1225 is a Triangular Number, a Perfect Square, and . . .

1 and 36 are both triangular numbers and square numbers. The next number that can make the same claim is 1225. Why is this so?

1225 is a triangular number that just happens to be the sum of two consecutive triangular numbers. Two consecutive triangular numbers can always be made into a perfect square. Here’s a gif that tries to illustrate these facts about 1225:

1225 Perfect Square and Triangular Number

GIFs like this at MakeaGif

1225 is the 49th triangular number because 49(50)/2 =1225

It is the 35th square number because 35² = 1225

That it is both a triangular number AND a square number is pretty remarkable. But guess what? 1225 is ALSO a hexagonal number.

1225 is the 25th hexagonal number because (25)(2·25 – 1)= 1225
In fact, 1225 is the smallest number greater than 1 that is a triangle, a square, AND a hexagon! (Yeah, 1 is also all three and a whole more, but does 1 dot REALLY look like a triangle, a square, a hexagon and everything else all at the same time?)
All of these facts are great reasons to get very excited about the number 1225. Here are some more facts about this number:
  • 1225 is a composite number.
  • Prime factorization: 1225 = 5 × 5 × 7 × 7, which can be written 1225 = 5²× 7²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1225 has exactly 9 factors.
  • Factors of 1225: 1, 5, 7, 25, 35, 49, 175, 245, 1225
  • Factor pairs: 1225 = 1 × 1225, 5 × 245, 7 × 175, 25 × 49, or 35 × 35
  • 1225 is a perfect square. √1225 = 35
Not only is 1225 a triangular number that is the sum of two other triangular numbers, but
1225 is also a square that is the sum of two other squares!
28² + 21² = 35² = 1225

1225 is the hypotenuse of two Pythagorean triples:
735-980-1225 which is (3-4-5) times 245
343-1176-1225 calculated from 28² – 21², 2(28)(21), 28² + 21²,
but it is also (7-24-25) times 49

1225 looks like a square in some other bases:
It’s 441 in BASE 17,
169 in BASE 32,
144 in BASE 33,
121 in BASE 34, and
100 in BASE 35

Stetson.edu reports that 1225 is the smallest number that can be written as the sum of four cubes three different ways. It looks like these are the three ways:
10³ + 6 ³ + 2³ + 1³ = 1225
9³ + 6³ + 6³ + 4³ = 1225
8³ + 7³ + 7³ + 3³ = 1225

1225 is, indeed, the smallest number that has all the special properties listed above!

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How I Knew Immediately that a Factor Pair of 1224 is . . .

12 = 3 × 4 and 24 is one less than 25. Those two facts helped me to know right away that 35² = 1225 and 34 × 36 = 1224. Study the patterns in the chart below and you will likely be able to remember all of the multiplication facts listed in it!

a² – b² = (a – b)(a + b)
You may remember how to factor that from algebra class. Here when b = 1, it has a practical application that can allow you to amaze your friends and family with your mental calculating abilities!

I’ve only typed a small part of that infinite pattern chart. For example, if you know that 19 × 20 = 380, then you can also know that 195² = 38025 and 194 × 196 = 38024.

Also because of that chart, I know that 3.5² = 12.25 and 3.4 × 3.6 = 12.24
(Also (3½)² = 12¼, but 2½  × 4½ = 11¼ because 3-1 = 2, 3+1 = 4, 12-1 = 11
thus 2.5 × 4.5 = 11.25 and 2½  × 4½ = 11¼)

You could also let b = 2 so b² = 4. Then 25 – 4 = 21, and you could know facts like
33 × 37 = 1221 or 193 ×  197 = 38021

I hope you have a wonderful time being a calculating genius!

