1733 and Its Prime Number Substrings

Today’s Puzzle:

Use logic to write each 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.

Factors of 1733:

  • 1733 is a prime number.
  • Prime factorization: 1733 is prime.
  • 1733 has no exponents greater than 1 in its prime factorization, so √1733 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1733 has exactly 2 factors.
  • The factors of 1733 are outlined with their factor pair partners in the graphic below.

How do we know that 1733 is a prime number? If 1733 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1733. Since 1733 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1733 is a prime number.

More About the Number 1733:

1733 is the sum of two squares:
38² + 17² = 1733.

1733 is the hypotenuse of a Pythagorean triple:
1155-1292-1733 calculated from 38² – 17², 2(38)(17), 38² + 17².

Here’s another way we know that 1733 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 17² = 1733 with 38 and 17 having no common prime factors, 1733 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1733. Since 1733 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1733 is a prime number.

1733 is also the difference of two squares:
867² – 866² = 1733.
That means 1733 is also the short leg of the Pythagorean triple calculated from
867² – 866², 2(867)(866), 867² + 866².

1733 is a palindrome in three bases:
It’s 2101012 in base 3 because
2(3⁶) +1(3⁵) + 0(3⁴) + 1(3³) + 0(3²) +1(3¹) + 2(3⁰) = 1733,
It’s 565 in base 18 because 5(18²) + 6(18) + 5(1) = 1733, and
it’s 4F4 in base 19 because 4(19²) + 15(19) + 4(1) = 1733.

The next fact I learned from Twitter:

What prime numbers can be made with the digits of 1733?
3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 313, 317, 331, 337, 373, 733, 1733.

The numbers in red are the 6 prime numbers that are substrings of 1733. I made a gif showing those 6 primes:

1733 Smallest Prime with 6 Prime Substrings

make science GIFs like this at MakeaGif
1733 is quite a fascinating number!

1729 Not a Dull Number

Today’s Puzzle:

A little more than a hundred years ago near Cambridge University G. H. Hardy took a taxi to visit his young friend and fellow mathematician, Srinivasa Ramanujan, in the hospital. Hardy couldn’t think of anything interesting about his taxi number, 1729, and remarked to Ramanujan that it appeared to be a rather dull number. But even the reason for his hospitalization could not prevent Ramanujan’s genius from shining through. He immediately recognized 1729’s unique and very interesting attribute: it is the SMALLEST number that can be written as the sum of two cubes in two different ways! Indeed,
12³ + 1³ = 1729, and
10³ + 9³ = 1729.

Today’s puzzle looks a little bit like a modern-day American taxi cab with the clues 17 and 29 at the top of the cab. The table below the puzzle contains all the Pythagorean triples with hypotenuses less than 100 sorted by legs and by hypotenuses. Use the table and logic to write the missing sides of the triangles in the puzzle. The right angle on each triangle is the only one that is marked. Obviously, none of the triangles are drawn to scale. No Pythagorean triple will appear more than once in the puzzle.

Sorted TriplesHere’s the same puzzle without all the added color:

Print the puzzles or type the solutions in this excel file: 10 Factors 1721-1729.

What taxi cab might Hardy have tried to catch next? He might have had to wait a long time for it, 4104.
16³ + 2³ = 4104, and
15³ + 9³ = 4104.

Factors of 1729:

  • 1729 is a composite number.
  • Prime factorization: 1729 = 7 × 13 × 19.
  • 1729 has no exponents greater than 1 in its prime factorization, so √1729 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 1729 has exactly 8 factors.
  • The factors of 1729 are outlined with their factor pair partners in the graphic below.

More About the Number 1729:

Did you notice these cool-looking factor pairs?
19 · 91 = 1729.
13 · 133 = 1729.

1729 is the hypotenuse of a Pythagorean triple:
665-1596-1729 which is 133 times (5-12-13).

1729 is the difference of two squares in four different ways:
865² – 864² = 1729,
127² – 120² = 1729,
73² – 60² = 1729, and
55² – 36² = 1729.

