# 1456 and Level 2

#### Today’s Puzzle:

Can you write each number from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues?

#### 1456 Factor Tree:

1456 is made from two multiples of 7, so we know it is divisible by 7. It’s last two digits are 56, so it is divisible by 4. Since it is divisible by 7 and by 4, it is divisible by 28. I used that fact to make this factor tree for 1456:

#### Factors of 1456:

• 1456 is a composite number.
• Prime factorization: 1456 = 2 × 2 × 2 × 2 × 7 × 13, which can be written 1456 = 2⁴ × 7 × 13
• 1456 has at least one exponent greater than 1 in its prime factorization so √1456 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1456 = (√16)(√91) = 4√91. The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1456 has exactly 20 factors.
• The factors of 1456 are outlined with their factor pair partners in the graphic below:

#### Interesting facts about the number 1456:

1456 is the difference of two squares in six different ways:
365² – 363² = 1456
184² – 180² = 1456
95² – 87² = 1456
59² – 45² = 1456
41² – 15² = 1456
40² – 12² = 1456

1456 is the hypotenuse of one Pythagorean triple:
560-1344-1456 which is (5-12-13) times 112

# 1455 and Level 1

### Today’s Puzzle:

All of the clues in today’s puzzle are divisible by the same number. Can you figure out what that number is? If you can, then you can solve this puzzle.

### Factors of 1455:

That was puzzle number 1455. In case you would like to know a little bit about that number, here are a few facts:

• 1455 is a composite number.
• Prime factorization: 1455 = 3 × 5 × 97
• 1455 has no exponents greater than 1 in its prime factorization, so √1455 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 1455 has exactly 8 factors.
• The factors of 1455 are outlined with their factor pair partners in the graphic below.

### Facts about the number 1455:

1455 is the difference of two squares in four different ways:
728² – 727² = 1455
244² – 241² = 1455
148² – 143² = 1455
56² – 41² = 1455

1455 is also the hypotenuse of four Pythagorean triples:
132-1449-1455 which is 3 times (44-483-485)
279-1428-1455 which is 3 times (93-476-485)
873-1164-1455 which is (3-4-5) times 291
975-1080-1455 which is 15 times (65-72-97)

# 135th Playful Math Education Blog Carnival

Ladies and gentlemen welcome to the Playful Math Education Blog Carnival! This month’s carnival features the versatile number 135. It is the smallest number whose digits are the first three odd numbers. Watch 135 perform these AMAZING stunts:
1351 x 3³ 5¹
135 = (1+3+5)(1×3×5)
135 = 1¹ + 3² +5³
135 made that last one look as easy as 1-2-3.

### Carnival Wait Times

Now before we get started with our playful math blog posts, I would like to address wait times. Every worthwhile amusement park and carnival has lines in which people must wait. The playful math carnival loves lines and is no exception, but is waiting really a bad thing? In Solving 100÷3 Mentally: a Surprise, Marilyn Burns teaches and assesses children’s understanding of mathematics. She explains that patiently waiting for kids’ mathematical thoughts can actually speed up their understanding and enjoyment. Hence, it is all worth the wait!

Mr. Mathematics also waits patiently for his students to think problems through themselves. When an inspector tried to interview him about issues he faces in his department, he turned the tables and talked about his students’ learning so much because he waits patiently for his students to problem solve.

### Art

Robert Loves Pi has makes gorgeous mathematical art. For example, he’s made this design with circles, triangles and a pentagon and a beautiful rotating arrangement of a pentagrammic prism.

Since it is February, Colleen Young collected a very nice assortment of hearts related to mathematical content.

Mike and his sons of Mike’s Math Page produced some lovely geometric designs in A Fun Zometool project with Decagons.