Now I’ll share some other facts about the number 1224:

  • 1224 is a composite number.
  • Prime factorization: 1224 = 2 × 2 × 2 × 3 × 3 × 17, which can be written 1224 = 2³ × 3² × 17
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1224 has exactly 24 factors.
  • Factors of 1224: 1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, 612, 1224
  • Factor pairs: 1224 = 1 × 1224, 2 × 612, 3 × 408, 4 × 306, 6 × 204, 8 × 153, 9 × 136, 12 × 102, 17 × 72, 18 × 68, 24 × 51 or 34 × 36
  • Taking the factor pair with the largest square number factor, we get √1224 = (√36)(√34) = 6√34 ≈ 34.98571

When a number has so many factors, I often will make a forest of factor trees for that number, but today I just want us to enjoy this one tree for 34 × 36 = 1224.

1224 is also the sum of two squares:
30² + 18² = 1224

1224 is the hypotenuse of a Pythagorean triple:
576-1080-1224 which is (8-15-17) times 72
That triple can also be calculated from 30² – 18², 2(30)(18), 30² + 18²

293 + 307 + 311 + 313 = 1224 making 1224 the sum of four consecutive prime numbers.

1219 is a Centered Triangular Number

If you look at an ordered list of centered triangular numbers, 1219 will be the 29th number on the list.

Study this graphic to see why:

Here’s more about the number 1219:

  • 1219 is a composite number.
  • Prime factorization: 1219 = 23 × 53
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1219 has exactly 4 factors.
  • Factors of 1219: 1, 23, 53, 1219
  • Factor pairs: 1219 = 1 × 1219 or 23 × 53
  • 1219 has no square factors that allow its square root to be simplified. √1219 ≈ 34.91418

1219 is the sum of the thirteen prime numbers from 67 to 127.

1219 is also the hypotenuse of a Pythagorean triple:
644-1035-1219 which is 23 times (28-45-53)

 

1207 The Risks of Tearing Down Walls

When we moved into our house twenty-five years ago, the professional movers somehow managed to get our bedroom dresser around our living room wall and upstairs where we wanted it. Nevertheless, after watching their tricky maneuvering, I knew that dresser wasn’t ever leaving upstairs so long as that wall remained.

Over the last sixteen years, my husband has longed for a new bedroom set. Every time he’s brought it up, I’ve pointed to that living room wall. Sometimes walls keep us from doing what we want to do. Taking down a wall can be risky, and it can be quite messy. In our case, when the wall came down, we also got a hole in our ceiling from one joist to another. Our air conditioning intake vent was in that wall so it had to be moved and holes in the floor needed to be repaired. On both ends of the wall, we had light switches that required moving. This seemingly simple wall removal required us to hire an electrician, a HVAC expert, and a drywall expert. Here’s how it looks today right after it was removed:

Not taking down walls also has risks. We decided that the benefits and risks of taking down the wall outweighed those of keeping it.

Many people have put up an anti-math wall in their lives.

Is such a wall keeping you from doing something you really want to do? Is it keeping you from getting a college degree or pursuing an occupation that would bring you fulfillment? Tearing down that wall may require you to hire some experts to help you patch up the holes you have in your skills. However, whatever you have to do, it is worth it if it helps you fulfill your dreams.

Sometimes if you look into what you thought was a boring topic, you might find something really interesting about it.

For example, 1207 might seem like a boring number, but I bet I can find at least one interesting fact about it:

  • 1207 is a composite number.
  • Prime factorization: 1207 = 17 × 71
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1207 has exactly 4 factors.
  • Factors of 1207: 1, 17, 71, 1207
  • Factor pairs: 1207 = 1 × 1207 or 17 × 71
  • 1207 has no square factors that allow its square root to be simplified. √1207 ≈ 34.74191

Did you notice that 17 × 71 = 1207?  Likewise 71 × 17 = 1207
Its prime factors are looking in the mirror at each other!

1207 is the sum of three consecutive prime numbers:
397 + 401 + 409 = 1207

1207 can also be written as the difference of two squares two different ways:

44² – 27² = 1207
604² – 603² = 1207

And guess what? I haven’t written everything that could be written about this number. You can actually learn more about it if you choose to break down the anti-math wall to find out more!

