DNA and Big Brother

When my husband was a little baby, his dad filled out the genealogy section in his baby book in his beautiful, distinct handwriting:

Even though most of the pages are blank, my husband has always cherished that book, and it has been extremely helpful in finding many other of his ancestors.

From additional research, we have learned that my husband’s grandfather, Frank Kovach, was born Kovács Ferenc in Szürte, Ung County, Hungary. That little town has had several border changes and is now part of Ukraine, but still only about eight miles from the Hungarian border. You can see a map showing the location of Szürte in a post I wrote a couple of years ago. Ferenc (Frank) was born 13 June 1883 to Kovács Péter and Péntek Mária, that’s their names in Hungarian name order. The baby book gives their names in English name order.

Many years ago when I tried to figure out Frank’s place of birth, I found three other people whose parents had the same names as his parents. Could they be Frank’s siblings? Could the two boys be his big brothers? (You will need to be logged into FamilySearch.org and Ancestry.com to see most of the links I’ve included in this post.)

  1. Julia Kovach (Kovács Juliánna) was born 12 Apr 1882 in Hungary (both of her parents were born in Ung County, Hungary!). She died 15 Jun 1940 in Cleveland, Ohio. Maybe Frank was also born in Ung County, I excitedly thought! Several years later I found a death record for one of Frank’s sons that gave the specific town in Ung county where Frank was born. Still years after that I found Frank’s petition for naturalization also confirming it.
  2. Steven Kovach (Kovács István) was born about 1874 in Hungary. He married Julia Csengeri on 22 Sept 1901 in New York.

    He MAY have died just a few short years later on 11 Dec 1918 in Union, Washington, Pennsylvania, but buried in Cleveland, Ohio.  The father on that death certificate was Pete Kovacs and the mother was Mary Pantik. The certificate says he is married, but there was no place to write the wife’s name on it. The informant was Steve Kovach, which just happens to be Julia Kovach’s husband’s name, so her husband might have actually been the informant. Julia and Steve lived in Cleveland, and the deceased, Steve, was buried in Cleveland even though he died in Pennsylvania.
  3. John Kovacs (Kovács János) was born 23 Jan 1870 in Hungary. He died 29 Oct 1943 in Cleveland. To fully appreciate the information for John, we need to look at his and his wife’s death certificates side by side.

Notice that the address for both John and Veronica is 9012 Cumberland, so that helps to establish that they were husband and wife even though the spellings of their last names are not exactly the same. This is important since there were MANY men named John Kovach in Cleveland. The couple’s shared tombstone confirms the dates given above. On Veronica’s death certificate, her father is listed as John Daniels and the informant is Dale Kovats. Further research establishes that Dale is John and Veronica’s son, and the 1940 census shows Dale and his wife, Rose at the bottom of the page, and their daughter, Joanne, and some of Rose’s relatives on the top of the next page. Dale is the key to this puzzle because Dale has a descendant who is a 3rd to 4th cousin DNA match to my husband! That means that John Kovacs is indeed Frank’s big brother, and I am in tears as I am finally able to positively make that statement.

Ancestry.com explains “Our analysis of your DNA predicts that this person you match with is probably your third cousin. The exact relationship however could vary. It could be a second cousin once removed, or perhaps a fourth cousin. While there may be some statistical variation in our prediction, it’s likely to be a third cousin type of relationship—which are separated by eight degrees or eight people. However, the relationship could range from six to ten degrees of separation.” (bold print added)

My husband, Steven, and this DNA match are separated by seven degrees.

Was big brother John also born in Szürte? It seems likely, but he may have also been born about 3 miles away in Kholmetz where a 4th-6th cousin DNA shared match traces her ancestry. If only I could get into the Szürte Reformed Church records and Kholmetz records to look for a Kovács János (John Kovacs) born on 23 Jan 1870 as well as the records for the others and certainly a few more siblings as well!

 

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Find the Factors (ax±b)(cx±d)

I liked making a puzzle using trinomials earlier today. This one will take more skill to solve even though it contains fewer trinomials. Some of the factors will have negative numbers, and the leading coefficients of the trinomials are not 1.

In this puzzle, you can see the number 24 twice. It needs to be factored to solve the puzzle. It might be 3 × 8 or 4 × 6, but it can’t be 1 × 24 or 2 × 12 because for this puzzle ALL of the factors of 24 have to be non-zero integers from -10 to +10.

Every factor must appear once in the first column and once in the top row. So if you put 2x + 5 in the top row, you will also have to put 2x + 5 somewhere in the first column as well.

Sometimes all of the terms in the trinomial have a common factor and can, therefore, be factored further, but don’t worry about that right now.

You will have to find all of the factors in the puzzle before you can figure out what the missing clue should be. That’s about all the mystery I can put in a puzzle like this. Good luck with it!

Since this is different than any other puzzle I’ve ever published, you can see the solution here:

Positive Trinomial Puzzle

Today on Twitter, Mr. Allen requested some good problem-solving resources for quadratics. He made up one himself.

