confession: I remember making that exact error in high school when I definitely should have known better. Using induction to prove something in mathematics is a valid technique, but if you use invalid equations like
(x + y)³ = x³ + y³, you will make invalid conclusions. Thus, today might also be a good day to review the binomial theorem and Pascal’s triangle. (Pascal’s triangle has numbers in its interior, not just 1’s going down the sides, after all.)

My post today was inspired by a post written by Sara Van Der Werf titled Why I’ve Started Teaching the FOIL Method Again. In her post, she not only plays a great April Fools’ joke on her readers, but she explains a tried and true way to multiply binomials and other polynomials.

I read her post exactly one year ago today, and since then, I have been waiting for April Fools’ Day to roll around again so that I could share this post with you. It is my hope that you will enjoy my little prank and learn a little mathematics from it as well.

Now I’ll write a little bit about the number 1371:

• 1371 is a composite number.
• Prime factorization: 1371 = 3 × 457
• 1371 has no exponents greater than 1 in its prime factorization, so √1371 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 1371 has exactly 4 factors.
• The factors of 1371 are outlined with their factor pair partners in the graphic below.

1371 is the hypotenuse of a Pythagorean triple:
504-1275-1371 which is 3 times (168-425-457)

OEIS.org informs us that 1² + 37² + 1² = 1371, and there’s no April Fooling about that!

# 1366 Fractions Acting Improperly

In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.

On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.

This first tweet has a link explaining why it is improper to use the term improper fraction:

I hope that you will consider not labeling any fraction as improper, as well!

Now I’ll write a little bit about the number 1366:

• 1366 is a composite number.
• Prime factorization: 1366 = 2 × 683
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1366 has exactly 4 factors.
• Factors of 1366: 1, 2, 683, 1366
• Factor pairs: 1366 = 1 × 1366 or 2 × 683
• 1366 has no square factors that allow its square root to be simplified. √1366 ≈ 36.95944

1366 is also the sum of the twenty-six prime number from 5 to 107. Do you know what all those prime numbers are?

# 1350 Logic is at the Heart of This Puzzle

### Today’s Puzzle:

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

If you need some tips on how to get started on this puzzle, check out this video:

### Factors of 1350:

• 1350 is a composite number.
• Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
• The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
• Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
• Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
• Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

### Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1350 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

### More about the Number 1350:

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

# Facts about and Factors of 2019

Here’s a countdown you can use to ring in the New Year:

make science GIFs like this at MakeaGif

2019 is the sum of consecutive numbers three different ways:
1009 + 1010 = 2019
672 + 673 + 674 = 2019
334 + 335 + 336 + 337 + 338 + 339 = 2019

There is one way that 2019 is the sum of consecutive odd numbers:
671 + 673 + 675 = 2019

2019 is the difference of two squares two different ways:
338² – 335² = 2019
1010² – 1009² = 2019

2019 is the sum of three squares nine different ways:
43² + 13² + 1² = 2019
43² + 11² + 7² = 2019
41² + 17² + 7² = 2019
41² + 13² + 13² = 2019
37² + 25² + 5² = 2019
37² + 23² + 11² = 2019
37² + 19² + 17² = 2019
35² + 25² + 13² = 2019
31² + 23² + 23² = 2019

2019 is the hypotenuse of a Pythagorean triple:
1155-1656-2019 so 1155² + 1656² = 2019²

2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ + 2⁰ = 2019

2019 is a palindrome in a couple of bases:
It’s 5B5 in BASE 19 (B is 11 base 10) because 5(19²) + 11(19) + 5(1) = 2019,
and 3C3 in BASE 24 (C is 12 base 10) because 3(24²) + 12(24) + 3(1) = 2019

Every year has factors that often catch people by surprise. Today I would like to give you my predictions for the factors of 2019:
2019 will have four positive factors: 1, 3, 673, and 2019
However, 2019 will also have four negative factors: -1, -3, -673, and -2019

Which factors, positive or negative, will be your focus in the coming year?

Finally, I’ll share some mathematics-related 2019 and New Year tweets that I’ve seen on twitter. Some of these tweets have links that contain even more facts about the number 2019.

That tweet inspired me to make my own 3 × 3 Magic Square where every number is different but every row, column and diagonal totals the same number:

But because 2019 is divisible by 3, it can also be a magic sum of 3 × 3 magic square:

5

5

And finally, here is my contribution to 2019 twitter:

# 810 Stick and Stone

Print the puzzles or type the solution on this excel file: 10-factors 807-814

One of my education professors taught that you can teach any concept with a picture book.

I recently read the book, Stick and Stone, to a class of 6th graders. Yes, 6th graders. You can get away with reading something way below grade level if you tell them before you start reading that you will use the book to introduce them to something that is definitely NOT below grade level. The first few pages of the book are shared by its publisher here:

As you can see, those first few pages equate stone as a zero and stick as a lonely number one.

The middle part of the book teaches about synergizing, working together to make life good and helping each other through tough times.

By the end of the book Stick and Stone know how to work very well together, “Stick, Stone. Together again. Stick, Stone. A perfect ten.”

The book pretty much ends there, but making a perfect ten is only the beginning of what these two characters can do together. I used this book to teach the class not only about getting along and working together, but also about base 2, or binary as it is also called. Every counting number we know can be represented by using just 1’s and 0’s. I wrote on the board the numbers from 1 to 16 and represented the first few of those numbers in base 2. Then I invited class members to come up with how to write the rest of the numbers in base 2. Some students caught on immediately while the others were able to learn how to do it by watching their classmates and listening to them. Eventually with at least 12 different student’s inputs, we came up with a chart that looked something like this:

Notice that the numbers from 9 to 15 are just 1000 plus the numbers directly across from them in the first column.