 

The factors of the hundred numbers just before 1201

I’ve made a simple chart of the numbers from 1101 to 1200, but it’s packed with great information. It gives the prime factorization of each of those numbers and how many factors each of those numbers have. The numbers written with a pinkish hue are the ones whose square roots can be simplified. Notice that each of those numbers has an exponent in its prime factorization.

I didn’t make a horserace from the amounts of factors this time because it isn’t a very close race. Nevertheless, you can guess which number appears most often in the “Amount of Factors columns” and see if your number would have won the race.

Now I’ll share some information about the next number, 1201. Notice the last entry in the chart above. It had so many factors that there weren’t very many left for 1201 to have. . .

  • 1201 is a prime number.
  • Prime factorization: 1201 is prime.
  • The exponent of prime number 1201 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1201 has exactly 2 factors.
  • Factors of 1201: 1, 1201
  • Factor pairs: 1201 = 1 × 1201
  • 1201 has no square factors that allow its square root to be simplified. √1201 ≈ 34.65545

How do we know that 1201 is a prime number? If 1201 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1201 ≈ 34.7. Since 1201 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1201 is a prime number.

Even though it doesn’t have many factors, 1201 is still a fabulous number:

25² + 24² = 1201

1201 is the 25th Centered Square Number because 25² + 24² = 1201, and 24 and 25 are consecutive numbers:

1201 is the hypotenuse of a primitive Pythagorean triple:
49-1200-1201 calculated from 25² – 24², 2(25)(24), 25² + 24²

Here’s another way we know that 1201 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 24² = 1201 with 25 and 24 having no common prime factors, 1201 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1201 ≈ 34.7. Since 1201 is not divisible by 5, 13, 17, or 29, we know that 1201 is a prime number.

The 120th Playful Math Carnival

Plinko is a fun carnival game of chance. This Plinko board is really just a portion of Pascal’s triangle. Stetson.edu informs us that 120 is the smallest number to appear six times in that triangle. Why did those six times happen?
120 = 10!/3!/7! That’s why it appears twice in the 10th row of the Plinko board below.
120 = 16!/2!/14! Which is why it appears twice in the 16th row as well.
120 will appear two more times in its 120th row.

Now step right up and learn some other incredible facts like
120 = 5! because 1·2·3·4·5 = 120

120 is also the smallest positive multiple of 6 that is neither preceded nor followed by a prime number!

What kind of shape is 120 in?
120 is the 15th triangular number because 15(16)/2 = 120,
it’s the 8th tetrahedral number because (8)(9)(10)/6 = 120 (That means 120 is the sum of the first eight triangular numbers), and
it is the 8th hexagonal number because (8)(2·8-1) = 120.

Math Journals and Creative Writing

Every Playful Math Carnival contains blog links about ways to play with math and insights into teaching math. Blogging about math helps clarify thoughts, document experiences, and share the joy math brings us. It is a lot like keeping a math journal. Denise Gaskins wrote a post about the benefits of math journalling and included some prompts to help students get writing. Whether writing about joys or frustrations, math journalling has its benefits.

Abhishek Pathania wrote a clever limerick titled Maths that uses mathematical terms such as chance, calculated guess, multiply and divide. I enjoyed the limerick and I bet your students will, too. Another blogger, Roland, shared Maths Limerick, which is quite a bit of fun, too.

I hesitate a little to share this next one. However, older students may enjoy reading a little satire from the Onion that was shared this month on the Bluebird of Bitterness, Young girls creeped out by older scientists constantly trying to lure them into STEM. It certainly could give you something to talk about.

The next stop at our carnival is a house of horrors that is simply terrifying to some people. It is known as . . . . .

Math Anxiety

In Life Cameo’s post Learning, a young girl goes from liking math to feeling significantly less confident and quietly suffering from math anxiety. Thankfully her teacher intervened and she is now just starting to understand it again.