1729 is also a pseudoprime number! For example, even though it is a composite number, it passes the quick prime number test I first wrote about in 341 is the smallest composite number that gives a false positive for this Quick Prime Number Test:

It is known as a Carmichael number because back in 1909, Robert D. Carmichael, an American mathematician, showed that it passed even more prime number tests than this one.

1727 A Happy Birthday Cake for Jo Morgan

Today’s Puzzle:

Today is Jo Morgan’s birthday, so I made her a cake!

Use logic to find the factors in the cake. There is only one solution. Yesterday’s puzzle was pretty easy at least to get started. This one won’t be. You’ll have to use logic just to figure out which set of numbers belongs in the top row, 1-12 OR 13-24, and which set belongs in the first column. Not only that but both 126 and 180 appear THREE times in the puzzle! Have a party figuring it out!

Factors of 1727:

1 – 7 + 2 – 7 = -11, a multiple of 11, so 1727 is divisible by 11.

  • 1727 is a composite number.
  • Prime factorization: 1727 = 11 × 157.
  • 1727 has no exponents greater than 1 in its prime factorization, so √1727 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 1727 has exactly 4 factors.
  • The factors of 1727 are outlined with their factor pair partners in the graphic below.

More About the Number 1727:

1727 is the hypotenuse of a Pythagorean triple:
935-1452-1727, which is 11 times (85-132-157).

From OEIS we learn that
1727 = 12³ – 1³, and
7271 = 20³ – 9³.
Thus 1727 forward and backward is the difference of two cubes!

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.)

Countdown to 2023

make science GIFs like this at MakeaGif
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:


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.


1722 Mikulás, Fill My Boot With Goodies!

Today’s Puzzle:

Today is December 5, so children in Hungary and some other countries will shine their boots and put them out for Mikulás (Saint Nick) to fill with candies. Can you solve this boot puzzle? Write the numbers 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write.

Here is a song children will sing in Hungary today. You are already familiar with the tune. This particular video includes subtitles.

Here are the lyrics in Hungarian and Google’s translation into English:

Hull a pelyhes fehér hó, jöjj el kedves Télapó!
Minden gyermek várva vár, vidám ének hangja száll.
Van zsákodban minden jó, piros alma, mogyoró,
Jöjj el hozzánk, várunk rád, kedves öreg Télapó.

Nagy szakállú Télapó jó gyermek barátja.
Cukrot,diót, mogyorót rejteget a zsákja.
Amerre jár reggelig kis cipőcske megtelik,
megtölti a Télapó, ha üresen látja!

The fluffy white snow is falling, come dear Santa!
Every child waits expectantly, the sound of happy singing is heard.
You have everything good in your bag, red apples, hazelnuts,
Come to us, we are waiting for you, dear old Santa Claus.

Santa Claus with a big beard is a good friend of children.
His sack hides sugar, walnuts and hazelnuts.
Wherever you go, your little shoe will be full by morning,
Santa will fill it if he sees it empty!

Factors of 1722:

  • 1722 is a composite number.
  • Prime factorization: 1722 = 2 × 3 × 7 × 41.
  • 1722 has no exponents greater than 1 in its prime factorization, so √1722 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 1722 has exactly 16 factors.
  • The factors of 1722 are outlined with their factor pair partners in the graphic below.

More About the Number 1722:

Since 1722 = 41·42, we know that 1722 is the sum of the first 41 EVEN numbers. It is two times the 41st triangular number.

1719 A Meaningful Statistics Project

What Is This Statistics Project?

FamilySearch.org and Ancestry.com have a statistics project that you can be involved with. It’s indexing the 1950 census. This project is unlike any other project I’ve seen. A computer has already indexed the census, and they just want humans to verify that the computer did it right. Also not only can you choose the state, but also the surnames that you verify. The project has been going on for about three months already, but I didn’t look into it until the middle of May.  My home state, Nevada, is 100% done so I missed out on verifying my family’s data. Instead, I tried to find my husband’s family. They lived in Ohio, but some of them moved to California in 1950. I wasn’t sure which month they moved. The program asked me if I wanted to find a particular surname. I chose Sallay a few times in both Ohio and California, and I indexed whatever Sallay person I saw and their entire household. Sometimes I did their neighbors, but most of the time I didn’t. On about the tenth try, this page came up:

I was so thrilled. The family in blue is my husband’s grandfather, grandmother, and Uncle Paul.  His grandparents had died before I ever met my husband, but I have read his grandfather’s journal, and I feel like I know him and his wife pretty well. My husband’s Uncle Paul was very near and dear to my heart. For twenty years he was always very kind to my family whenever we visited him, and it was my privilege to move him into my home and be his primary caregiver for the last 7 1/2 years of his life.