Paula Beardell Krieg has published several blog posts on Frieze symmetry. Here is the start of that series of posts:

### Education

I’ve seen the visual below several times before, but I didn’t realize its magnificence until I read Sara VanDerWerf post describing how she has used it to inspire herself, her students, and other teachers:

Ben Orlin has tips from four math teachers on what makes a teacher great:

### Food Court at the Math Carnival

Here’s a tasty blog post:

### Geometry

Did you know that 81² + 108² = 135²? That’s simply 27 times 3² + 4²= 5², the most famous example of the Pythagorean Theorem. The scarecrow from the Wizard of Oz sounds impressive when he recites something that sounds a little like the Pythagorean Theorem. Is the formula he gives true? Watch this short movie clip and tell me what you think:

Check out these geometry blog posts I saw on twitter:

### Language Arts and Math

If you just asked students to write something about math, they might not have much to say, but if you gave them some of the wonderful prompts suggested by CLopen Mathdebater in Explore Math with Prompts, you might just be as pleasantly surprised as she was with her student’s writing and mathematical thinking!

Intersections–Poetry with Mathematics writes about a particular subject we’ve all seen in both word problems and nightmares: Those Trains in Word Problems–Who Rides Them?

Kelly Darke of Math Book Magic has been reading picture books about counting with her child. She explains why they found the picture book, 1-2-3 Peas, to be magical. She loved watching her child trace the illustrated numerals as they explored bigger numbers like 80 and made connections between numerals and the letters of the alphabet.

### Math News Room

Here are some news articles related to math that I enjoyed reading this month:

### Museum of Math

Pat’s Blog can tell you what mathematical event happened on today’s date. For example, here is mathematical history for February 10.

Life Through a Mathematician’s Eyes shares the highlights of a history of mathematics for January.

The following images would fit in perfectly in any math museum:

### Probability and Statistics

Fraction Fanatic has been sharing resources every week since the beginning of 2020. In the first post of the year, This week, Number 1, we see ways to make leaf and stem plots and to make predicting probability both pertinent and fun.

Joseph Nebus regularly shares mathematically themed comics on his blog. In this one, he shares a comic that points out that those probabilities pertain to you personally, not just everybody else.

### Puzzles

Mathtuition shares a math puzzle that will get kids in primary school thinking in Marbles Math Question.

Here are some other mathematical puzzles from blogs that I saw on twitter:

### Strategies

K-8 Math Specialist Jenna Leib writes about learning and having fun with ten frames and tiny polka dots in Kindergarten Debate:  Building Appreciation for Ten Frames. By the way, it’s the kindergarteners who are debating, not adults!

Given two ordered pairs, how do you find their midpoint? Don’t use some cheap trick. Math Chat has a midpoint strategy that students will remember forever, and it promotes mathematical understanding.

The writer behind Math QED earned a 770 on the math portion of the SAT and lists strategies to help you get a perfect score on that part of the test as well.

Here are a couple more blog posts about strategies that were shared on twitter:

### More Carnivals

I hope you enjoyed all the attractions at this month’s Playful Math Education Blog Carnival.

Last month the carnival was hosted by Math Misery?, but so far no one has volunteered to host the carnival for the end of March. Perhaps you will consider hosting it! It is a lot of fun exploring other people’s blogs and selecting what to share, so do think about hosting it next month or some other time in the future. Click here to volunteer to host or to submit one of your posts to the carnival.

You may also want to check out February’s Carnival of Mathematics hosted by Stormy at Storm Bear World.

# 1454 Happy Valentines’ Day

I hope that today and always you have lots of love in your heart and that you feel so much love from others. Have a wonderful Valentine’s Day!

#### A Valentines Puzzle

Here is a Valentines themed mystery level puzzle from me to you:

#### Factors of 1454

• 1454 is a composite number.
• Prime factorization: 1454 = 2 × 727
• 1454 has no exponents greater than 1 in its prime factorization, so √1454 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 1454 has exactly 4 factors.
• The factors of 1454 are outlined with their factor pair partners in the graphic below.

#### Interesting Fact about the Number 1454:

From Stetson.edu we learn that 11+444+555+444=1454.

#### Mathematical Valentine Tweets I’ve Seen Today:

I hope those tweets help fuel your love for mathematics. Have a very happy Valentines’ Day!

# 1453 Happy Birthday, Jo Morgan

#### Who Is Jo Morgan?

Jo Morgan is an inspiring mathematics teacher, collaborator, tweeter, blogger, podcast guest, and author. Today is her birthday!