Alyssa lets you take a peek into the world of one who suffers from math anxiety in her post What Does Math Anxiety Look Like?

A young man named Dave blogged about his lifelong struggles with math in Dealing with Learning Disabilities in Math. Although he occasionally used a strong word to voice his frustrations, his is an important point of view that ought to be shared. This school year I am working with students who need specialized help with mathematics so this post gave me some food for thought.

Preventing and Treating Math Anxiety

So as you can see Math Anxiety is a real concern. What can you do about it? Josh Rappaport of Math Chat advises How to talk about math without scaring children off.

And of course, Denise Gaskins’ Let’s Play Math Blog is filled with ways to PLAY with math. Play can relieve a lot of anxiety. Recently Denise posted a quote by Rózsa Péter about math being worthy of our time and how Rózsa’s class of twelve-year-old girls begged her to let them explore the Euclidean algorithm. These girls felt no fear; it was a joyful experience for them the entire time.

Which method is better for children to learn math, discovery or traditional? The Intrepid Mathematician suggests a combination of the two and how to implement that teaching in A third path for early math education.

Mathematical Art on Exhibit

The average preschooler/kindergartner only gets 58 seconds of math instruction a day. Those who get Paula Krieg to teach them for one fascinating hour a week are really fortunate! You can see what I mean by reading her post, Little Hands, Little Books, Folds, & Math.

Number Loving Beagle shares a raw, personal story of years of yearning for artistic talent in Math is Beautiful (and other lies). Math really can make beautiful, frameable art as demonstrated in that post, but too often math has become nothing more than misery-inducing, anxiety producing, seemingly worthless calculations. Which math will you choose for yourself and your children?

Su Leslie created a beautiful piece of fractal art in Pretty Maths. Su’s work could inspire others to see the beauty in mathematics.

Rachel Shey shares some more mathematical art and thoughts in the post Math and Art. I also liked her thoughts about two fields intersecting.

One great way to make mathematical art is to use mirrors as demonstrated in these photos by Annie Fetter when she went to Math on a Stick.

Robert Loves Pi once again has created some beautiful, rotating 3-dimensional mathematical art for us all to enjoy.

The Math Museum featuring Calculators, Castles, and Puzzles

Simona Prilogan of Let’s Math regularly posts a number puzzle on her blog, Let’s Math. Some of the puzzles may be easier to solve than others, but I’m sure students will be able to figure these two out.  Boats Tuesday Maths Puzzle and Sunshine Thursday Maths Puzzle. That second one actually contains a few carnival pictures!

I visited a type of museum inside Romania’s Corvin Castle in Hunedoara this summer. Although I didn’t know when I visited, Hunedoara is Simona Prilogan’s hometown! I was delighted to find a post she published about the castle in her poetry blog less than a month before I wrote a post with some mathematical pictures from inside the castle. I am amazed at how small the world of mathematics can be!

Life Through a Mathematicians Eyes also grew up in Romania and offers a guided tour of Calculators That Made History. When I took the tour I was amazed at how old some of those calculators are. I’m sure you will enjoy the tour very much!

Colleen Young has several different mathematical examples in her post Here’s the diagram. What’s the question? What better way could there be to learn any of those topics frontward and backward than make it feel like solving a puzzle?

I took a photo at a Hungarian museum village and turned it into a mathematical puzzle/lesson for young ones by asking a couple of simple questions. How Are They the Same? How are They Different?

Logic

BloggingIsAResponsibility wrote a post titled Is Math Meaningless, and Is That an Insult? If you’re introducing syllogisms in your geometry class, you might want to try some of these effective but meaningless arguments!

Life Through a Mathematician Eyes offers thoughts and study videos on more advanced Logic Problems beginning with Studying Logic – Day 1.

Science Book A Day reviewed mathematician Eugenia Cheng’s book, The Art of Logic: How to Make Sense in a World That Doesn’t.

