1415 and Level 6

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Very likely when you look at this puzzle common factors of 40 and 10, 8 and 16, 9 and 18, and 20 and 40 will pop into your head. Will they be the right common factors that work with all the other clues in the puzzle to produce a unique solution? Let logic be your guide when finding the factors.

Now I’ll tell you something about the puzzle number, 1415:

  • 1415 is a composite number.
  • Prime factorization: 1415 = 5 × 283
  • 1415 has no exponents greater than 1 in its prime factorization, so √1415 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 1415 has exactly 4 factors.
  • The factors of 1415 are outlined with their factor pair partners in the graphic below.

1415 is also the hypotenuse of a Pythagorean triple:
849-1132-1415 which is (3-4-5) times 283.

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1414 Your Math Education Post Will Add So Much to This Month’s Carnival!

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Have you written a blog post that would bring delight to a preschool, K-12 or homeschool mathematics teacher or student? Then submit it to this month’s Playful Math Education Blog Carnival or message me on Twitter by Friday, September 20th! I’m hosting the carnival this month, and I would love to read your post. So come join the fun!

Today’s puzzle looks a little like a wild, but fun? carnival ride. The numbers 36 and 12 went together on the ride. They managed to stay with each other but the ride went so fast, you can see 36 and 12 in two different places at the same time. There’s also poor number 40. You can see it in THREE places at the same time.

Oh my! Can you use logic to find where the numbers 1 to 10 need to go in both the first column and the top row so that this wild ride will behave like a multiplication table? It’s a level 5 so it won’t be easy to find its unique solution. Are you brave enough to try?

That puzzle’s number is 1414. Let me tell you a little about that number:

  • 1414 is a composite number.
  • Prime factorization: 1414 = 2 × 7 × 101
  • 1414 has no exponents greater than 1 in its prime factorization, so √1414 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 1414 has exactly 8 factors.
  • The factors of 1414 are outlined with their factor pair partners in the graphic below.

1414 is also the hypotenuse of a Pythagorean triple:
280-1386-1414 which is 14 times (20-99-101)

 

DNA Evidence at Ellis Island

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About the middle of August, Ancestry.com contacted my husband informing him that he had a new DNA match who was his second or third cousin. I was very excited to look into it. This new match is my husband’s second-best match. The two of them share 204 centimorgans (cM) across 3 DNA segments. There were several shared matches between them, and based on them, I was confident that the DNA they shared was from his mother’s side of the family. The surname on the match was Kovacs (Equivalent to Smith in English), and I was hopeful that there would be a connection to one of the known or probable siblings of my husband’s grandfather, Frank Kovach.

I immediately looked at the match’s pedigree. The names of the living were not given, but it appears that the match was the grandchild of a Mr. Kovacs and Betty Baker who were married on 26 February 1960 in Trumbull, Ohio. Mr. Kovacs was the son of William Ray Kovacs Sr and Barbara Bernice Jennings who were married 13 April 1937 in Pomeroy, Washington.  That marriage record indicated that William Sr’s parents were Samuel Kovacs and Elizabeth Jenney. I didn’t find any other records for Samuel, but I did find several for Sandor Kovacs and Elizabeth Jeney. Perhaps, the clerk had mistakenly written Samuel instead of Sandor on that marriage record. The 1940 Census shows a William R Kovacs, his wife, Barbara, and their two children living in Trumbull County, Ohio. That’s where Sandor eventually settled.

Samuel or Sandor.  I was hoping to see John, Stephen, or Julia. I was feeling a little disappointed that I wasn’t seeing the connection I had hoped for. I looked at this DNA match’s ethnicity tab on Ancestry.com. My husband is 98% Eastern Europe and 2% Baltic States. This match was only 4% Eastern Europe, 0% Baltic States, and 96% other places. Doubt crept in. How could these two possibly be 2nd or 3rd cousins? That just seemed too close with so little shared ethnicity.

After I got over my initial disappointment, I looked at my husband’s grandfather’s 1938 petition for naturalization. It stated that he, Frank Kovach, was born in Szürthe, Czechoslovakia (previously Hungary, but currently part of Ukraine) and that he immigrated to the United States on 16 June 1902. I was able to find this page of the 16 June 1902 New York arrival manifest for the ship Vaderland when he arrived at Ellis Island.  I had not seen this manifest before, and it gave me some wonderful information:

  • Ferencz Kovach is the fourth name from the bottom of the manifest. (Ferencz is the Hungarian equivalent of Frank.)
  • The ship, Vaderland, set sail from Antwerp, Belgium on 7 June 1902. It was nine days later when Ferencz got to Ellis Island. (The ship probably arrived at New York sooner than nine days, but each ship had to wait its turn in the harbor for its passengers to be processed.)
  • When he arrived at Ellis Island, Ferencz was a 19-year-old single male in good health, yet he had only one dollar in his pocket. He told officials that his occupation was a laborer. He came here to work!
  • There were a few other Hungarians listed on this same page of the manifest, but Ferencz was the only one from Szürte. Still, he had at least a few people he could speak to in Hungarian on the voyage.
  • It was his first trip to the United States. His brother, Alexander Kovacs, paid for his passage. Ferencz was going to McKeesport, Pennsylvania where his brother lived at 817 Jerome Street. Alexander is the English equivalent of the Hungarian given name Sándor! That meant that Sandor Kovacs was Ferencz’s big brother, AND he was the one who helped him get to America! It also means that the third great grandfather of my husband’s DNA match was indeed named Sandor and not Samuel.

