1445 Virgács for Your Boots Tonight

Tomorrow is Mikulás (Saint Nicholas Day) in Hungary. Children will awake to find candy, fruit, or nuts in their polished shoes or boots because every boy and every girl has been at least a little bit good all year long.

Because they have also been at least a little bit naughty, they will also find virgács in those same shoes or boots. Virgács are little twigs that have been spray-painted gold and tied together at the top with red ribbon.

Santa is so busy this time of year, that I thought I would give him a helping hand. I’ve made some virgács for YOUR boots or shoes!

Start at the top of the puzzle and work your way down cell by cell to solve this Level 3 puzzle. Oh, but I’ve been just a little bit naughty making this puzzle: you will need to look at later clues to figure out what factors to give to 40. Will clue 40 use a 5 or a 10? Look at clues 60 and 90, and you will have only one choice for that answer. Then you can forgive my tiny bit of naughtiness.

Now I’ll tell you a few facts about the puzzle number, 1445:

  • 1445 is a composite number.
  • Prime factorization: 1445 = 5 × 17 × 17, which can be written 1445 = 5 × 17²
  • 1445 has at least one exponent greater than 1 in its prime factorization so √1445 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1445 = (√289)(√5) = 17√5
  • 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 1445 has exactly 6 factors.
  • The factors of 1445 are outlined with their factor pair partners in the graphic below.

1445 is the sum of two squares in THREE different ways:
31² + 22² = 1445
34² + 17² = 1445
38² + 1² = 1445

1445 is the hypotenuse of SEVEN Pythagorean triples:
76-1443-1445 calculated from 2(38)(1), 38² – 1², 38² + 1²
221-1428-1445 which is 17 times (13-84-85)
477-1364-1445 calculated from 31² – 22², 2(31)(22), 31² + 22²
612-1309-1445 which is 17 times (36-77-85)
680-1275-1445 which is (8-15-17) times 85
805-1200-1445 which is 5 times (161-240-289)
867-1156-1445 which is (3-4-5) times 289 and can
also be calculated from 34² – 17², 2(34)(17), 34² + 17²

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1444 Christmas Wrapping Paper

A fourth of the products in this multiplication table puzzle are already there just because I wanted the puzzle to have a wrapping paper pattern. Can you figure out what the factors are supposed to be and what all the other products are?

  • 1444 is a composite number and a perfect square.
  • Prime factorization: 1444 = 2 × 2 × 19 × 19, which can be written 1444 = 2²× 19²
  • 1444 has at least one exponent greater than 1 in its prime factorization so √1444 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1444 = (√4)(√361) = (2)(19) = 38
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1444 has exactly 9 factors.
  • The factors of 1444 are outlined with their factor pair partners in the graphic below.

Square number 1444 looks a lot like another square number, 144.
If we keep adding 4’s to the end, will we continue to get square numbers?
No.

However, in different bases, 1444 looks like several other square numbers:
It’s 484 in BASE 18,
400 in BASE 19,
169 in BASE 35,
144 in BASE 36,
121 in BASE 37, and
100 in BASE 38.

1442 A Birthday Present for My Brother, Andy

My brother, Andy, has a birthday today. He’s very good at solving puzzles, so I made this Challenge puzzle as a  present for him.  Have a very happy birthday, Andy!

You can try to solve it too. If the box and ribbon are too distracting, here’s a copy of the puzzle without the added color. Click on it to see it better.

Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

That was puzzle number 1442. Here are some facts about that number:

  • 1442 is a composite number.
  • Prime factorization: 1442 = 2 × 7 × 103
  • 1442 has no exponents greater than 1 in its prime factorization, so √1442 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 1442 has exactly 8 factors.
  • The factors of 1442 are outlined with their factor pair partners in the graphic below.

2(103)(7) = 1442, making it a leg in this Pythagorean triple:
1442-10560-10658 calculated from 2(103)(7), 103² – 7², 103² + 7²

Why is 1441 a Star?

