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.

Puzzles

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


Fractions

Geometry

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.

Statistics

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).

1705 The Seat Numbers from Jimmy Fallon’s Twelve Days of Christmas Sweaters

Today’s Puzzle:

Jimmy Fallon’s Twelve Days of Christmas Sweaters tradition has become something I look forward to each December. The sweaters are one-of-a-kind masterpieces. I love when the sweaters are revealed. Jimmy reaches into a bright red Christmas stocking and randomly pulls out a number, the seat number of the winner of the sweater. Miraculously,  the winner of each sweater looks fabulous in it, no matter how big or small the winner is. I love this tradition, the sweaters, the winners modeling the sweaters, but I also love hearing the seat numbers. Each seat number has something special about it. (Just because it is a number!) By day three, I knew I wanted to blog about the numbers this year. The seat numbers were 295, 257, 314, 270, 419, 126, 256, 417, 433, 242, 232, and 120. I immediately knew something special about several of the numbers, but some of them I had to research. Can you figure out what is so special about each one?

Three of the seat numbers were primes. Which three?

One of those primes is both the fourth Fermat prime and the second-largest known Fermat prime. Which prime number is that?

Two of the numbers were palindromes (numbers that read the same forward and backward). Which two?

One of the seat numbers is equal to 1 × 2 × 3 × 4 × 5. Mathematicians write that as 5! Which seat number is equal to 5!?

One of the numbers is 10π rounded. Which one?

How Do Some of the Seat Numbers Shape Up?

Two of the numbers were decagonal numbers. Which two?

126 is not only a decagonal number, but it is also a pyramid formed by stacking the first six pentagonal numbers on top of each other.
1 + 5 + 12 + 22 + 35 + 51 = 126.

120 comes in THREE shapes.

One of the seat numbers is a star:

Something Special About Each Seat Number:

I’ll explain some of these reasons below.

Three of the Numbers Were the First Numbers to do Something Special:

242 is the smallest number whose square root can be simplified that is followed by three other numbers whose square root can also be simplified. Also, all four numbers have exactly six factors. Numbers with exactly six factors always have simplifiable square roots.

  • 242 = 2·11²; its six factors are 1, 2, 11, 22, 121, 242.
  • 243 = 3⁵; its six factors are 1, 3, 9, 27, 81, 243.
  • 244 = 2²·61; its six factors are 1, 2, 4, 61, 122, 244.
  • 245 = 7²·5; its six factors are 1, 5, 7, 35, 49, 245.

Square roots 242 - 245

417 is the smallest number that is the first of four consecutive integers that are divisible by a different number of primes.

419 is one less than 420, the smallest number divisible by 1, 2, 3, 4, 5, 6, and 7. As a consequence of that, 419 is the smallest number that leaves a remainder of 1, when it is divided by 2, a remainder of 2, when it is divided by 3, a remainder of 3, when it is divided by 4, a remainder of 4, when it is divided by 5, a remainder of 5, when it is divided by 6, and a remainder of 6, when it is divided by 7. This next graphic is a different way to make the same point.

What Do I Mean by Sum-Difference?

Two of the seat numbers have factor pairs that make sum-difference: the numbers in one of its factor pairs add up to a particular number and the numbers in a different factor pair subtract to the same number. Coincidentally, both of the seat numbers are related to 30, another number that makes sum-difference.

I’ve so enjoyed discovering what made all those seat numbers special, and I hope that you have enjoyed reading about them as well!

Since this is my 1705th post, I’ll write a little about that number, as well.

Factors of 1705:

  • 1705 is a composite number.
  • Prime factorization: 1705 = 5 × 11 × 31.
  • 1705 has no exponents greater than 1 in its prime factorization, so √1705 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 1705 has exactly 8 factors.
  • The factors of 1705 are outlined with their factor pair partners in the graphic below.

More About the Number 1705:

1705 is the hypotenuse of a Pythagorean triple:
1023 1364 1705, which is (3-4-5) times 341.

1703 A Wreath to Hang on Your Door

Today’s Puzzle:

A wreath is a lovely decoration to hang on your door at Christmastime. This one might have a few thorns in it, but if you are careful, they won’t bother you in the least. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues create a multiplication table.

Factors of 1703:

  • 1703 is a composite number.
  • Prime factorization: 1703 = 13 × 131.
  • 1703 has no exponents greater than 1 in its prime factorization, so √1703 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 1703 has exactly 4 factors.
  • The factors of 1703 are outlined with their factor pair partners in the graphic below.

More About the Number 1703:

Did you notice a cool pattern in 1703’s prime factorization?
13·131 =1703.

1703 is the hypotenuse of a Pythagorean triple:
655-1572-1703, which is (5-12-13) times 131.

1703 is the difference of two squares in two different ways:
852² – 851² = 1703, and
72² – 59² = 1703.

1702 A Puzzle Idea from @mathequalslove Tweaked into a Subtraction Puzzle That Directs You to a Post from NebusResearch

Today’s Puzzle:

Joseph Nebus is nearly finished with all the posts in his Little 2021 Mathematics A to Z series. Every year he requests that his readers give him mathematical subjects to write about. At my suggestion, he recently wrote about subtraction, and how it is a subject that isn’t always as elementary as you might expect.  With a touch of humor, we learn that subtraction opens up whole new topics in mathematics.