I did not index the record. I had my husband do it. It was a sweet experience. Perhaps YOU have family members who were alive in the United States in 1950 that you could index. It could be one of the most meaningful statistics projects of your life!

How did the computer do indexing my husband’s family members? It got their names right, but it completely got their street name wrong. Since I knew the street name, I was able to change it from something like Ainberland to the correct name of Cumberland.

My father, mother, two sisters, and a brother are in the 1950 Census. They lived in the house that I would come home to shortly after my birth in a few more years. The thing that amazed me the most was our house number. I’ve known the street name my entire life, but I didn’t know the house number until today. For the first 5 years of my life, my house number was 2535, and that also just happens to be my house number for these last 28 1/2 years, too, and I don’t have plans to move anytime soon. I was so stunned at this revelation, that I called my sister who has been to my house many times. She knew the house numbers were the same, but never mentioned it to me because she figured I was so good with numbers that I already knew.

What will surprise you when you look at the 1950 Census? Indexing your own family can prevent errors. I wish I had indexed my family. I would not have said my sister was already a widow at five years of age or that my brother was born in Cavada. Find your family in the census and get their information right!

Now I’ll write a little about the number 1719 because this is my 1719th post.

Factors of 1719:

  • 1719 is a composite number.
  • Prime factorization: 1719 = 3 × 3 × 191, which can be written 1719 = 3² × 191.
  • 1719 has at least one exponent greater than 1 in its prime factorization so √1719 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1719 = (√9)(√191) = 3√191.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1719 has exactly 6 factors.
  • The factors of 1719 are outlined with their factor pair partners in the graphic below.

More About the Number 1719:

1719 is the difference of two squares in three different ways:
860² – 859² = 1719,
288² – 285² = 1719, and
100² – 91² = 1791.

1791 is the sum of nine consecutive odd numbers:
183 + 185 + 187 + 189 + 191 + 193 + 195 + 197 + 199 = 1719.
The numbers in red are prime numbers that form a prime decade.

The 153rd Playful Math Education Blog Carnival!

Welcome to the 153rd Playful Math Education Carnival! Thanks to those who blogged and/or tweeted about math, we have another fun-filled carnival this month. Since a picture is worth 1000 words, and tweets usually have lovely pictures and captions included with them, I’ve embedded a lot of tweets in this carnival. Many of the tweets include links to blog posts. You can be transported directly to any area of the carnival you desire by clicking one of the following links:

Ukraine and Math

I have been very upset about the recent events in Ukraine and wondered how I could possibly publish a cheery, playful carnival at this time.

I decided to publish the carnival but include a couple of blog posts that link math and Ukraine.

This first post is a poem about the current situation: Evil Adds Up.

The second post is about Voroinoi Diagrams and Ukrainian mathematician, Georgii Voronoi. I am glad to know a little bit about Goergii Voronoi now after reading that post.

A Fishing Pond

You can learn some fun math facts by reading blogs. A few years ago I read a post on the Math Online Tom Circle blog that made the number 153 unforgettable for me. 153 is known as the St. Peter Fish Number.

Now fishing pond booths are often a part of a traditional carnival so this 153rd edition of the Playful Math Education Blog Carnival just has to have one, too. Its fishing booth has 153 fish in it representing the 153 fish Simon Peter caught in John 21: 9-14.

I made the fish tessellate because tessellation is a cool mathematical concept. The fish form a triangle because 153 is a triangular number. I colored the fish to show that
153 = 5! + 4! + 3! + 2! + 1! I like that 5! is also a triangular number so I put it at the top of the triangle, but 1! and 3! are triangular numbers, too. Can you use addition on the graphic to show that 153 = 1³+5³+3³?