Jo recently published her first book, A Compendium of Mathematical Methodsand it is all the rage on twitter. Here is a small sampling of tweets expressing excitement for her book:

I can hardly wait until February 4th when Amazon makes it available in the United States!

Jo has enjoyed solving some of my puzzles, so to commemorate her birthday, I’ve made one especially for her. To solve this puzzle, write the numbers 1 to 10 in each of the four sections outlined in bold so that those numbers are the factors of the product clues given in each of the four mixed-up multiplication tables that make up the puzzle. Use logic to solve the puzzle, but I’m warning you, it won’t be easy.

Happy birthday, Jo! I hope you enjoy the puzzle!

#### Find the Factors 1 – 10 Birthday Challenge Puzzle:

Print the puzzles or type the solution in this excel file: 12 Factors 1443-1453

#### Factors of 1453:

It is convenient for puzzles to be numbered, and this puzzle number is 1453. Here are a few facts about that number:

• 1453 is a prime number.
• Prime factorization: 1453 is prime.
• 1453 has no exponents greater than 1 in its prime factorization, so √1453 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1453 has exactly 2 factors.
• The factors of 1453 are outlined with their factor pair partners in the graphic below.

How do we know that 1453 is a prime number? If 1453 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1453. Since 1453 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1453 is a prime number.

#### Other Facts about the Number 1453:

1453 is the sum of two squares:
38² + 3² = 1453

That means 1453 is the hypotenuse of a Pythagorean triple:
228-1435-1453 calculated from 2(38)(3), 38² – 3², 38² + 3².

Here’s another way we know that 1453 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 3² = 1453 with 38 and 3 having no common prime factors, 1453 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1453. Since 1453 is not divisible by 5, 13, 17, 29, or 37, we know that 1453 is a prime number.

# Countdown to 2020

#### Countdown to 2020:

It seems that every New Year’s Eve, mathematicians come up with equations with the new year in it. Some of those equations will be a countdown. Here is the equation that I found and made into a gif:

make science GIFs like this at MakeaGif

#### Factors of 2020:

What will the factors be in the year 2020? I’m here ready with my predictions and the reasons that you can rely on them:
• 2020 is a composite number.
• Prime factorization: 2020 = 2 × 2 × 5 × 101, which can be written 2020 = 2² × 5 × 101
• 2020 has at least one exponent greater than 1 in its prime factorization so √2020 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √2020 = (√4)(√505) = 2√505
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 2020 has exactly 12 factors.
• The factors of 2020 are outlined with their factor pair partners in the graphic below.

#### A 2020 Factor Tree and a 2020 Factor Cake:

A 2020 factor tree may interest you:

Or a 2020 Factor Cake:

#### Other Facts about the number 2020:

2020 is palindrome 4C4 in BASE 21 (C is 12 in base 10):
4(21²) + 12(21) + 4(1) = 4(441) + 12(21) + 4 = 1764 + 252 + 4 = 2020

42² + 16² = 2020
38² + 24² = 2020

2020 is the hypotenuse of FOUR Pythagorean triples:
400-1980-2020 which is 20 times (20-99-101)
868-1824-2020 calculated from 38² – 24², 2(38)(24), 38² + 24²
1212-1616-2020 which is (3-4-5) times 404
1344-1508-2020 calculated from 2(42)(16), 42² – 16², 42² + 16²

2020 is the difference of two squares two different ways:
506² – 504² = 2020
106² – 96² = 2020
Can you calculate when 2020 is a leg in a Pythagorean triple from those equations? (The Pythagorean triples can be calculated from a² – b², 2(a)(b),  a² + b²)

Then can you calculate other times 2020 is a leg in a Pythagorean triple from these facts? (The Pythagorean triples can be calculated from 2(a)(b), a² – b²,  a² + b²)
2(1010)(1) = 2020
2(505)(2) = 2020
2(202)(5) = 2020
2(101)(10) = 2020
Did any of those equations produce the same Pythagorean triples that the difference of two squares produced?

2020 is the sum of four squares in MANY different ways. Here is how I found two of those ways:

That’s a lot of love for the number 2020. Have a wonderful year, everybody!

