1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding.

Now I’ll share a few facts about the puzzle number, 1413:

  • 1413 is a composite number.
  • Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
  • 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
  • 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 1413 has exactly 6 factors.
  • The factors of 1413 are outlined with their factor pair partners in the graphic below.

1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

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1412 and Level 3

If you know the greatest common factor of 56 and 48, then you have taken the first step in solving this puzzle. Once you put the factors of 56 and 48 in the appropriate cells, work down from the top of the puzzle to the bottom, cell by cell, until you have put all the numbers from 1 to 10 in both the first column and the top row.

Here are a few facts about the puzzle number, 1412:

  • 1412 is a composite number.
  • Prime factorization: 1412 = 2 × 2 × 353, which can be written 1412 = 2² × 353
  • 1412 has at least one exponent greater than 1 in its prime factorization so √1412 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1412 = (√4)(√353) = 2√353
  • 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 1412 has exactly 6 factors.
  • The factors of 1412 are outlined with their factor pair partners in the graphic below.

1412 is the sum of two squares:
34² + 16² = 1412

1412 is the hypotenuse of a Pythagorean triple:
900-1088-1412 calculated from 34² – 16², 2(34)(16), 34² + 16²

1411 and Level 2

Four of the fourteen clues, 18, 24, 16, and 40, appear twice in this puzzle, but do they lead you to the same factors? Where do the factors from 1 to 10 belong that will make this puzzle function like a multiplication table?

Now I’ll write a little bit about the puzzle number, 1411:

  • 1411 is a composite number.
  • Prime factorization: 1411 = 17 × 83
  • 1411 has no exponents greater than 1 in its prime factorization, so √1411 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 1411 has exactly 4 factors.
  • The factors of 1411 are outlined with their factor pair partners in the graphic below.

1411 is the hypotenuse of a Pythagorean triple:
664-1245-1411 which is (8-15-17) times 83.

1410 and Level 1

Start the school year off right with a quick review of the multiplication table. You can actually construct an entire 10 × 10 table with only the nine clues in this puzzle. Figure out where the numbers 1 to 10 go in both the first column and the top row and amaze yourself with how much you remember!

  • 1410 is a composite number.
  • Prime factorization: 1410 = 2 × 3 × 5 × 47
  • 1410 has no exponents greater than 1 in its prime factorization, so √1410 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 1410 has exactly 16 factors.
  • The factors of 1410 are outlined with their factor pair partners in the graphic below.

1410 is the hypotenuse of a Pythagorean triple:
846-1128-1410 which is (3-4-5) times 282.

1409’s Super Power

 

Stetson.edu informed me that 1409⁸ is the ONLY known 8th power that is the sum of EIGHT 8th powers. Wow! That seems to me to give 1409 quite the superpower!

What were those eight 8th powers that are included in the sum? That’s a puzzle more suited for a computer than a human, but Wolfram Mathworld Diophantine came to my rescue with this POWERFUL fact: 1324⁸+1190⁸+1088⁸+748⁸+524⁸+478⁸+223⁸+90⁸=1409⁸.

Go ahead and check it out on your computer’s calculator. It’s true! Notice also that two of those eighth powers are permutations of each other!

I was so intrigued with 1409 that I had to make this cape so everyone can see how super 1409 is:

Sometimes 1409 wears a more modest super cape because 1409² is also the sum of TWO squares:
159² +1400² = 1409² 

Here are some more super facts about the number 1409:

  • 1409 is a prime number.
  • Prime factorization: 1409 is prime.
  • 1409 has no exponents greater than 1 in its prime factorization, so √1409 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1409 has exactly 2 factors.
  • The factors of 1409 are outlined with their factor pair partners in the graphic below.

How do we know that 1409 is a prime number? If 1409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1409. Since 1409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1409 is a prime number.

1409 is the sum of two squares:
28² + 25² = 1409

1409 is the hypotenuse of a primitive Pythagorean triple:
159-1400-1409 calculated from 28² – 25², 2(28)(25), 28² + 25²

Here’s another way we know that 1409 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 25² = 1409 with 28 and 25 having no common prime factors, 1409 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1409. Since 1409 is not divisible by 5, 13, 17, 29, or 37, we know that 1409 is a prime number.