Math Literature and Books

Musings of a Mathematical Mom blogged about a mathematical adventure her children enjoyed. They counted and divided using Christopher Danielson’s book How Many. Her children even drew pictures afterward that would allow them to count and think about even more fractions. Who could ask for anything more?

Life Through a Mathematicians Eyes reviews three books that teens and teachers can most certainly enjoy in New Book Discoveries. The books reviewed are Weird Maths: At the Edge of Infinity and Beyond by David Darling and Agnijo Banerjee, Your Daily Maths: 366 Number Puzzles and Problems to Keep you Sharp by Laura Laing, and 50 Maths Ideas You Really Need to Know By Tony Crilly.

Susan mentioned Ramanujan and the book The Man Who Knew Infinity when she wrote a blog post she called The Story of the Locked Box and the Key of Dreams. Her title sounds like a mathematical fairy tale, but it is not a storybook at all. It gives a vivid description of her lucid mathematical dreams, her struggles with dyscalculia, and her triumphs in learning math. Ramanujan also had wonderful mathematical dreams, so she is in good company.

Crafts, Fashion, Souvenir booths

At this next carnival booth, you can buy a variety of clothing items. Should you buy any of them? Fashion Math-Thinking about the Cost Per Wear shares a formula created to help you make that decision.

TerifiCreations by Teri Lewis asks, “Has anyone ever written an article encouraging quilters to do math?” If any quilters out there struggle with the math, she will gladly help out.

Mathemagic or Carni Game?

How to get super-rich; millionaire math suggests 13 different ways to get to a million and would be a fun way to increase number sense for students who already know how to multiply.

Sometimes students come up with ridiculous answers to word problems. DC Gilbert shares a disastrous story and concludes, “Mathematics! It is Really That Important!

Using statistics to tell lies: Open Mind gives an example in USA Temperatures: Can I Sucker You?

Winning Mathematical Game Skills

The son of one of Math Mammoth’s customers created a flash program that helps second-grade students practice simple addition and subtraction facts. Skills require practice so check it out!

Resourceaholic offers some fun beginning-of-the-school-year activities for year 7 students.

Dealing with histograms might seem as treacherous as getting through an obstacle on American Ninja Warrior, but Math Only Math gives step by step histogram instructions to help middle and high school students navigate through those different-height rectangles in record time.

If you’re teaching the Fundamental Counting Principle, I’m sure you can find a way to use Wrong Hands’ clever/funny comic Lesser super-hero movie title generator.

Chris McMullen can answer your students’ question, Which Calculus Skills are most essential, practical?

How do you prove that e is an irrational number? Mjlawler tackles that problem in Walking through the proof that e is irrational with a kid.

The Carnival of the Future

Joseph Nebus, who will host the carnival in September at his blog, NebusResearch mentioned some comics that could lessen geometry anxiety in Reading the Comics, Ragged Ends Edition.

Joseph also writes a humor blog that sometimes has gems like the Venn Diagram he made for his post Statistics Saturday: Trivia Night Questions, by Kind.

I can tell that Joseph is pretty pumped about writing the carnival next month. Read The Mathematics Carnival is coming! and enjoy his enthusiasm.

You can also enjoy the August 2018 edition of the Carnival of Mathematics.

Finally, no matter where or how you teach mathematics, remember these words Jennie penned in  An Open Letter to Teachers, “You have to share your love and passions.  That’s your joy.  In that way, you are sharing you.  And, all that children want to know is that you love them and love what you are teaching.  If they know that, the floodgates will open to learning.”

The future of mathematics education is in YOUR hands. Have fun!

 

 

1199 and Level 1

Here’s a puzzle that even someone just learning to multiply and divide can solve. That means you can solve it, too!

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are some facts about the number 1199:

  • 1199 is a composite number.
  • Prime factorization: 1199 = 11 × 109
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1199 has exactly 4 factors.
  • Factors of 1199: 1, 11, 109, 1199
  • Factor pairs: 1199 = 1 × 1199 or 11 × 109
  • 1199 has no square factors that allow its square root to be simplified. √1199 ≈ 34.62658

1199 is the sum of the fifteen prime numbers from 47 to 109. That last one just happens to be one of its prime factors, too!