A Little Magic:

That same Tom Circle blog post also revealed the magician’s secret behind a potential math magic trick:
Pick a number, ANY number. Multiply it by 3. Then find the sum of the cubes of its digits. Find the sum of the cubes of the digits of that new number. I might have you repeat that last step a few times. I predict your final number is. . .

No matter what number you choose, I can accurately predict what your final number will be. If you open the sealed envelope in my hand, you will see that I did indeed predict your final number, 153.

Pat’s Blog teaches about another magical number property in Squares That Parrot Their Roots.

Creative Learning AfrikA+ writes about the secret of performing well on tests in It’s not Math Magic, It’s Consistency.

The Enchanted Tweeting Room

Speaking of the number 153, Jo Morgan recently published her 153rd Mathsgem post with many ideas from the Twitterverse:

In her 95th Monday Must-Reads blog post, Sara Carter shared some great ideas she saw on Twitter: a math word wall, some Desmos Gingerbread Houses, a Find-the-Imposter Spiderman Surds activity, A Polynomial Two Truths and a Lie game, and much more.

Math Art

Leonardo DeVinci and many other famous artists were also famous for their mathematics. Mathematics used to be considered a liberal art. Denise Gaskins encourages us to bring back the joy of learning math in part one of Rediscover the Liberal Art of Mathematics.

RobertLovesPi uses enneagrams, regular hexagons, and other polygons to make a lovely artistic design. He also creates a shimmering 3-D shape in A Faceted Rhombicosidocecahedron with 540 Faces.

Paula Beardell Krieg wrote about the experience of directly teaching paper folding and indirectly teaching mathematics over zoom for six weeks in A Lovely Experiment.

We can use Desmos to create stunning artwork:

Math Games

Every week Denise Gaskins shares a new game on her blog post Math Game Monday.

Julie Naturally shares some Awesome Free Math Games for Kindergartners, no electronics required.

Children at St Margaret’s Lee Church of England Primary School have been playing a domino game called the Mexican Train game. They like it so much that they’ve expressed the desire to play it at home with their families.

How much do children enjoy playing mathy board games? Just read this post by Jenorr73 of One Good Thing: Math Game Joy.

Ben Orlin of Math with Bad Drawings wrote about one of my favorite games, SET, in A Theory of Trios.

Claire Kreuz of NBC’s KARK.com blog reports that a 14-year-old High School student has developed a math game everyone can play even those with special needs.


Colleen Young blogs about a new publication by Jonny Griffiths in A-Level Starters.

This puzzle is my contribution to the carnival:

Here are some puzzles I found on Twitter:

Sarah Carter regularly shares puzzles that help us play with math:


Children’s Literature

You will want to read the responses to this next tweet. MANY biographies of mathematicians are mentioned:

Mathematical Poetry

What’s a Fib? It’s a poetry style beautifully explained by the Kitty Cats blog.

Molly Hogan of Nix the Comfort Zone wrote thoughtful poems about the Number Zero and How Many Snowflakes Were Seen out her window.

Catherine Flynn of Reading to the Core taught me about the Fibonacci style of writing poems in Poetry Fridays: Fractals, Fibonacci, and Beyond.

Count the syllables in Heidi Mordhorst of My Juicy Little Universe’s poem Jealousy and you will have counted down from nine to one.

MaryLee Hahn of A(another) Year of Reading makes us ponder our footprint in The Mathematics of Consumerism.

Number Sense

The Year One Class had a wonderful time playing with numbers as they put the numbers from 0 to 50 in their proper places on a number line and talked about number patterns.

Norah Colvin (Live, Laugh, Learn . . . Create the Possibilities blog) used easily stackable pancakes to help students have a better sense of how much 1000 is.

In Important ideas about addition, Tad Watanabe reminds us that children don’t necessarily understand concepts such as 30 being three tens. Students sometimes erroneously think of multi-digit numbers as “simply a collection of single-digit numbers that are somehow glued together.” He talks about what to do about these and a few other issues students face learning mathematics.

Jenna Laib of Embrace the Challenge observed that one of her students was having difficulty understanding negative numbers. Read what happened when she played a Tiny Number Game with her.