 

1408 Powers of 2 in the Multiplication Table

number, puzzle, factors, factor pairs, prime factorization,

I have a 10 × 10 multiplication table poster in my classroom to help students who haven’t memorized the times’ table yet. We have to spend our time going over more advanced topics. One student struggled with the idea of raising two to a power. I went to the poster and boxed in all the powers of two on it. While I boxed them in, I recited, “2⁰ = 1, 2¹ = 2, 2² = 2×2 = 4, 2³ = 2×2×2 = 8, 2⁴ = 2×2×2×2= 16, 2⁵ = 2×2×2×2×2= 32, 2⁶ = 2×2×2×2×2×2=64.”

I liked the pattern those powers of two made on the poster so I made this 32×32 multiplication chart on my computer and continued the pattern.

I expect the chart has many things for you to notice and wonder about. You could also do it with powers of 3, or another number, but you would need to use a much bigger multiplication table to show as many powers.

Now I’ll tell you a little bit about the number 1408.

1408 is not a power of 2, but it is 11 times a power of 2, specifically, it is 11 × 2⁷.

  • 1408 is a composite number.
  • Prime factorization: 1408 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 11, which can be written 1408 = 2⁷ × 11
  • 1408 has at least one exponent greater than 1 in its prime factorization so √1408 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1408 = (√64)(√22) = 8√22
  • The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1408 has exactly 16 factors.
  • The factors of 1408 are outlined with their factor pair partners in the graphic below.

Here is a festive multilayered factor cake for 1408:

So delicious! And here is a nicely balanced factor tree showing all of its prime factors:

 

1407 Please Stop Making Excuses for My Dear Aunt Sally

Please Excuse My Dear Aunt Sally. You’ve heard math teachers say that phrase many times. Supposedly, Aunt Sally is supposed to help you remember Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction as the correct order to do operations when simplifying math problems.

I say, please stop making excuses for my dear Aunt Sally!

My Dear Aunt Sally. People think they know her, but too often they really don’t. Lots of people have tried to please her. Sometimes they succeed, but just as often they fail. She seems to relish the fact that so many people misunderstand her.

I clearly remember my first year teaching a classful of seventh graders at a new school. I was trying to develop a good relationship with my students and be the best teacher I could be. One of the first lessons I was supposed to teach them was order-of-operations.

I wish I knew about the better mnemonic PEMA back then, but I didn’t. Instead, I brought my dear Aunt Sally to class with me: I introduced her to my students and tried to make it clear that multiplication and division were equals so they must be done in order from left to right whichever one comes first. The same is true of addition and subtraction.

“That’s not what we learned last year!” students responded. Their teacher last year brought Aunt Sally to class, too. but she gave them the impression that all multiplication was supposed to be done before any division, and the same for addition and subtraction. Yeah, Aunt Sally went to class their previous year and didn’t say a word when their teacher gave them misinformation. Now that I was telling them the truth about her, she didn’t speak up and tell them I was right either. Instead, she allowed me to lose credibility with my students that day as I insisted on sticking with the truth. If I had retold the lie, the students would have believed me more. I also discovered that for some problems in the textbook, you would get it right either way.

I seriously couldn’t believe that their teacher from the last year would have given them the wrong information. Surely the students misunderstood what had been taught. However, since that day, I have heard more than one teacher incorrectly tell students to do all the multiplication, division, addition, and subtraction in that order from left to right. Those teachers put the students’ next teachers in a catch-22:

That is why I prefer to keep “my dear Aunt Sally” away from kids. She always shows up at the beginning of the school year when students and teachers are trying to start off on the right foot.  She torments students and immediately causes them to feel bad about themselves or mathematics. She makes them question the teaching of their current teacher or their past teachers. She gets a kick out of making children and even adults feel like there’s no way to understand math:

Why do we allow Aunt Sally to abuse children like this? I want to shout, “please, stop making excuses for my dear, Aunt Sally!”

Let me tell you the story of when I decided not to introduce this abusive aunt to children every again.  It was 2016. I was substituting in a 5th-grade class. I wrote an expression I saw on twitter on the board and told the students it was my favorite order-of-operations problem. Here’s what I wrote:

10 + 9 + 8 × 7 × 6 × 5 – 4 + 321 = 

I, along with my dear Aunt Sally,  encouraged the students to figure it out. The students knew that 8 × 7 was 56. I watched them struggle to multiply 56 by 6 and then by 5. When I mentioned that they could multiply the 6 and the 5 first to get 56 × 30 to make the problem easier, they argued that doing that wasn’t allowed. They said that the order-of-operations demanded that the multiplication be done in ORDER from left to right.