Some of the sixth grade students had already heard of binary, so I showed them a little more about base 2: I wrote a bunch of 1’s and 0’s “off the top of my head” onto the board and added the headings to show place values: 1’s place, 2’s place, etc.

Then I told them to sum up the place values that contained a one:

Stick and Stone are the main two characters, but the book has one other character, Pinecone. At first Pinecone bullied Stone, but after Stick stood up to him, the three of them were eventually able to become friends. You might enjoy finding out more about Pinecone by listening to Sean Anderson read the entire book to his children, one of which seems to really enjoy numbers.

Children might also enjoy representing all the numbers from 1 to 31 with one hand:

If you used a unique symbol to represent Pinecone, it could look like a 2. Then you also could use the symbols 0, 1, and 2 to represent every counting number in base 3. That’s another concept the picture book Stick and Stone could be used to introduce!

To make a chart for base 3, start with these 3 columns of numbers with 3 numbers in each:

Since this is base 3, where should 10 and 100 go? The bottom of the first column and the bottom of the third column both MUST look like a power of 10. The rest of the chart is easy to fill out. Notice the 1 and 2 look exactly the same in base 10 and base 3. Also since 4 = 3 + 1, 5 = 3 + 2, and 6 = 3 + 3, we can easily fill in the 2nd column. Two more addition facts will finish the third column: 7 = 6 + 1, and 8 = 6 + 2.

Now add what you learned about 4, 5, 6, 7, 8, and 9 to column 1 and put the numbers 10 – 18 in the base 10 second column and numbers 19 – 27 in the base 10 third column. Again the bottom of the first column and the bottom of the third column both MUST look like a power of 10, so we now know where to put 1000.

To fill in the rest of the chart simply add 100 to the base 3 numbers in column 1 to get the the base 3 numbers in column 2. Then add 200 to the base 3 numbers in column 1 to get the remaining base 3 numbers in column 3.

You could do this process again to determine the first 81 counting numbers in base 3 with 81 being represented by 10000.

For base 4, you could do something similar with 4 columns. However, for counting in bases 4, 5, 6, 7, 8, and 9 I would suggest using the very versatile hundred chart. You can give instructions without even mentioning the concept of differing bases. For example, cross out every number on the hundred chart that has 7, 8, or 9 as one or more of its digits. Can you tell even before you get started how many numbers will get crossed out? (100 – 7²) What pattern do the cross-outs make? If you arrange the remaining numbers in order from smallest to largest, then you will have the first 49 numbers represented in base 7. With a minimal amount of cutting and taping you could have a “hundred” chart in base 7. Easy peasy.

This excel file not only has several puzzles, including today’s, but also a hundred chart and even a thousand chart because I know some of you might want to play with 3-digit numbers, too.

Now let me tell you a little bit about the number 810:

• 810 is a composite number.
• Prime factorization: 810 = 2 x 3 x 3 x 3 x 3 x 5, which can be written 810 = 2 × 3⁴ × 5
• The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 x 5 x 2 = 20. Therefore 810 has exactly 20 factors.
• Factors of 810: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810
• Factor pairs: 810 = 1 x 810, 2 x 405, 3 x 270, 5 x 162, 6 x 135, 9 x 90, 10 x 81, 15 x 54, 18 x 45 or 27 x 30
• Taking the factor pair with the largest square number factor, we get √810 = (√81)(√10) = 9√10 ≈ 28.4604989.

Since 810 has so many factors, it has MANY possible factor trees. If most people made a factor tree for 810, they would probably start with 81 × 10 or 9 x 90. NOT ME! Here are two less-often-used factor trees for 810:

Finally, here is an easy way to express 810 is in a different base:

1. Make a cake in which you divide 810 by the base number repeatedly, keeping track of the remainders, including zero, as you go.
2. Keep dividing until the number at the top of the cake is 0.
3. List the remainders in order from top to bottom and indicate the base you used to do the division.
4. This method is illustrated for BASE 2 and BASE 3 below:

That’s all pretty good work for a stone, a stick, and a pine cone!

By the way, using that method will also produce the following results:

• 810 is 30222 BASE 4
• 810 is 11220 BASE 5
• 810 is 3430 BASE 6 and so forth.

And just so you’ll know, 810 is the sum of consecutive primes 401 and 409.

# 807 and Level 1

What can I say about the number 807?

807 is palindrome 151 in BASE 26 because 1(26²) + 5(26) + 1(1) = 807.

Anything else? Well, I can figure out a few other things because 807’s has two prime factors, 3 and 269:

We can write ANY number (unless it’s a power of 2) as the sum of consecutive numbers in at least one way. 807 has three different ways to do that:

• 403 + 404 = 807 because 807 isn’t divisible by 2.
• 268 + 269 + 270 = 807 because it is divisible by 3.
• 132 + 133 + 134 + 135 + 136 + 137 = 807 since it is divisible by 3 but not by 6.

I know that one of 807’s factors, 269, is a hypotenuse of a Pythagorean triple, so 807 is also. Thus. . .

• (3·69)² + (3·260)² = (3·269)², or in other words, 207² + 780² = 807²

Since 807 has two odd sets of factor pairs, I know that 807 can be written as the difference of two squares two different ways:

• 136² – 133² = 807
• 404² – 403² = 807

I don’t usually do this, but today’s puzzle has something in common with 807. Can you tell what it is?

Print the puzzles or type the solution on this excel file: 10-factors 807-814

• 807 is a composite number.
• Prime factorization: 807 = 3 x 269
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 807 has exactly 4 factors.
• Factors of 807: 1, 3, 269, 807
• Factor pairs: 807 = 1 x 807 or 3 x 269
• 807 has no square factors that allow its square root to be simplified. √807 ≈ 28.4077454