Here is a descendant chart showing how my husband is connected to this DNA match. I would have expected the DNA match to have 12.5% Eastern European ethnicity, so 4% is remarkably low. Ancestry.com says there is only a 2% chance that two people sharing their amount of DNA would only be 2nd cousins, twice removed. We each get 50% of our DNA from both parents, but the 50% we get isn’t necessarily evenly distributed from every previous generation!

Now I wanted to know all I could about this Alexander/Sandor Kovacs! I found out that Sandor and his wife welcomed a new baby boy into their family just a few months earlier. They named him Chas, and he was born on 23 November 1901. Sandor was a miner at the time, a very dangerous occupation. Note that Chas’s birth was not registered until 6 January 1902.  That may be why his birth year was mistakenly listed as 1902 on his birth certificate. His birth certificate lists his father’s birthplace as Szürte and his mother’s birthplace as Gönc. I was so happy to see those birthplaces!

When Ferencz arrived at Ellis Island, he must have been very excited to see his brother, his wife, Elizabeth, and their 6 1/2-month-old baby boy.

I constructed a table of the household of Sandor Kovacs from 1910, 1930, and the 1940 Censuses.  The dates of birth were found in other records that are included at the bottom of this post.

The April 1910 Census had Alexander Kovacs employed as a helper in the steelworks industry and living at 917 Chestnut Street in Duquesne, Allegheny, Pennsylvania. The census indicated that he immigrated to the United States in 1895 and was now a naturalized citizen. It also includes his brother-in-law, John Jeney, who was an engineer in the Steelworks industry.

That census record led me to the manifest showing Sándor Kovács at Hamburg on 21 August 1895 as he traveled to Amerca. His is the sixth surname from the bottom on the right side of the manifest. Szürte is in Ung county, the previous residence listed for him on the manifest.

The 1910 census record stated that AlexanderJr was born in Hungary in 1905. What was that all about? I found 1905 civil registration records from Gönc, Hungary for this family!

In the margin of the right side is the civil registration of their marriage, we learn that Kovács Sándor and Jenei Erzsébet were married in the Reformed Hungarian Church in Pittsburgh on 6 November 1900 and that Jenei Erzsébet had been born 11 July 1878 in Gönc.  I wondered if I could get a copy of the marriage record from the church in Pittsburgh. Then it occurred to me that it might be in the Family History Library in downtown Salt Lake City.  It was! I went to the library the first day I could after work and found it! Click on it to see it better.

Indeed, in Pittsburgh on 6 November 1900, 28-year-old Kovács Sándor, the son of the late Kovács Péter and Péntek Mária wed 22-year-old Jeney Erzsébet, the daughter of Jeney János and Laczkó Mária. He was born in Szürte and she was born in Göncz. I did not know before I saw this record that Sandor and Ferencz’s father, Péter, had died before Ferencz left Szürte to go to America.

I would have preferred to have the entire page from the anyakönyv, but the projector at the library didn’t focus very well when I tried to get the entire page, and I could only get a blurred copy of the full page below.

Thus, DNA led me to Ellis Island where I found my husband’s wonderful great uncle. I am beyond thrilled! I can tell that he was a very kind man because he paid for his little brother’s passage to America and he allowed his grown children to live with him in 1940 as the country was getting over the Great Depression.

Here are the family records that I found for this family:

Károly Kovács (AKA Carl, Chas, Charles) born 23 November 1901 in West Virginia. The record indicates that both Sándor and Erzsébet were living in Gönc in 1905 when this civil registration occurred.

Sándor Kovács (AKA Alex, Alexander) born 3 January 1905 in Gönc, Hungary. 

1930 Census

1940 Census Page 1 includes his daughter Helen Haught and her husband Terrance Haught.

1940 Census Page 2 includes granddaughter Helen Haught and his daughter, Mary Kovacs Collins, who lived next door with her husband and three children. (See Grave Stone and Obituary for Mary Kovacs Collins born 18 Jun 1908 and died 18 Jul 1987).

8 Jan 1953 Death record of Sandor Kovacs from his Find A Grave Memorial.

Edna Kovacs Staub born 19 Feb 1911, married Wayne Staub 11 Jan 1930

Alex Kovacs married Mary Ann Rusky 20 Dec 1948 in Mt Clemens, Macomb, Michigan

19 Oct 1995 Warren, Trumbull, Ohio Death Record for James Kovacs who was born on 29 Nov 1909

4 Sept 2004 death index of Helen Kovacs Waldron who was born 24 Mar 1912, daughter of Kovacs and Jenei

21 Mar 2000 Death of William R. Kovacs Sr who was born 29 June 1914

 

 

 

 

 

 

 

 

 

1413 and Level 4

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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)

1412 and Level 3

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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

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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

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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

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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

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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

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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.