1441 is the 16th star number. If you look at the tiny squares that make up each triangle in the star, you might notice that each of those triangles is made with 15 rows of squares. Thus each triangle represents the 15th triangular number.

A six-pointed star is really just a 12-sided figure better known as a dodecagon.

Since I recently wrote that figurate numbers can be found using triangular numbers, I wondered if centered figurate numbers can make the same claim. Yes, they can!

Centered Figurate Numbers are also called Centered Polygonal Numbers and can be easily calculated from the triangular numbers.

Here are some more facts about the number 1441:

  • 1441 is a composite number.
  • Prime factorization: 1441 = 11 × 131
  • 1441 has no exponents greater than 1 in its prime factorization, so √1441 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 1441 has exactly 4 factors.
  • The factors of 1441 are outlined with their factor pair partners in the graphic below.

1441 is a palindrome with an even number of digits so it is divisible by 11.

1441 is also a palindrome in some other bases, namely, base 6, base 30, base 32, and base 36.

Taking the Mystery out of 1440

The number 1440 has thirty-six factors. That’s a lot! The only number smaller than it with that many factors is 1260. Why do they have so many?

Would solving this Mystery Level puzzle be an easier mystery to solve?

Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

  • 1440 is a composite number.
  • Prime factorization: 1440 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5, which can be written 1440 = 2⁵ × 3² × 5
  • 1440 has at least one exponent greater than 1 in its prime factorization so √1440 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1440 = (√144)(√10) = 12√10
  • The exponents in the prime factorization are 5,2, and 1. Adding one to each exponent and multiplying we get (5 + 1)(2 + 1)(1 + 1) = 6 × 3 × 2 = 36. Therefore 1440 has exactly 36 factors.
  • The factors of 1440 are outlined with their factor pair partners in the graphic below.

There you have it. The exponents of the prime factorization determine the number of factors a number has, and (5 + 1)(2 + 1)(1 + 1) = 6 × 3 × 2 = 36. How do we find its prime factorization? We can divide it over and over by the smallest prime number factor until we make an easy-to-read and very delicious cake!

Or you could make a factor tree such as the one below. Then rake up the prime number factors afterward and hope that you see them all:

I prefer cake to yard work any day!

As I mentioned before, 1440 is NOT the smallest number with 36 factors:
Because (2 + 1)(2 + 1)(1 + 1)(1 + 1) = 3 × 3 × 2 × 2 = 36 also, we get
2²× 3²× 5 × 7 = 1260, which just happens to be smaller than 1440, so 1260 gets the smallest-number-with-36-factors prize. 1440 is the SECOND smallest number with 36 factors.

1440 is the sum of the interior angles of a decagon. Why?
Because 180(10-2) = 180(8) = 1440
So what? 1260 is the sum of the interior angles of a nonagon.

1440 is the hypotenuse of only one Pythagorean triple:
864-1152-1440 which is (3-4-5) times 288
Yeah? 1260 is the hypotenuse of only one Pythagorean triple as well.

Look, 1440 is not second best because it has this one other claim to fame:
From Stetson.edu we learn that 1440 deserves a lot of exclamation points since
1440 = 2!3!5!

There you have it. We’ve taken the mystery out of the number 1440, AND it is a fabulous number!

 

1431 is a Triangular Number and a Hexagonal Number

If you only look at a list of triangular numbers or a list of hexagonal numbers, you might miss the relationship that figurate numbers have with each other.

1378 is the 52nd triangular number, and you can use it to find the 53rd triangular number (1431), the 53rd square number, the 53rd pentagonal number, and so forth.

351 is the 26th triangular number, and you can use it to find the 27th triangular number, the 27th square number, the 27th pentagonal number, the 27th hexagonal number (1431), and so forth.

See the relationship in the graphic below:

Should you get excited that 1431 is BOTH a triangular number and a hexagonal number? Not really. It turns out that every hexagonal number is also a triangular number. (But not every triangular number is a hexagonal number.)

