The 168th Playful Math Education Blog Carnival

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A Couple of Carnival Puzzles for You to Solve:

Puzzle one: There are 168 prime numbers less than 1000, and four of them are consecutive primes that add up to 168. Can you figure out what those four prime numbers are? Hint: The average of the four prime numbers will be 168÷4.

Puzzle two: How many pips (dots) are on this regular set of 28 dominoes?

I’ve grouped the dominoes into groups with six dots. How many groups of six are there? Multiply six by the number of groups and you will know the number of dots. You can also count them by using this cool triangular display from Wikipedia.

Five Amazing Facts about the Number 168:

168 isn’t perfect, but it is the product of the first two perfect numbers, 6 and 28.

Most of us recognize that 168 is one less than a perfect square, so 13² – 1² = 168, but did you know that 168 is also the difference of two squares in three other ways?
43² – 41² = 23² – 19² = 17² – 11² = 168.

Here’s another square way to make 168:
18² – 17² + 16² – 15² + 14² – 13² + 12² – 11² + 10² – 9² + 8² – 7² + 6² – 5² + 4² – 3² = 168.

2¹⁶⁸ = 374144419156711147060143317175368453031918731001856. Look it over. You won’t see a “2” anywhere in that 50-digit number. 2¹⁶⁸ is the largest known power of 2 not to contain each of the 10 digits at least once.

168 is a repdigit in several other bases:
168₁₀ = CC₁₃ = 88₂₀ =77₂₃ = 66₂₇ = 44₄₁ = 33₅₅ = 22₈₃ = 11₁₆₇ = 168₁₀.

More Domino Fun:

The DewWool blog has made ten free domino worksheets for kindergarten and first-grade students to practice addition.

You can also use a double nine set of dominoes to solve an algebra problem:

And what about using dominoes to add fractions?

Autumn Math:

If your town has a scarecrow trail, you can add some learning while you take the kids around. They can learn about many subjects including mathematics as they do the activities suggested in this blog post by Curious Kids 101.

Apples have several lessons inside of them. AdamPetersonEducation suggests using apple seeds as manipulatives in Apple Activities in the Classroom.

Math Story Books:

Writer’s Rumpus’s post about a brand-new picture book, Smarty Ants, by Corey Rosen Schwartz and Kirsti Call, illustrated by Erin Taylor. The blog post includes pictures that clearly show some of the math involved in what looks like a delightful story. There’s even a song for your students to enjoy.

Way Past Embarrassed by Hailey Adelman is a picture book that will help students who are too embarrassed to ask for help in math class to speak up to get the help they need.

Tara Lazar is a children’s book writer, but she also shares other books on her blog that her readers will enjoy. Read her post and you will know what inspired Erin Dealey to write a great introduction to fractions, The Half Birthday Book. The whimsical illustrations were drawn by Germán Blanco.

Bound for Escapes Reviews The Math Kids: An Incorrect Solution, and says that “whether a child likes math or not, this is a fun story.”

Pages Unbound Reviews Talia’s Codebook for Mathletes by Marissa Moss. This is a book that is sure to appeal to students in middle grades as Talia navigates middle school, refusing to give in to those who doubt her mathematical abilities simply because she is a girl.

A Kid’s Book a Day reviewed The Probability of Everything by Sara Everett. This book will engage 4th – 7th graders in discussions about probable and improbable events. Pelicans and Prose reviewed the same book and described The Probability of Everything as one of the most powerfully written books one could read.

The Christian Fiction Girl reviewed Calculated, by Nova McBee. It is a young adult novel that’s a blend of Mission Impossible and The Count of Monte Cristo.

Numbers on the Number Line:

Positlive compares where we are in life to additions and subtractions of positive or negative numbers on a number line in Life’s Simple Equation: Number Lines, Arithmetic, and Five Life Lessons.

Poetry for Finding Meaning in the Madness, Just Poetry wrote Odd, a poem about the life of an odd number.

You can read or listen to Keith’s Ramblings story, Unlucky Numbers. How many ways can the number 13 be partitioned anyway?

Help from Heaven has a completely different take on that number in 13 is My Lucky Number.

You’ll have to scroll down to get to the 4th-grade math sections in this post from Teaching to the Beat of a Different Drummer, but it will be well worth it. On day 3 there is a Me in Numbers activity, on day 4 a rounding to the nearest hundred activity, on day 5 a doubling single digit numbers game, and on day 6 a mind-blowing partial sums strategy.