# Let’s Make a Factor Cake for 2020

We often celebrate special occasions with a cake!

Coincidentally, there is a method to find the prime factorization of a number that is called the cake method.

Let’s make a factor cake for the year 2020 to celebrate its arrival!

make science GIFs like this at MakeaGif
The factor cake shows that the prime factorization of 2020 is 2 × 2 × 5 × 101. We can write that more compactly: 2020 = 2² × 5 × 101.
In case you would like a still picture of the cake instead of the gif, here it is:
I will write more about the number 2020 before tomorrow. Enjoy saying good-bye to 2019 and getting ready for the new year!

# 1452 Poinsettia Plant Mystery

Merry Christmas, Everybody!

The poinsettia plant has a reputation for being poisonous, but it has never been a part of a whodunnit, and it never will. Poinsettias actually aren’t poisonous.

Multiplication tables might also have a reputation for being deadly, but they aren’t either, except maybe this one. Can you use logic to solve this puzzle without it killing you?

To solve the puzzle, you will need some multiplication facts that you probably DON’T have memorized. They can be found in the table below. Be careful! The more often a clue appears, the more trouble it can be:

Notice that the number 60 appears EIGHT times in that table. Lucky for you, it doesn’t appear even once in today’s puzzle!

Now I’d like to factor the puzzle number, 1452. Here are a few facts about that number:

1 + 4 + 5 + 2 = 12, which is divisible by 3, so 1452 is divisible by 3.
1 – 4 + 5 – 2 = 0, which is divisible by 11, so 1452 is divisible by 11.

• 1452 is a composite number.
• Prime factorization: 1452 = 2 × 2 × 3 × 11 × 11, which can be written 1452 = 2² × 3 × 11²
• 1452 has at least one exponent greater than 1 in its prime factorization so √1452 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1452 = (√484)(√3) = 22√3
• The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 1452 has exactly 18 factors.
• The factors of 1452 are outlined with their factor pair partners in the graphic below.

To commemorate the season, here’s a factor tree for 1452:

Have a very happy holiday!

# 1451 Star of Wonder

An important part of Mathematics is noticing patterns. I love it when mathematicians ask students, “What do you notice? What do you wonder?”

Those are questions you can ponder as you gaze on this star of wonder made from several different graphs.

To help distinguish the graphs, the dotted lines are exponential functions, the dashed lines are natural logarithm functions, and the solid lines are linear functions.

What do you notice? What do you wonder?

Now I’ll tell you a little bit about the post number, 1451:

• 1451 is a prime number.
• Prime factorization: 1451 is prime.
• 1451 has no exponents greater than 1 in its prime factorization, so √1451 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1451 has exactly 2 factors.
• The factors of 1451 are outlined with their factor pair partners in the graphic below.

How do we know that 1451 is a prime number? If 1451 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1451. Since 1451 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1451 is a prime number.

# 1450 A Pair of Factor Trees

On today’s puzzle, there are two small Christmas trees. Will two smaller trees on the puzzle be easier to solve than one big one? You’ll have to try it to know!

Every puzzle has a puzzle number to distinguish it from the others. Here are some facts about this puzzle number, 1450:

• 1450 is a composite number.
• Prime factorization: 1450 = 2 × 5 × 5 × 29, which can be written 1450 = 2 × 5² × 29
• 1450 has at least one exponent greater than 1 in its prime factorization so √1450 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1450 = (√25)(√58) = 5√58
• The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1450 has exactly 12 factors.
• The factors of 1450 are outlined with their factor pair partners in the graphic below.

In case you are looking for factor trees for 1450, here are two different ones:

1450 is the hypotenuse of SEVEN Pythagorean triples:
170-1440-1450 which is 10 times (17-144-145)
240-1430-1450 which is 10 times (24-143-145)
406-1392-1450 which is (7-24-25) times 58
666-1288-1450 which is 2 times (333-644-725)
728-1254-1450 which is 2 times (364-627-725)
870-1160-1450 which is (3-4-5) times 290
1000-1050-1450 which is (20-21-29) times 50