1348 Coloring Paula Krieg’s Polar Rose

Paula Beardell Krieg recently wrote about using Desmos to create designs that can be colored by hand or by computer programs like Paint. I like using Paint so with her permission I took a design she made and colored it so I could present it here in this post. I chose colors that make me think of spring because, frankly, I’m ready for winter to be over!

Now I’ll write a little bit about the number 1348:

1348 is the sum of two squares:
32² + 18² = 1348

1348 is the hypotenuse of a Pythagorean triple:
700-1152-1348 which is 32² – 18², 2(32)(18), 32² + 18²

1348 is also the short leg in a primitive Pythagorean triple:
1348-454275-454277

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1314 Desmos Art

A teacher at my school had his students graph some polynomials and their inverses. I got to help some of his students with their graphs. After seeing the beautiful symmetry of the graphs together, I excitedly exclaimed to a few of the students, “Isn’t this a cool assignment?”

During my lunch, I put one of the graphs, its inverse, and some of their translations on Desmos and made a simple but lovely piece of art in the process. 

Before I was done, I showed it to a couple of students. One of them asked, “Are you saying that math can create art?” I loved replying, “Yes, it can!” Now that student wants to create some works of art, too. It was a privilege to show her how to use Desmos.

These are the inequalities I used to make my work of art:

MANY teachers have figured out that students could learn a lot about functions and their graphs by using Desmos to create drawings, pictures, or artwork. For example, look at this tweet and link shared by Chris Bolognese:

Here’s the final @desmos coloring book from my amazing Precalculus students. We are headed to the lower school next week to join them in some coloring fun. Pictures to come! https://t.co/iejf2RYVWX @ColumbusAcademy @CAVikesSTEAM @mathequalslove #iteachmath— Chris Bolognese (@EulersNephew) December 8, 2018

Now I’ll share some facts about the number 1314:

  • 1314is a composite number.
  • Primefactorization: 1314= 2 × 3 × 3 × 73,which can be written 1314 = 2 × 3² × 73
  • The exponents inthe prime factorization are 1, 2, and 1. Adding one to each and multiplying weget (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1314has exactly 12 factors.
  • Factors of 1314:1, 2, 3, 6, 9, 18, 73, 146, 219, 438, 657, 1314
  • Factor pairs: 1314= 1 × 1314,2 × 657, 3 × 438, 6 × 219, 9 × 146, or 18 × 73 
  • Taking the factorpair with the largest square number factor, we get √1314= (√9)(√146) = 3√146 ≈ 36.24914

1314 is the sum of two squares:
33² + 15² = 1314

1314 is the hypotenuse of a Pythagorean triple:
864-990-1314 which is 18 times (48-55-73) and
can also be calculated from 33² – 15², 2(33)(15), 33² + 15²

1277 Strată Bolyai János in Timișoara, Romania

 

Around the turn of the 20th century, Bolyai Farkás taught mathematics at a university in Transylvania.  One day he was too sick to teach, so he sent his mathematically gifted 13-year-old son, János, to teach his classes! As you might imagine, János became quite the mathematician in his own right.

Ninety-five years ago today Bolyai János went to Timișoara, Romania to announce his findings concerning geometry’s fifth postulate. For centuries it was argued that this parallel lines postulate could probably be proved using the previous four of Euclid’s postulates, and it should, therefore, be considered a theorem rather than a postulate. Bolyai János proved that it is indeed something that must be assumed rather than proven, because, by assuming it wasn’t necessary, he was able to create a new and very much non-Euclidean geometry, now known as hyperbolic geometry or Bolyai–Lobachevskian geometry.

Last summer I was walking with some family members through a shopping area behind the opera house in Timișoara, Romania. Suddenly my son, David, excitedly shouted, “Mom, look!” There we stood in front of a street sign marking the strată named for Bolyai János! Here is a picture of me in front of that street sign.

Under his image are several plaques. The first is a replica of part of his proof. Underneath are plaques with a quote from him translated into several languages. Perhaps your favorite language is among them. Here is a close-up of the plaques:

The plaque at the bottom is in English, “From nothing I have created a new and another world. It was with these words that on November 3, 1823, Janos Bolyai announced from Timișoara the discovery of the fundamental formula of the first non-Euclidean geometry.”