Writing from the Heart with Brian notes that 50000 doesn’t look much different from 50000000 until you can actually relate something to each of the numbers. This insight may be helpful to your students.

The Internet Effect also discusses relative place value in The Illusory  Arithmetic: A Tale of $30 K and $300 K.

Exponential Growth:

A Berg’s Eye View shares a priceless meme to help your students remember the meaning of Exponential Growth.

Lino Matteo will open students’ eyes with the role of exponential growth in Financial Literacy: Housing.

Sumant shows his work for this SAT question on Exponential Growth.

Mathematical Games and Puzzles:

Math File Folder Games has directions to play a drawing game that will reinforce several geometry concepts AND lists the standards that the game can cover.

The Bad Mathematics blog discusses the card game 24 where the players must use addition, subtraction, multiplication, and/or division on the 4 cards they are dealt to arrive at the number 24.

The Napier Local Arithmetic Board will feel like a game as students use it as they peg to multiply numbers.

Puzzle a Day invites you to determine where is the only safe place to stand in 1500 People in a Circle. Any guess is more likely fatal than not.

A safer puzzle to answer is Puzzle a Day’s The Mystery Middle Digit.

Quadratablog.blogspot has made a sudoku puzzle that is mathier than most in a Small Multiples Sudoku.

Geometry:

The world’s geography in geometric shapes:

The Craft of Coding very much enjoyed writing code to calculate the volume of a gugelhupf pan (what I would call an angel food cake pan.) The post explains the advantages of baking when the volume of such a pan is known.

Graphs and The Coordinate Plane:

Here’s my contribution to the carnival: I made a polygonal elephant using the Cartesian coordinate system. Using Desmos, I was able to do several reflections in mere seconds: I reflected the elephant over the x-axis and the y-axis, and then reflected the reflections into the 3rd quadrant.

Nicholas C. Rossis blogged a little about the Cartesian coordinate system and shared a hilarious video in The Shape of Stories, According to Kurt Vonnegut.

Roro’s Adventurous Blog discusses vectors in Fun with Random Numbers and Matrices.

The Golden Ratio and Fibonacci:

Puzzle a Day asks that we analyze the numbers in the Fibonacci sequence in a Fibonacci puzzle.

Nature Notes the Arb shares pictures of Fibonacci numbers in nature.

On a more serious note, HTT Network has written a beautiful post Unveiling the Enigmatic Golden Ration: Unlocking the Secrets of Beauty and Harmony.

The Graphxhub blog gives us How Math in Graphic Design Adds up to Stunning Visuals. This post helped me understand the golden ratio in art much better, but it covers so much more than that.

Lego Math:

Shapes in Blue has created some lovely blue geometric Lego art you’ll want to see.

Alison Kiddle used Legos to make 31 math conversation starters, one for each day of the month of August. These three are only a sample. Check them all out!

Multiplication:

How about a Multiplication Magic Show from ActiveWordZ to help students learn the 2 and 3 times tables? It sounds pretty exciting to me!

The Reflections and Tangents blog teaches Area Arrangements going from one-digit multiplication problems to multiplying numbers with several place values, to fraction multiplication, and even to multiplying polynomials.

Mathematics History Museum:

Learn the history of the Rubik’s cube as you walk through MillenialMatriarch’s post on Rubik’s Magic Cube.

Sheryl, a Hundred Years Ago explains how recommended calories were calculated in 1923 and compares it to the method used today.

Cedric School of Thought describes the difficult life of self-taught mathematician, Nicolo Tartaglia, in the 16th century.

Visit Live Life King Size for a lovely presentation, What is the History of Maths? It is perfect for our Museum of Mathematics.

Math Through Music:

I saved my favorite for last:

Want more? Playlistideas has compiled a playlist of artist-produced songs about math just for us all to enjoy!

Mathematical Optical Illusions and Photography:

What carnival would be complete without a house of mirrors?

Zsolt Zsemba wrote a post explaining why a few mathematical designs turn into optical illusions.

Mathematics and art combine to make a rug that looks dangerous to step on.

Click on the video at Terryorism’s blog and watch some squares look like they’re moving, but they aren’t.

Enchanted Seashells shared these squares that are drawn to look like a stairway that changes direction while you’re looking at it.

Let’s Write… saw a picture of real stairs that look like they could be going up just as easily as they are going down and wrote about it.

