1679 and Level 6

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Today’s Puzzle:

Can you complete this multiplication table puzzle? It’s a level 6, so some of the clues might be a little tricky. Use logic every step of the way, and everything will work out for you!

Factors of 1679:

  • 1679 is a composite number.
  • Prime factorization: 1679 = 23 × 73.
  • 1679 has no exponents greater than 1 in its prime factorization, so √1679 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 1679 has exactly 4 factors.
  • The factors of 1679 are outlined with their factor pair partners in the graphic below.

More About the Number 1679:

From OEIS.org we learn that
1 + 6 + 7 + 9 = 23, AND 1679 is divisible by 23.
This is cool: 1679 is the smallest multiple of 23 that can make that claim!

1679 is the hypotenuse of a Pythagorean triple:
1104-1265 -1679, which is 23 times (48-55-73).

1678 and Level 5

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Today’s Puzzle:

Using logic, write the numbers from 1 to 10 in both the first column and the top row so that this puzzle will function like a multiplication table.

Factors of 1678:

  • 1678 is a composite number.
  • Prime factorization: 1678 = 2 × 839.
  • 1678 has no exponents greater than 1 in its prime factorization, so √1678 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 1678 has exactly 4 factors.
  • The factors of 1678 are outlined with their factor pair partners in the graphic below.

More about the number 1678:

Because 2(839)(1) = 1678, you can calculate the only Pythagorean triple that contains the number 1678:
The smaller leg will be 2(839)(1) = 1678.
The longer leg will be 839² – 1² = 703920.
The hypotenuse will be 839² + 1² = 703922.

1677 and Level 4

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Today’s Puzzle:

Using logic, write the numbers from 1 to 10 in both the first column and the top row so that this puzzle forms a multiplication table.

Factors of 1677:

  • 1677 is a composite number.
  • Prime factorization: 1677 = 3 × 13 × 43.
  • 1677 has no exponents greater than 1 in its prime factorization, so √1677 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 1677 has exactly 8 factors.
  • The factors of 1677 are outlined with their factor pair partners in the graphic below.

More About the Number 1677:

The last digit of every prime factor of 1677 is 3.

1677 is the hypotenuse of a Pythagorean triple:
645-1548-1677, which is (5-12-13) times 129.

1676 Really, How Well Can You Read a Table?

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Today’s Puzzle:

Reading a table is an important mathematical skill. Can you read names from a table even if the table is written in a different language? Note: in Hungary, the surnames are written before the given names.

Debreczeni Eszter was born on 23 February 1855. I know this because that was the date written by the minister on her church’s marriage index.

Today’s puzzle: Look at this table of baptisms and determine the names of Debreczeni Eszter’s parents.

If like me, you quickly determined from entry 79 that Debreczeni Eszter was indeed born on February 23 and baptized on March 28, and her parents’ names were Debreczeni János and Rácz Erzsébet, you will feel quite confused when you look at her 1876 marriage record:

Her father’s name was Debreceni Sándor? What? The birth year of 1855 would be right for a 20-year-old marrying in January of 1876, but when I looked for her on FamilySearch, I didn’t find anyone with her name with that father’s name. Did she lie about her age when she married? Did the minister get it wrong on either the marriage record or the marriage index? She was a member of the Reformed Church when she married, but perhaps she was Lutheran or Jewish when she was born? (Those records haven’t been indexed yet.)

I looked at the marriage index again.

You can see their entry on the top by the date jan. 5. As you can see, the minister did not write in dates of birth for all those getting married. Did he get this 1855-ii-23 birthdate wrong?

Knowing that if she lived into the 20th century, there was a good chance the names of her husband as well as both of her parents would appear on her death record, I looked through years of not-yet-indexed death records, and I finally this Debreczeni Eszter record that gives a quick snapshot of her life!

She died 1913 Oct 31 at 2:00. Her name Finta Andrásné (Mrs. András Finta), Debreczeni Eszter. She was 58 years old (born about 1855) when she died. Her husband was Finta András and her parents were the late Debreczeni Sándor and the late N. Nagy Eszter.