We did not get to visit the university named for Bolyai János, but I am thrilled that my son spotted this historic location!

Now I’ll write a little about the number 1277:

  • 1277 is a prime number.
  • Prime factorization: 1277 is prime.
  • The exponent of prime number 1277 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1277 has exactly 2 factors.
  • Factors of 1277: 1, 1277
  • Factor pairs: 1277 = 1 × 1277
  • 1277 has no square factors that allow its square root to be simplified. √1277 ≈ 35.73514

How do we know that 1277 is a prime number? If 1277 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1277 ≈ 35.7. Since 1277 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1277 is a prime number.

1277 is the sum of two squares:
34² + 11² = 1277

1277 is the hypotenuse of a Pythagorean triple:
748-1035-1277 calculated from 2(34)(11), 34² – 11², 34² + 11²

Here’s another way we know that 1277 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 11² = 1277 with 34 and 11 having no common prime factors, 1277 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1277 ≈ 35.7. Since 1277 is not divisible by 5, 13, 17, or 29, we know that 1277 is a prime number.

1266 is a Centered Pentagonal Number

1266 is the 23rd centered pentagonal number because 5(22)(23)/2 + 1 = 1266. The graphic below shows 1266 tiny dots arranged into a pentagonal shape and that 1266 is one more than five times the 22nd triangular number.

Here are some more facts about the number 1266:

  • 1266 is a composite number.
  • Prime factorization: 1266 = 2 × 3 × 211
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1266 has exactly 8 factors.
  • Factors of 1266: 1, 2, 3, 6, 211, 422, 633, 1266
  • Factor pairs: 1266 = 1 × 1266, 2 × 633, 3 × 422, or 6 × 211
  • 1266 has no square factors that allow its square root to be simplified. √1266 ≈ 35.58089

1266 is also the sum of ten consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1266

 

1261 Can You Make a Star out of a Hexagon?

Can you make a star out of a hexagon? If you have 1261 tiny squares arranged as a centered hexagon, you can rearrange those 1261 tiny squares into a six-pointed star as illustrated below!

37 was the last centered hexagonal number that was also a star number.

Here are some more facts about the number 1261:

  • 1261 is a composite number.
  • Prime factorization: 1261 = 13 × 97
  • 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 1261 has exactly 4 factors.
  • Factors of 1261: 1, 13, 97, 1261
  • Factor pairs: 1261 = 1 × 1261 or 13 × 97
  • 1261 has no square factors that allow its square root to be simplified. √1261 ≈ 35.51056

1261 is the sum of two squares two different ways:
30² + 19² = 1261
35² + 6² = 1261

1261 is the hypotenuse of FOUR Pythagorean triples:
420-1189-1261 calculated from 2(35)(6), 35² – 6², 35² + 6²
485-1164-1261 which is (5-12-13) times 97
539-1140-1261 calculated from 30² – 19², 2(30)(19), 30² + 19²
845-936-1261 which is 13 times (65-72-97)

Haunted Forest with 1260 Factor Trees

1260 is the smallest number with 36 factors. That’s a new record. (32 was the old record and was held by both 840 and 1080.)

Often when a number has a lot of factors, we will visit a forest of its factor trees. 1260 certainly deserves such a forest. Since it is just before Halloween, It happens to be a haunted forest. Do you dare to go into such a forest? These three trees are scary enough for me! However, there are MANY more factor trees in that haunted forest! Perhaps if you are brave, you can find some of those factor trees in the haunted forest yourself.

Here’s more about the number 1260:

  • 1260 is a composite number.
  • Prime factorization: 1260 = 2 × 2 × 3 × 3 × 5 × 7, which can be written 1260 = 2² × 3² × 5 × 7.
  • The exponents in the prime factorization are 2, 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) (1 + 1) = 3 × 3 × 2 × 2  = 36. Therefore 1260 has exactly 36 factors.
  • Factors of 1260: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
  • Factor pairs: 1260 = 1 × 1260, 2 × 630, 3 × 420, 4 × 315, 5 × 252, 6 × 210, 7 × 180, 9 × 140, 10 × 126, 12 × 105, 14 × 90, 15 × 84, 18 × 70, 20 × 63, 21 × 60, 28 × 45, 30 × 42 or 35 × 36
  • Taking the factor pair with the largest square number factor, we get √1260 = (√36)(√35) = 6√35 ≈ 35.49648

21 × 60 = 1260 The same digits are used on both sides of that equation and that makes 1260 the 19th Friedman number.