Robert’s Snap Spot shared An Optical Illusion at Shelter Cove with some cool shadows. This photo really exemplifies perspective in art as well as in mathematics.

Travel with me takes us to a photo gallery highlighting triangles in each frame.

The Other Life of This Math Teacher is Photography: See the geometry in
Make Friends, Have Fun,
in Penta-licious,
and in a Chinese Pagoda.

Paper Folding, Origami, and Tangrams:

Carter in the Classroom reported to an NBC affiliate how A Texas Teacher Uses Origami to Teach Math (and a Growth Mindset).

Paula Beardell Krieg has produced some wonderful blogposts on paper folding this summer:

Paula Beardell Krieg also wrote several wonderful posts on exploring geometry through tangram play.

Geometric Designs and Tessellations:

RobertLovesPi regularly makes and shares tessellations of his blog. His tessellation of octogons, rhombi, and darts is very pretty. Be sure to check out his tessellation which includes heptagons! I don’t think they show up in other people’s tessellations very often.

You may have heard of the newly discovered tessellating hat, or maybe even the new tessellating turtle, but have you heard of David Smith’s tessellating Spectre?

Pascal’s Prism teaches us about the hidden math behind art: the fascinating geometry of tessellations.

Geometric designs might be done with plastic, paper, or cloth. Pieced quilts can certainly bring pleasure to those who make them as well as those who look at them. Take a look at these recent blog posts featuring some lovely quilted geometric designs and tessellations:

  1. Sailing Ships made from circles and squares,
  2. Intersecting Circles,
  3. Triangles,
  4. A Sampler with a variety of designs, any of which could tessellate an entire quilt.
  5. A Design made with squares in a couple of different sizes.
  6. Parallelograms (Scroll down a little to see them.),
  7. A Square and Triangle design with a free pattern and instructions,
  8. A Flower made with mini hexagons,
  9. Maple Leaf (very nice for autumn),
  10. A Mystery Halloween Quilt,
  11. A Halloween Lap Quilt,

Sets

Read and Play All Day will help us all put objects into sets with Sorting and Ordering by Size: A Precursor of Ordering Numbers.

Elorine takes 5th-graders on a few adventures. The first is Unleashing the Power of Sets, A Magical Math adventure.

The second is a virtual field trip, Exploring the Wonderful World of Sets. Find Treasures Within.

The third is Exploring the Magic of Union Sets in Mathematics.

The fourth is The Marvelous World of Equivalent Sets.

Probability and Statistics:

Wind Kisses discusses the likelihood of certain events and shares some inspirational quotes in Likely or Unlikely.

There are 23 players on a soccer team, what is the probability that at least two of the players have the same birthday? That question was examined at the Women’s World Cup, and the answer may surprise you!

Statistical Odds and Ends asks, “What is the probability of getting all heads on multiple coin flips?”

Every Saturday Nebushumor publishes a funny statistics post. For example, take a look at How I Use My Recreational Time. You and/or your students can probably relate to it. Or how about Fun, by Decade. I think he’s spot on.

Thoughts on Teaching Mathematics to All Students:

Denise Gaskins is very impressed with a fully developed, revolutionary program that teaches Algebra before Arithmetic and wishes she had access to it 30 years ago when her kids were young.

You can read the importance of a growth mindset from a parent’s perspective in The Power of Yet.

Heidi Allum asks if Math Play can be a Part of Trauma-Informed Care? I’m sure you will be interested in learning how math play affects all students including those who have experienced trauma.

Don’t let the solved calculus problem at the beginning of the post intimidate you, but Alternative Amie has several insights into the benefits of melding the arts with the teaching of mathematics.

Other Carnivals with Mathematics:

Every month The Aperiodical blog also coordinates a math carnival that includes college-level mathematics. September’s carnival is at Reflections and Tangents.

The last Playful Math Carnival was at Learning Well at Home. The next one will be at Math Hombre. How about a future carnival being on your blog? Volunteers are needed and welcome! Coordinate with Denise Gaskins for a month that is convenient to you by going to the Playful Math Carnival Volunteer Page.

1766 A Polygonal Elephant for World Elephant Day

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Today’s Puzzle:

Today many people are reflecting on the plight of elephants throughout the world and what we can do to protect them and their habitats. Yesterday I created an image of a lone elephant in Desmos:

Today’s puzzle is to take the Desmos Elephant image and transform it as I have done below. There is no need to retype all the ordered pairs that made the elephant. One of the reflections was made simply by typing “polygon(x1,-y1)” below all the order pairs in the Desmos image. Can you determine which reflection that was? What was typed inside the parenthesis to make the other two elephants?