Here are a few more facts about the number 1431:

  • 1431 is a composite number.
  • Prime factorization: 1431 = 3 × 3 × 3 × 53, which can be written 1431 = 3³ × 53
  • 1431 has at least one exponent greater than 1 in its prime factorization so √1431 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1431 = (√9)(√159) = 3√159
  • 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 1431 has exactly 8 factors.
  • The factors of 1431 are outlined with their factor pair partners in the graphic below.

1431 is the hypotenuse of a Pythagorean triple:
756-1215-1431 which is 27 times (28-45-53)

1430 Is a Catalan Number

1430 is the eighth Catalan number because it is equal to (2⋅8)!÷((8+1)!8!):
10 × 11 × 12 × 13 × 14 × 15 × 16 ÷( 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 ) = 1430.

For example, the vertices of a decagon can also be the vertices of eight triangles. Those eight triangles can be drawn in 1430 different ways. Here are a few of those ways:

Some of the 1430 ways are rotations and/or reflections of the ways illustrated above. Many of the 1430 ways are NOT represented in that graphic at all. It would be mind-boggling to draw all 1430 ways!

Here are some other facts about the number 1430:

  • 1430 is a composite number.
  • Prime factorization: 1430 = 2 × 5 × 11 × 13
  • 1430 has no exponents greater than 1 in its prime factorization, so √1430 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1430 has exactly 16 factors.
  • The factors of 1430 are outlined with their factor pair partners in the graphic below.

1430 is the hypotenuse of FOUR Pythagorean triples:
352-1386-1430 which is 22 times (16-63-65)
550-1320-1430 which is (5-12-13) times 110
726-1232-1430 which is 22 times (33-56-65)
858-1144-1430 which is (3-4-5) times 286

 

1428 Factor Trees in Autumn

I recently decided that I wanted to make some factor trees in various fall colors. 1428 has plenty of factors so it has MANY different factor trees. Here are just eleven of them, each initially factored by a different factor pair.

Here are some more facts about the number 1428:

  • 1428 is a composite number.
  • Prime factorization: 1428 = 2 × 2 × 3 × 7 × 17, which can be written 1428 = 2² × 3 × 7 × 17
  • 1428 has at least one exponent greater than 1 in its prime factorization so √1428 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1428 = (√4)(√357) = 2√357
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1428 has exactly 24 factors.
  • The factors of 1428 are outlined with their factor pair partners in the graphic below.

1428 is the hypotenuse of a Pythagorean triple:
672-1260-1428 which is (8-15-17) times 84.

1426 is a Pentagonal Number

I remembered that 1426 is a pentagonal number, but I didn’t have the formula for pentagonal numbers memorized. While I was waiting to do something else, I tried to come up with the formula myself.

The formula I derived isn’t what you usually see, but I rather like it! This formula can be extended to any figurate number as the chart below shows.

I love that the distance between consecutive figurate numbers on the chart is 465 which is the 30th triangular number!

Now if you asked me how many little dots does the 31st 1000-gonal number have, I also would be able to tell you that it has 998(465) + 31 = 464101 dots!

But today’s post is asking for the 31st pentagonal number. Here is what that looks like:

Can you see that it is three times the 30th triangular number plus 31?

Here are some more facts about the number 1426:

  • 1426 is a composite number.
  • Prime factorization: 1426 = 2 × 23 × 31
  • 1426 has no exponents greater than 1 in its prime factorization, so √1426 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 1426 has exactly 8 factors.
  • The factors of 1426 are outlined with their factor pair partners in the graphic below.

Since 1426 = 2(31)(23) and 2(713)(1), it is a leg in two Pythagorean triples:
432-1426-1490 calculated from 31² – 23², 2(31)(23), 31² + 23²
1426-508368-508370 calculated from 2(713)(1), 713² – 1², 713² + 1²

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!

131, a permutable prime number

make science GIFs like this at MakeaGif
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.

Factoring Quadratics

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!