1457 and Level 3

Today’s Puzzle:

What numbers are common factors of 35 and 30? That’s the first question you need to solve this puzzle. All that’s left afterward is to look at the clues in the puzzle from top to bottom and write their factors in the factor column and the factor row. Remember, only use numbers from 1 to 10 as factors.

Factors of 1457:

  • 1457 is a composite number.
  • Prime factorization: 1457 = 31 × 47
  • 1457 has no exponents greater than 1 in its prime factorization, so √1457 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 1457 has exactly 4 factors.
  • The factors of 1457 are outlined with their factor pair partners in the graphic below.

Facts about the number 1457:

1457 is the difference of two squares in two different ways:
729² – 728² = 1457
39² – 8² = 1457

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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

 

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:

How about these optical illusion?

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!

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!
2020 Factor Cake

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!

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.

1449 Christmas Star

If you’ve ever wished you knew the multiplication table better, then make that wish upon this Christmas star. If you use logic and don’t give up,  then you can watch your wish come true!

I number the puzzles to distinguish them from one another. That star puzzle is way too big for a factor tree made with its puzzle number:

Here’s more about the number 1449:

  • Prime factorization: 1449 = 3 × 3 × 7 × 23, which can be written 1449 = 3² × 7 × 23
  • 1449 has at least one exponent greater than 1 in its prime factorization so √1449 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1449 = (√9)(√161) = 3√161
  • 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 1449 has exactly 12 factors.
  • The factors of 1449 are outlined with their factor pair partners in the graphic below.

1449 is the difference of two squares in 6 different ways:
725² – 724² = 1449
243² – 240² = 1449
107²-100² = 1449
85² – 76² = 1449
45² – 24² = 1449
43² – 20² = 1449

1448 Christmas Factor Tree

Here’s a puzzle that looks a little like a Christmas tree. Some of the clues might give you a little bit of trouble. For example, the common factor of 60 and 30 might be 5, 6, or 10. Likewise, the common factor of 8 and 4 might be 1, 2, or 4.

Which factor should you use? Look at all the other clues and use logic. Logic can help you write each of the numbers 1 to 12 in both the first column and the top row so that the given clues and those numbers behave like a multiplication table. Good luck!

I have to number every puzzle. It won’t help you solve the puzzle, but here are some facts about the number 1448:

The number made by its last two digits, 48, is divisible by 4, so 1448 is also divisible by 4. That fact can give us the first couple of branches of 1448’s factor tree:

  • 1448 is a composite number.
  • Prime factorization: 1448 = 2 × 2 × 2 × 181, which can be written 1448 = 2³ × 181
  • 1448 has at least one exponent greater than 1 in its prime factorization so √1448 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1448 = (√4)(√362) = 2√362
  • 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 1448 has exactly 8 factors.
  • The factors of 1448 are outlined with their factor pair partners in the graphic below.

1448 is also the hypotenuse of a Pythagorean triple:
152-1440-1448 which is 8 times (19-180-181)

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

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.