How would the elephant be transformed if you typed “polygon(-y1,x1)”, “polygon(x1+5,y1)” or  “polygon(2×1,2y1)” below the ordered pairs in Desmos? Can you figure out a way to shrink the elephant and make it upside down in the first quadrant? I hope you will take the time to check out these transformations and experiment with some of your own.

Factors of 1766:

  • 1766 is a composite number.
  • Prime factorization: 1766 = 2 × 883.
  • 1766 has no exponents greater than 1 in its prime factorization, so √1766 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 1766 has exactly 4 factors.
  • The factors of 1766 are outlined with their factor pair partners in the graphic below.

More About the Number 1766:

1766 looks interesting in some other bases:
It’s 6E6 in base 16 because 6(16²) + 14(16) + 6(1) = 1766,
272 in base 28 because 2(28²) + 7(28) + 2(1) = 1766, and
123 in base 41 because 1(41²) + 2(41) + 3(1) = 1766.

1765 On This Memorial Day

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Today’s Puzzle:

This weekend I laid a bouquet of red and white flowers on my husband’s grave and decided to make a red rose Memorial Day puzzle for the blog as well. It is a mystery-level puzzle.

Write the number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. There is only one solution.

Factors of 1765:

  • 1765 is a composite number.
  • Prime factorization: 1765 = 5 × 353.
  • 1765 has no exponents greater than 1 in its prime factorization, so √1765 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 1765 has exactly 4 factors.
  • The factors of 1765 are outlined with their factor pair partners in the graphic below.

More About the Number 1765:

1765 is the sum of two squares in two different ways:
42² + 1² = 1765, and
33² + 26² = 1765.

1765 is the hypotenuse of FOUR Pythagorean triples:
84 1763 1765, calculated from 2(42)(1), 42² – 1², 42² + 1²,
413 1716 1765, calculated from 33² – 26², 2(33)(26), 33² + 26²,
1059-1412-1765, which is (3-4-5) times 353, and
1125-1360-1765, which is 5 times (225-272-353).

1765 is a digitally powerful number:
1⁴ + 7³ + 6⁴ + 5³ = 1765.

1765 is a palindrome in a couple of different bases:
It’s A5A base 13 because 10(13²) + 5(13) + 10(1) = 1765, and
it’s 1D1 base 36 because 1(36²) + 13(36) + 1(1) = 1765.

1764 Perfect Squares Are Amazing!

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Today’s Puzzle:

1764 is a perfect square, and its last two digits are a perfect square, too. The same thing was true of the last perfect square, 1681. And it will be true of the next perfect square, 1849. These perfect squares got me curious about perfect squares in general, and I noticed something. Can you notice it, too?

Look at this graphic of perfect squares. What do you notice? What do you wonder?

I have known for years that the last digits of perfect squares follow a palindromic
0-1-4-9-6-5-6-9-4-1-0 pattern. The first time the backward repeating happens is at 5¹.

I had no idea that the last two digits followed a palindromic pattern, too! Notice that the first time the backward repeating of the last two digits happens is at (5²)² or 25². It also happens at 75², 125², and so forth. The forward repeating happens at 0², 50², 100², and so forth. Note that (-n)² = n², so this pattern goes on in both directions FOREVER!

Also if the last two digits of a number aren’t either 00, 25, or (even number)1, (even number)4, (even number)9,  or (odd number)6, then no way is it a perfect square!

I wondered if the last three digits would backward repeat at (5³)² or 125². It doesn’t. 🙁
But at 250², it DOES! Rather than make a graphic with all 250 rows, I just made one with the last 25 rows:

Will the last FOUR numbers start backward repeating at 2500²? I know the answer, but perhaps you would like to find out for yourself!

Factors of 1764:

  • 1764 is a composite number and a perfect square.
  • Prime factorization: 1764 = 2 × 2 × 3 × 3 × 7 × 7, which can be written 1764 = 2² × 3² × 7².
  • 1764 has at least one exponent greater than 1 in its prime factorization so √1764 can be simplified. √1764 = √(42 × 42) = 42.
  • The exponents in the prime factorization are 2, 2, and 2. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(2 + 1) = 3 × 3 × 3 = 27. Therefore 1764 has exactly 27 factors. (Only perfect squares can have an odd number of factors.)
  • The factors of 1764 are outlined with their factor pair partners in the graphic below.