Multiplication and Factoring



Look at the pictures. You can tell that Mrs. Bracken’s class enjoyed exploring and discussing ways to display four squares and their reflections.

Check out Pat’s Blog So You Thought You Knew Everything About Equilateral Triangles.

Math History Museum

Health and First Aid Station

Math Anxiety can be a real health issue:

Math Teachers can experience a different type of Math Anxiety:

The Math Teacher Experience

What if a math lesson is fun but the concepts won’t be a major part of the end-of-year test? Pay attention to Melissa D of the Dean of Math blog post, It’s a fun unit, but it’s not really necessary.

Anna of iamamathteacher.blogspot.com shares how her school year has gone so far.

Robert Talbert discusses going from a grading-to-upgrading in an upper-level math course.

Football and Math

In Inequalities on the Gridiron, Dick Lipton and Ken Regan talk about why the Buffalo Bills weren’t in the Superbowl and whether or not the c in the inequality
a² – b² = c is positive or negative.

Check out Eric Eager’s article: Football’s lessons about mathematics, academia, and industry.

The Quillette has an interesting, although possibly controversial read: It’s Time to Start Treating High School Math Like Football.

If a math student, football player, or anyone else feels like a zero, they could benefit from Fran Carona inspiring You Are Not a Zero about Cooper Kupp, the Super Bowl MVP.


Stanleyavestaff room 6 students enjoyed the lollipop statistics assignment so much fun that they didn’t even know they were doing math.

Joseph Nebus of Another Blog, Meanwhile made a humorous pie chart in Statistics Saturday: My Schedule for Doing Things. Many students and even adults can probably relate to it.

Calculus and Higher Math

Joseph Nebus of Nebusresearch has been reading a biography of Pierre-Simon LaPlace, so naturally, he blogs about monkeys, typewriters, and William Shakespeare in Some Progress on the Infinitude of Monkeys.

Math Wordle

Wordle has recently taken the world by storm. Got some math vocabulary words for your students? No matter how long the words are, your students can try to guess such words when they’re presented as wordles that you’ve made with the help of mywordle.strivemath.com. I made the one below. I told my son it was a math term and asked him to solve it:

There are also numerous wordles based on numbers rather than letters:

Stand-up Comedy Show

Archon’s Den shared some clever Math One-liners that will make you and your students either roll on the floor with laughter or roll your eyes.

Other Carnivals

The 201st edition of the Carnival of Mathematics can be found at the team at Ganit Charcha.

Last month the 152nd Playful Math Carnival was hosted by Denise Gaskins. Perhaps you would like to host the next carnival or one later in the year. We need more volunteers! To volunteer to host the carnival go to Denise Gaskins’ Carnival Volunteer Page.

1718 Factor Fits Valentine

Today’s Puzzle:

Happy Valentine’s Day! I hope you enjoy my Valentine to YOU!

Other Mathematical Valentines:

Colleen Young of Mathematics, Learning and Technology has a nice collection of mathematical hearts For Valentine’s Day.

Factors of 1718:

  • 1718 is a composite number.
  • Prime factorization: 1718 = 2 × 859.
  • 1718 has no exponents greater than 1 in its prime factorization, so √1718 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 1718 has exactly 4 factors.
  • The factors of 1718 are outlined with their factor pair partners in the graphic below.

More About the Number 1718:

1718 is the sum of four consecutive numbers:
428 + 429 + 430 + 431 = 1718.


1711 is a Triangular Number So The Taxman Wants His Share

Today’s Puzzle:

Imagine this puzzle is made up of 58 envelopes, each containing the amount of money printed in bold type on its front. You can’t take any of the envelopes without the Taxman also taking a share. The Taxman will take EVERY available envelope that has a factor of the number you take on it. When you have taken all the cash you can, the Taxman gets ALL the leftover cash, and the game is over. You want the Taxman to get as little money as possible.

How much money is at stake? (58 × 59)/2 = 1711. That means 1711 is a triangular number, the sum of all the numbers from 1 to 58. Thus, the total amount of money you will be splitting with the Taxman is 1711.