1260 is also the sum of the interior angles of a nine-sided polygon. Convex or concave, that is the sum. The concave nonagon below is a good illustration of that fact:

1260 is also the hypotenuse of a Pythagorean triple:
756-1008-1260 which is (3-4-5) times 252

1247 Is a Pentagonal Number

Two factors of 1247 make it the 29th pentagonal number. Here’s why:

29(3·29-1)/2 = 29(86)/2 = 29(43) = 1247

Here is an illustration of this pentagonal number featuring a different, but equivalent, formula.  Seeing the pentagonal numbers less than 1247 in the illustration won’t be difficult either.

Here are some more facts about the number 1247:

  • 1247 is a composite number.
  • Prime factorization: 1247 = 29 × 43
  • 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 1247 has exactly 4 factors.
  • Factors of 1247: 1, 29, 43, 1247
  • Factor pairs: 1247 = 1 × 1247 or 29 × 43
  • 1247 has no square factors that allow its square root to be simplified. √1247 ≈ 35.31289

1247 is the sum of consecutive prime numbers two different ways:
It is the sum of the twenty-three prime numbers from 11 to 103.
It is also the sum of seven consecutive primes:
163 + 167 + 173 + 179 + 181 + 191 + 193 = 1247

1247 is the hypotenuse of a Pythagorean triple:
860-903-1247 which is (20-21-29) times 43

 

1242 is a Decagonal Number

If you had 1242 tiny little squares you could arrange them into a decagon, just as I did for the graphic below.

18 is a factor of 1242. Since 18(4·18-3) = 18(69) = 1242, it is the 18th decagonal number.

Here are a few more facts about the number 1242:

  • 1242 is a composite number.
  • Prime factorization: 1242 = 2 × 3 × 3 × 3 × 23, which can be written 1242 = 2 × 3³ × 23
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 2 × 2 = 16. Therefore 1242 has exactly 16 factors.
  • Factors of 1242: 1, 2, 3, 6, 9, 18, 23, 27, 46, 54, 69, 138, 207, 414, 621, 1242
  • Factor pairs: 1242 = 1 × 1242, 2 × 621, 3 × 414, 6 × 207, 9 × 138, 18 × 69, 23 × 54, or 27 × 46
  • Taking the factor pair with the largest square number factor, we get √1242 = (√9)(√138) = 3√138 ≈ 35.24202

1242 is the sum of consecutive prime numbers two different ways:
It is the sum of the eighteen prime numbers from 31 to 107, and
it is also the sum of the sixteen prime numbers from 43 to 109.

1240 is a Square Pyramidal Number

1240 is the 15th square pyramidal number because
1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² + 15² = 1240

We can know that 1240 is the 15th square pyramidal number because
15(15 + 1)(2·15 + 1)/6
= 15(16)(31)/6
= (5)(8)(31)
= (40)(31)
= 1240

Here are some more facts about the number 1240:

  • 1240 is a composite number.
  • Prime factorization: 1240 = 2 × 2 × 2 × 5 × 31, which can be written 1160 = 2³ × 5 × 31
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1240 has exactly 16 factors.
  • Factors of 1240: 1, 2, 4, 5, 8, 10, 20, 31, 40, 62, 124, 155, 248, 310, 620, 1240
  • Factor pairs: 1240 = 1 × 1240, 2 × 620, 4 × 310, 5 × 248, 8 × 155, 10 × 124, 20 × 62, or 31 × 40
  • Taking the factor pair with the largest square number factor, we get √1240 = (√4)(√310) = 2√310 ≈ 35.21363


1240 is the hypotenuse of a Pythagorean triple:
744-992-1240 which is (3-4-5) times 248