More About the Number 1764:

Square number, 1764,  looks square in some other bases, too:
It’s 900 in base 14 because 9(14²) + 0(14) + 0(1) = 1764,
484 in base 20 because 4(20²) + 8(20) + 4(1) = 1764,
400 in base 21 because 4(21²) + 0(21) + 0(1) = 1764,
169 in base 39 because 1(39²) + 6(39) + 9(1) = 1764,
144 in base 40 because 1(40²) + 4(40) + 4(1) = 1764,
121 in base 41 because 1(41²) + 2(41) + 1(1) = 1764, and
100 in base 42 because 1(42²) + 0(42) + 0(1) = 1764.

1763 Daffodil Puzzle

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Today’s Puzzle:

Spring has sprung and perhaps flowers are blooming in your area. I think my favorite flowers are daffodils. I love the way they are shaped and their vibrant colors.

This daffodil puzzle is a great way to welcome spring. It may be a little bit tricky, but I think if you carefully use logic you will succeed! Just write each of the numbers 1 to 12 in the first column and again in the top row so that those numbers are the factors of the given clues. As always there is only one solution.

Here’s the same puzzle if you’d like to print it using less ink:

Factors of 1763:

  • 1763 is a composite number and the product of twin primes.
  • Prime factorization: 1763 = 41 × 43.
  • 1763 has no exponents greater than 1 in its prime factorization, so √1763 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 1763 has exactly 4 factors.
  • The factors of 1763 are outlined with their factor pair partners in the graphic below.

More About the Number 1763:

1763 is the difference of two squares in two different ways:
882² – 881² = 1763, and
42² – 1² = 1763. (That means the next number will be a perfect square!)

1763 is the hypotenuse of a Pythagorean triple:
387-1720-1763, which is (9-40-41) times 43.

1763 is palindrome 3E3 in base 22 because
3(22²) + 14(22) + 3(1) = 1763.

Lastly and most significantly: 15, 35, 143, 323, 899, and 1763 begin the list of numbers that are the product of twin primes. 1763 is just the sixth number on that list! If we include the products of two consecutive primes whether they are twin primes or not, the list is still fairly small. How rarely does that happen?

When it was 2021, did you realize how significant that year was?

1762 Happy Saint Patrick’s Day!

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Today’s ♣ Puzzle:

Here’s a much easier puzzle than yesterday’s for you to enjoy on this Saint Patrick’s Day. The diagonal lines on the corner boxes are only to help define the leaves of the shamrock.

Factors of 1762:

  • 1762 is a composite number.
  • Prime factorization: 1762 = 2 × 881.
  • 1762 has no exponents greater than 1 in its prime factorization, so √1762 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 1762 has exactly 4 factors.
  • The factors of 1762 are outlined with their factor pair partners in the graphic below.

More About the Number 1762:

1762 is the sum of two squares:
41² + 9² = 1762.

1762 is the hypotenuse of a Pythagorean triple:
738-1600-1762 calculated from 2(41)(9), 41² – 9², 41² + 9².
It is also 2 times (369-800-881).

1762 is palindrome 7C7 in base 15
because 7(15²) + 12(15) + 7(1) = 1762.

1761 Irish Harp

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Today’s Puzzle:

This mystery-level puzzle was meant to look a little like an Irish harp. Using logic write the numbers 1 to 12 in the first column and again in the top row so that those numbers and the given clues make a multiplication table. There is only one solution.

Notice that the clues 16 and 24 appear THREE times in the puzzle. In each case, you will need to determine if the common factor is 2, 4, or 8. You will have to get the common factor for each one in the right place or it will cause trouble for another clue. Consider what problems each of the following scenarios bring to other clues. For example, 48 must be either 4 × 12 or 6 × 8, but both possibilities are impossible in at least one of these scenarios:

Once you determine the only scenario that doesn’t present a problem for any other clue, you will be able to begin the puzzle.

Here’s the same puzzle without any added color:

Factors of 1761:

1 + 7 + 6 + 1 = 15, a number divisible by 3, so 1761 is divisible by 3. Since 6 is divisible by 3, we didn’t have to include it in our sum: 1 + 7 + 1 = 9, so 1761 is divisible by 3.