To play this game, you can print the cards from this excel file: Taxman & 1537-1544. The factors of a number is printed in small type at the top of its card.

Here below I show the order I selected the cards when I played. For example, I took 53, and all the Taxman got was 1. I took 49, and the only card left for the Taxman to take was 7, and so forth. The last card I could take was 42, and the Taxman got 21, but since I couldn’t take anymore cards, the Taxman also got 31, 34, 37, 41, 43, and 47.

To win the game, you must get over half of the 1711 cash, but of course, you will want the Taxman to get much less than nearly half the money.

I didn’t want to make a long addition problem to find out how much I kept, so I came up with an easier strategy: I pushed the Taxman’s share to the side, and looked for ways to make 100 from my cards. (It is almost the 100th day of school, after all.) I found 9 ways to make 100, so I clearly kept more than half of the 1711.

I added that final 53 + 50 + 39 + 36 in my head by thinking that if I took 3 away from 53 and gave it to the 36, it would be the same as 50 + 50 + 39 + 39, a fairly easy sum.

Factors of 1711:

  • 1711 is a composite number.
  • Prime factorization: 1711 = 29 × 59.
  • 1711 has no exponents greater than 1 in its prime factorization, so √1711 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 1711 has exactly 4 factors.
  • The factors of 1711 are outlined with their factor pair partners in the graphic below.

1711 is a Shape-Shifting Number:

  • 1711 is the 58th Triangular Number because (58·59)/2 = 1711.
  • It is the 20th Centered Nonagonal Number because it is one more than nine times the 19th triangular number: 9(19·20)/2 + 1 =1711. AND
  • It is the 19th Centered Decagonal Number because it is one more than 10 times the 18th triangular number: 10(18·19)/2 +1 = 1711.

That last figure I’ve illustrated below with ten triangles circled around the center dot:

Now get this: Not only is 1711 the 19th Centered Decagonal Number, but similar-looking 17111 is the 59th Centered Decagonal Number! (A mere coincidence, but the 59th is even cooler because 59·29 = 1711.)

More About the Number 1711:

1711 is the hypotenuse of a Pythagorean triple:
1180-1239-1711, which is (20-21-29) times 59.

1711 is the difference of two squares in two different ways:
856² – 855² = 1711, and
44² – 15² = 1711.

1711 is a fun number to explore!

1706 Can You Make the Factors Fit?

Today’s Puzzle:

It’s 2022. Happy New Year! I wanted to make a puzzle that has 20 and 22 as clues and thought about what I could do.

I continue to be inspired by an old addition puzzle Sarah Carter @mathequalslove shared on Twitter:

I decided to tweak that puzzle into a multiplication puzzle. I ran into a problem, however. Having products in every triangle made the puzzle way too easy. How do I fix that? I removed some of the product clues. Can you use logic and factoring to know where each factor from 1 to 12 belongs? Can you determine the missing products? I hope you have lots of fun finding the puzzle’s only solution! And I hope you make the factors fit instead of having a fit trying!

Here’s something I haven’t told you before: I made lots of multiplication-table puzzles years before I started blogging. I wanted to give the puzzles a good name. At first, I called them “Turn the Tables on Multiplication” or “Turn the Tables” for short. I thought that title was clever but a little bit unwieldy. For a short time, I called the puzzles “Factor Fits.” It was a play on words because all the factors fit, but they might give you fits as you try to find them. I finally settled on “Find the Factors.” That title doubled as instructions for the puzzles. I still liked the name “Factor Fits,” and this puzzle lets me give new life to that name.

Factors of 1706:

  • 1706 is a composite number.
  • Prime factorization: 1706 = 2 × 853.
  • 1706 has no exponents greater than 1 in its prime factorization, so √1706 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 1706 has exactly 4 factors.
  • The factors of 1706 are outlined with their factor pair partners in the graphic below.

More About the Number 1706:

1706 is the sum of two squares:
41² + 5² = 1706.

1706 is the hypotenuse of a Pythagorean triple:
410-1656-1706, calculated from 2(41)(5), 41² – 5², 41² + 5².
It is also 2 times (205-828-853).