1199 is the hypotenuse of a Pythagorean triple:
660-1001-1199 which is 11 times (60-91-109)

1199 looks cool in base 10, and it’s palindrome
2F2 in BASE 21 (F is 15 base 10)

 

1198 Challenge Puzzle

You can solve this Find the Factors 1 – 10 puzzle if you use logic. Guessing and checking will likely only frustrate you. Go ahead and give logic a try!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some facts about the number 1198:

  • 1198 is a composite number.
  • Prime factorization: 1198 = 2 × 599
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1198 has exactly 4 factors.
  • Factors of 1198: 1, 2, 599, 1198
  • Factor pairs: 1198 = 1 × 1198 or 2 × 599
  • 1198 has no square factors that allow its square root to be simplified. √1198 ≈ 34.61214

1198 is also palindrome 262 in BASE 23

1188 How Many Triangles? How Many Factors?

Manually counting ALL the triangles in the graphic below could get confusing. Try counting all the triangles that point up first then all the triangles that point down.

Since the number of triangle rows is an even number, you could just use this formula:

n(n+2)(2n+1)/8 = the number of triangles

In this case, n = 16, so the number of triangles is 16×18×33/8 = 1188.

(If the number of rows is an odd number, the formula produces a number with a decimal. In that case, just ignore the decimal and everything after it to get the number of triangles.)

Here’s some more information about the number 1188:

  • 1188 is a composite number.
  • Prime factorization: 1188 = 2 × 2 × 3 × 3 × 3 × 11, which can be written 1188 = 2² × 3³ × 11
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = 24. Therefore 1188 has exactly 24 factors.
  • Factors of 1188: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 27, 33, 36, 44, 54, 66, 99, 108, 132, 198, 297, 396, 594, 1188
  • Factor pairs: 1188 = 1 × 1188, 2 × 594, 3 × 396, 4 × 297, 6 × 198, 9 × 132, 11 × 108, 12 × 99, 18 × 66, 22 × 54, 27 × 44 or 33 × 36
  • Taking the factor pair with the largest square number factor, we get √1188 = (√36)(√33) = 6√33 ≈ 34.46738

1188 has MANY possible factor trees. Here are four of them:

1188 is an interesting-looking base 10 number, but it also looks interesting when it is written in some other bases:
It’s 2244 in BASE 8,
990 in BASE 11 because 9(11²) + 9(11) = 9(132) = 1188,
543 in BASE 15,
4A4 in BASE 16 (A is 10 base 10),
XX in BASE 35 (X is 33 base 10) because 33(35) + 33(1) = 33(36) = 1188,
and it’s X0 in BASE 36 because 33(36) = 1188

1183 is the 13th Pentagonal Pyramidal Number

 

1183 is the 13th pentagonal pyramidal number. Here’s an attempt to illustrate that fact. (Try to think 3-dimensionally.):

Here are some more facts about the number 1183:

  • 1183 is a composite number.
  • Prime factorization: 1183 = 7 × 13 × 13, which can be written 1183 = 7 × 13²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 1183 has exactly 6 factors.
  • Factors of 1183: 1, 7, 13, 91, 169, 1183
  • Factor pairs: 1183 = 1 × 1183, 7 × 169, or 13 × 91
  • Taking the factor pair with the largest square number factor, we get √1183 = (√169)(√7) = 13√7 ≈ 34.39476

1183 is the hypotenuse of two Pythagorean triples:
833-840-1183 which is 7 times (119-120-169)
455-1092-1183 which is (5-12-13) times 91

I like 1183 in a couple of different bases:
It’s palindrome 1121211 in BASE 3 because
3⁶ + 3⁵ + 2(3⁴) + 3³ + 2(3²) + 3 + 1 = 1183,
and it’s 700 in BASE 13 because 7(13²) = 1183