  • 1761 is a composite number.
  • Prime factorization: 1761 = 3 × 587.
  • 1761 has no exponents greater than 1 in its prime factorization, so √1761 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 1761 has exactly 4 factors.
  • The factors of 1761 are outlined with their factor pair partners in the graphic below.

More About the Number 1761:

1761 is the difference of two squares in two different ways:
881² – 880² = 1761, and
295² – 292² = 1761.

1761 is palindrome 1N1 in base 32
because 1(32²) + 23(32) + 1(1) = 1761.

1760 Pots of Gold and Rainbows

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Today’s Puzzle:

Write the numbers 1 to 12 in the first column and again in the top row so that those numbers are the factors that make the given clues. It’s a level 6, so it won’t be easy. Finding a leprechaun’s pot of gold isn’t easy either. Still, if you can solve this puzzle, then you will have found some real golden nuggets of knowledge.

They say there’s a pot of gold at the end of the rainbow. Where’s the rainbow?

Factors of 1760:

Puzzle number, 1760, has many factors. It makes a very big factor rainbow!

  • 1760 is a composite number.
  • Prime factorization: 1760 = 2 × 2 × 2 × 2 × 2 × 5 × 11, which can be written 1760 = 2⁵ × 5 × 11.
  • 1760 has at least one exponent greater than 1 in its prime factorization so √1760 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1760 = (√16)(√110) = 4√110.
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1760 has exactly 24 factors.
  • The factors of 1760 are outlined with their factor pair partners in the graphic below.

More About the Number 1760:

1760 is the hypotenuse of a Pythagorean triple:
1056-1408-1760 which is (3-4-5) times 352.

1760 is the difference of two squares in eight different ways:
441² – 439² = 1760,
222² – 218² = 1760,
114² – 106² = 1760,
93² – 83² = 1760,
63² – 47² = 1760,
54² – 34² = 1760,
51² – 29² = 1760,  and
42² – 2² = 1760. (That means we are only four numbers away from the next perfect square!)

1760 is palindrome 2102012 in base 3
because 2(3⁶)+1(3⁵)+ 0(3⁴)+2(3³)+0(3²)+1(3¹)+2(3º) = 1760.

1759 Pie Over Two

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Today’s Puzzle:

Today in the United States many students will celebrate pi day by eating pie. Today’s puzzle looks a little like a pie that has been cut in half, so I’m calling it pie over two, abbreviated as “π/2”.

Write the numbers 1 to 12 in the first column and again in the top row so that those numbers and the given clues make a multiplication table. Be sure to use logic every step of the way.

Factors of 1759:

  • 1759 is a prime number.
  • Prime factorization: 1759 is prime.
  • 1759 has no exponents greater than 1 in its prime factorization, so √1759 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1759 has exactly 2 factors.
  • The factors of 1759 are outlined with their factor pair partners in the graphic below.

How do we know that 1759 is a prime number? If 1759 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1759. Since 1759 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1759 is a prime number.

More About the Number 1759:

Like every other odd number, 1759 is the difference of two squares:
880² – 879² = 1759.

OEIS.org informs us that 1759 is only the 17th Eisenstein-Mersenne prime number.

1758 Two-Shillelagh O’Sullivan

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Today’s Puzzle:

When I was looking for the song about the shillelagh for my previous post, I found another one called Two-Shillelagh O’Sullivan also by Bing Crosby. It wasn’t a song from my childhood, but it inspired me to make a puzzle with two shillelaghs anyway. In the song, O’Sullivan wears these walking sticks in a holster and can draw them quicker than anyone can draw a gun. He was impossible to beat.

This two-shillelagh puzzle is also a bit difficult to beat. You’re not going to let that stop you from trying, are you? Just use logic and your knowledge of the multiplication table.

Write the numbers from 1 to 12 in the first column and again in the top row so that those numbers are the factors of the given clues.

Here’s the same puzzle in black and white:

Factors of 1758:

  • 1758 is a composite number.
  • Prime factorization: 1758 = 2 × 3 × 293.
  • 1758 has no exponents greater than 1 in its prime factorization, so √1758 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 1758 has exactly 8 factors.
  • The factors of 1758 are outlined with their factor pair partners in the graphic below.

More About the Number 1758:

1758 is the hypotenuse of a Pythagorean triple:
408-1710-1758, which is 6 times (68-285-293).

1758 is palindrome 8E8 in base 14
because 8(14²) + 13(14) + 8(1) = 1758.