They thought that order-of-operation makes multiplication no longer commutative?!!  How do you counteract that misinformation? After that day, not only do I not invite my dear Aunt Sally to meet my students, but I also avoid the phrase “order-of-operations”!

Order-of-operations is just an ALGORITHM! It doesn’t trump the commutative property, and it doesn’t even have to be used to solve these kinds of problems!

Jo Boaler’s tweet especially applies to this kind of problem and this algorithm.

Besides, are these kinds of problems still necessary since typing on a computer no longer has the same limitations as typing on a typewriter? I hope you think about that! If you insist on using an algorithm, I suggest you use PEMA instead.

Since this is my 1407th post, I’d like to tell you a little bit about that number:

  • 1407 is a composite number.
  • Prime factorization: 1407 = 3 × 7 × 67
  • 1407 has no exponents greater than 1 in its prime factorization, so √1407 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 1407 has exactly 8 factors.
  • The factors of 1407 are outlined with their factor pair partners in the graphic below.

1407 looks interesting when it is written in some other bases:
It’s 111333 in BASE 4,
21112 in BASE 5, and
727 in BASE 14.

1406 Has a Very Cool 4th Root

To find the 4th root of 1406, all you need to do is take its square root twice. The square root of 1406 is 37.4966665185. . .

Take the square root of that and you get a decimal starting with 6.12345…

That’s pretty cool. I’m glad Stetson.edu let me know about it!

Here’s a little more about the number 1406:

  • 1406 is a composite number.
  • Prime factorization: 1406 = 2 × 19 × 37
  • 1406 has no exponents greater than 1 in its prime factorization, so √1406 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 1406 has exactly 8 factors.
  • The factors of 1406 are outlined with their factor pair partners in the graphic below.

1406 is the sum of the first 37 EVEN numbers because 37 × 38=1406.

1406 is the hypotenuse of a Pythagorean triple:
456-1330-1406 which is (12-35-37) times 38

1405 is the Sum of Squares

I knew that 1405 was the sum of two consecutive squares, but Stetson.edu let me know that it was the sum of even more consecutive squares, ELEVEN to be exact!

Because it is the sum of the 26th and the 27th squares, 1405 is also the 27th centered square number. Here are 1405 tiny squares illustrating that fact:

Here’s more about the number 1405:

  • 1405 is a composite number.
  • Prime factorization: 1405 = 5 × 281
  • 1405 has no exponents greater than 1 in its prime factorization, so √1405 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 1405 has exactly 4 factors.
  • The factors of 1405 are outlined with their factor pair partners in the graphic below.

I’ve mentioned one of these before, but 1405 is the sum of TWO squares in TWO ways:
27² + 26² = 1405
37² + 6² = 1405

1405 is also the hypotenuse of FOUR Pythagorean triples:
53-1404-1405 calculated from 27² – 26², 2(27)(26), 27² + 26²
444-1333-1405 calculated from 2(37)(6), 37² – 6², 37² + 6²
800-1155-1405 which is 5 times (160-231-281)
843-1124-1405 which is (3-4-5) times 281

1404 Texas Tessellation

I recently visited family members in Texas. My daughter-in-law is awesome at both mathematics and quilting. My photo does not do her work justice, but Texas is tessellated in this quilt! She also carefully chose the fabrics she pieced together. Do they remind you of anything for which Texas is famous?

Someone else designed the pattern, but piecing these pieces together was not the easiest sewing project.

I wondered if anyone else had thought to tessellate Texas and found a couple of examples on twitter. As this first one asked, should we call this Texellation?

Now I’ll tell you something about the number 1404:

  • 1404 is a composite number.
  • Prime factorization: 1404 = 2 × 2 × 3 × 3 × 3 × 13, which can be written 1404 = 2² × 3³ × 13
  • 1404 has at least one exponent greater than 1 in its prime factorization so √1404 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1404 = (√36)(√39) = 6√39
  • The exponents in the prime factorization are 2, 3, and 1. Adding one to each exponent and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = 24. Therefore 1404 has exactly 24 factors.
  • The factors of 1404 are outlined with their factor pair partners in the graphic below.

1404 is the hypotenuse of a Pythagorean triple:
540-1296-1404 which is (5-12-13) times 108

Since 1404 has so many factors, it also has MANY different factor trees. Here are four of them mixed in with some Texas tessellations!