I wanted to make a puzzle to commemorate his post. I gave it some thought and remembered a tweet from Sarah Carter @mathequalslove:

That puzzle originated from The Little Giant Encylopedia of Puzzles by the Diagram Group. I wondered how the puzzle would work if it were a subtraction puzzle instead of an addition puzzle, and here’s how I tweaked it:

 

There is only one solution. I hope you will try to find it! If you would like a hint, I’ll share one at the end of this post.

Factors of 1702:

  • 1702 is a composite number.
  • Prime factorization: 1702 = 2 × 23 × 37.
  • 1702 has no exponents greater than 1 in its prime factorization, so √1702 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 1702 has exactly 8 factors.
  • The factors of 1702 are outlined with their factor pair partners in the graphic below.

More About the Number 1702:

1702 is the hypotenuse of a Pythagorean triple:
552-1610-1702, which is (12-35-37) times 46.

1702² = 2896804, and
2197² = 4826809.
Do you notice what OEIS.org noticed about those two square numbers?

Puzzle Hint:

Here’s how I solved the puzzle: I let the rightmost box be x. Then using the values in the adjacent triangles and working from right to left, I wrote the values of the other boxes in terms of x.

x – 5 went in the box that is second to the right,
x – 5 + 2 = x – 3 went in the next box,
x – 3 + 5 = x + 2,
x + 2 – 6 = x – 4,
x – 4 + 5 = x + 1, and so on until I had assigned a value in terms of x for every box.

Think about it, and this hint should be enough for you to figure out where the numbers from 1 to 9 need to go.

1701 Is a Decagonal Number

Today’s Puzzle:

There is a pattern to the decagonal numbers. Can you figure out what it is?

Factors of 1701:

1701 is divisible by nine because 1 + 7 + 0 + 1 = 9.

  • 1701 is a composite number.
  • Prime factorization: 1701 = 3 × 3 × 3 × 3 × 3 × 7, which can be written 1701 = 3⁵ × 7.
  • 1701 has at least one exponent greater than 1 in its prime factorization so √1701 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1701 = (√81)(√21) = 9√21.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1701 has exactly 12 factors.
  • The factors of 1701 are outlined with their factor pair partners in the graphic below.

More About the Number 1701:

1701 is the difference of two squares in SIX different ways.
851² – 850² = 1701,
285² – 282² = 1701,
125² – 118² = 1701,
99² – 90² = 1701,
51² – 30² = 1701, and
45² – 18² = 1701.

1701 is the 21st decagonal number because
21(4·21 – 3) =
21(84-3) =
21(81) = 1701.

There is decagonal number generating function:
x(7x+1)/(1-x)³ = x + 10x² + 27x³ + 52x⁴ + 85x⁵ + . . .

The 21st term of that function is 1701 x²¹.

 

1700 Time for a Horse Race!

Today’s Puzzle:

I’ve made a table of all the numbers from 1601 to 1700, their prime factorizations, and how many factors each of those numbers has.

Each number from 1601 to 1700 has 2, 3, 4, 6, 8, 10, 12, 16, 18, 24, 30 or 40 factors. Do more numbers have 2 factors, 3 factors, or a different number? Sure, you could use the table to count, but it will be more fun if we make a horse race out of it. What pony will you pick, 2, 3, 4, 6? Make your prediction of the winner, and then watch the horse race to see if you were right.

Here are all the horses lined up at the gate. Pick your pony!

And they’re off!

1601 to 1700 Horse Race

make science GIFs like this at MakeaGif
The first half of the race was VERY interesting to me. Three different horses held the lead, and the leaders in the race was even neck and neck some of the time. After you know the winner, watch the race again, following the winner from start to finish.

Factors of the Number 1700:

  • 1700 is a composite number.
  • Prime factorization: 1700 = 2 × 2 × 5 × 5 × 17, which can be written 1700 = 2² × 5² × 17.
  • 1700 has at least one exponent greater than 1 in its prime factorization so √1700 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1700 = (√100)(√17) = 10√17.
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1700 has exactly 18 factors.
  • The factors of 1700 are outlined with their factor pair partners in the graphic below.

More About the Number 1700:

1700 is the sum of two squares THREE different ways:
40² + 10² = 1700,
38² + 16² = 1700, and
32² + 26² = 1700.

Those sum of two squares mean 1700 is the hypotenuse of some Pythagorean triples, SEVEN to be exact:

  1. 260-1680-1700, which is 20 times (13-84-85),
  2. 348-1664-1700, which is 4 times (87-416-425), but it can also be calculated from 32² – 26², 2(32)(26), 32² + 26²
  3. 476-1632-1700, which is (7-24-25) times 68,
  4. 720-1540-1700, which is 20 times (36-77-85),
  5. 800-1500-1700, which is (8-15-17) times 100, but it can also be calculated from 2(40)(10), 40² – 10², 40² + 10²,
  6. 1020-1360-1700, which is (3-4-5) times 340,
  7. 1188-1216-1700, which is 4 times (297-304-425), but it can also be calculated from 38² – 16², 2(38)(16), 38² + 16².

As OEIS.org informs us, 1700 is a Catalan number. It is found in the C(13,4) position, as shown below. With the exception of the lone 1 in the 0th row of the triangle, every number in Catalan’s triangle is the sum of the number above it and the number to its left. For example, 1700 = 1260 + 440.

I hope you have enjoyed learning about the number 1700.