I decided to make one as well. It is similar to my other Find the Factors puzzles. You will have to use logic to solve it, but in many ways, it will be easier to solve than most of my regular puzzles. Like always, there is only one solution.

Every term is positive so if you already know how to factor trinomials it should be relatively easy to solve. All the factors from (x + 1) to (x + 9) need to appear exactly one time in both the first column and the top row of the puzzle.  Once all the factors are found, the puzzle is solved, but you can find all the products of those factors and write them in the body of the puzzle if you want.

Easter Basket Challenge

Occasionally,  we hear that the number of Easter eggs that are found is one or two less than the number of eggs that were hidden. Still most of the time, all the eggs and candies do get found. You really have no trouble finding all those goodies, and the Easter Egg Hunt seems like it is over in seconds.  You can find Easter Eggs but can you find factors? Here’s an Easter Basket Find the Factors 1 – 10 Challenge Puzzle for you. I guarantee it won’t be done in seconds. Can you find all the factors? I dare you to try!

1365 Shamrock Mystery

Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Now I’ll share some information about the number 1365:

  • 1365 is a composite number.
  • Prime factorization: 1365 = 3 × 5 × 7 × 13
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
  • Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
  • Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
  • 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591

1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:

1363 and Level 6

The common factors of 60 and 30 allowed in the puzzle are 5, 6, and 10. Which one is the logical choice? Look at the other clues in the puzzle and you should be able to eliminate two of the choices.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few thoughts about the puzzle number, 1363:

  • 1363 is a composite number.
  • Prime factorization: 1363 = 29 × 47
  • 1363 has no exponents greater than 1 in its prime factorization, so √1363 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 1363 has exactly 4 factors.
  • The factors of 1363 are outlined with their factor pair partners in the graphic below.

1363 is the hypotenuse of a Pythagorean triple:
940-987-1363 which is (20-21-29) times 47

1361 and Level 5

If you carefully use logic instead of guessing and checking you can find the unique solution to this puzzle without tearing your hair out!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1361:

  • 1361 is a prime number.
  • Prime factorization: 1361 is prime.
  • 1361 has no exponents greater than 1 in its prime factorization, so √1361 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1361 has exactly 2 factors.
  • The factors of 1361 are outlined with their factor pair partners in the graphic below.

How do we know that 1361 is a prime number? If 1361 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1361. Since 1361 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1361 is a prime number.

1361 is the first prime number after 1327. That was 34 numbers ago!

1361 is the sum of two squares:
31² + 20² = 1361

1361 is the hypotenuse of a Pythagorean triple:
561-1240-1361 calculated from 31² – 20², 2(31)(20), 31² + 20²

Here’s another way we know that 1361 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 20² = 1361 with 31 and 20 having no common prime factors, 1361 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1361. Since 1361 is not divisible by 5, 13, 17, or 29, we know that 1361 is a prime number.

1360 and Level 4

There is only one way to write the numbers 1 to 12 in the top row and the first column of this puzzle so that the puzzle could turn into a multiplication table. Can you find the solution?

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1360:

  • 1360 is a composite number.
  • Prime factorization: 1360 = 2 × 2 × 2 × 2 × 5 × 17, which can be written 1360 = 2⁴ × 5 × 17
  • 1360 has at least one exponent greater than 1 in its prime factorization so √1360 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1360 = (√16)(√85) = 4√85
  • The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1360 has exactly 20 factors.
  • The factors of 1360 are outlined with their factor pair partners in the graphic below.

1360 is the sum of two squares in two different ways:
36² + 8² = 1360
28² + 24² = 1360

1360 is the hypotenuse of FOUR Pythagorean triples:
208-1344-1360 which is 16 times (13-84-85)
576-1232-1360 which is 16 times (36-77-85)
640-1200-1360 which is (8-15-17) times 80
816-1088-1360 which is (3-4-5) times 272

1359 and Level 3

The first row with a clue has a 6 in it. Use that 6 and another clue that goes with it, to figure out where to put the factors of 6 in this puzzle. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few facts about the number 1359:

  • 1359 is a composite number.
  • Prime factorization: 1359 = 3 × 3 × 151, which can be written 1359 = 3² × 151
  • 1359 has at least one exponent greater than 1 in its prime factorization so √1359 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1359 = (√9)(√151) = 3√151
  • 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 1359 has exactly 6 factors.
  • The factors of 1359 are outlined with their factor pair partners in the graphic below.

1359 is the difference of two squares in three different ways:
680² – 679² = 1359
228² – 225² = 1359
80² – 71² = 1359

1358 and Level 2

If you are familiar with a basic 12 × 12 multiplication table, then you can solve this puzzle. The clues aren’t in the same order as they are in the table, but that only makes it a little more challenging.

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here’s a little bit about the number 1358:

  • 1358 is a composite number.
  • Prime factorization: 1358 = 2 × 7 × 97
  • 1358 has no exponents greater than 1 in its prime factorization, so √1358 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 1358 has exactly 8 factors.
  • The factors of 1358 are outlined with their factor pair partners in the graphic below.

910-1008-1358 which is 14 times (65-72-97)

Stetson.edu informs us that 1358!!!! + 1 is a prime number.