I was still puzzled. Searching for Debreczeni Eszter in 1855 through FamilySearch brought up only the Eszter that was a daughter of János and another Eszter, the daughter of Imre. The table of Túrkeve Reformed Church 1855 christenings was 52 pages long and had 324 entries. Perhaps her entry had been indexed incorrectly. I searched again using only first names and found a possible candidate, Nagy Eszter, who was baptized on March 1. I looked at the 1855 baptismal record again. And then I saw it. The minister didn’t get it wrong, the bride didn’t lie about her age: I needed to read the table better! It turns out two baby girls named Debreczeni Eszter were born on February 23rd, but I hadn’t looked past the first one listed. Look at the last entry in the table below. It is the christening record I was looking for!

Eszter, entry number 33, was born on 23 February and baptized on 1 March. Her parents were listed as Debr. Nagy Sándor and Nosza Nagy Eszter. (Having more than one surname was common in Hungary.) When this baptism was indexed by FamilySearch, the parents were understandably indexed as Nagy Sándor and Nagy Eszter, which also let them hide from me easier.

How did you do with this puzzle? You may have been faster than I was, but I knew something was wrong with my findings, and I stuck with it until I figured it out. Those are also important mathematical skills!

Factors of 1676:

Since this is my 1676th post, I’ll write a little about the number 1676.

1676 happens to be 200 years before the marriage I wrote about above.

  • 1676 is a composite number.
  • Prime factorization: 1676 = 2 × 2 × 419, which can be written 1676 = 2² × 419.
  • 1676 has at least one exponent greater than 1 in its prime factorization so √1676 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1676 = (√4)(√419) = 2√419.
  • 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 1676 has exactly 6 factors.
  • The factors of 1676 are outlined with their factor pair partners in the graphic below.

More About the Number 1676:

From OEIS.org, we learn that 1676 = 1¹ + 6²  + 7³ + 6⁴ .

1676 is the difference of two squares:
420² – 418² = 1676.

1675 You CAN Solve This Level 3 Puzzle!

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Great news for math enthusiasts, students, and teachers everywhere! The #148 Playful Math Education Carnival was published today at Math Book Magic!

Today’s puzzle:

I am confident that you can solve this level 3 puzzle! Here’s how: Using only the numbers from 1 to 10, write the factors of 27 and 6 in the appropriate cells. Next, write 18’s factors. Then, since this is a level 3 puzzle, write the factors of 30, 50, 40, 36, 14, 56, and 8, in that order, until all the numbers from 1 to 10 appear in the first column as well as in the top row. As always, there is only one solution.

What did I tell you? You could solve it!

Factors of 1675:

  • 1675 is a composite number.
  • Prime factorization: 1675 = 5 × 5 × 67, which can be written 1675 = 5² × 67.
  • 1675 has at least one exponent greater than 1 in its prime factorization so √1675 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1675 = (√25)(√67) = 5√67.
  • 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 1675 has exactly 6 factors.
  • The factors of 1675 are outlined with their factor pair partners in the graphic below.

More About the Number 1675:

Did you notice that 1675 = 5 × 67 × 5?

How many quarters are in $16.75?
Well, 17 × 4 quarters = 68 quarters = $17.00.
Subtracting one quarter from both sides of the equation, we get
67 quarters = $16.75.

1675 is the hypotenuse of TWO Pythagorean triples:
469-1608-1675, which is (7-24-25) times 67, and
1005-1340-1675, which is (3-4-5) times 335.

1675 is the difference of two squares in THREE different ways:
838² – 837² =  1675,
170² – 165² =  1675, and
46² – 21²  =  1675.

1674 and Level 2

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Today’s Puzzle:

Make a multiplication table out of this puzzle. Can you see how to do it? The factors won’t be in the usual order, but I’m sure you can figure it out!

Factors of 1674:

  • 1674 is a composite number.
  • Prime factorization: 1674 = 2 × 3 × 3 × 3 × 31, which can be written 1674 = 2 × 3³ × 31.
  • 1674 has at least one exponent greater than 1 in its prime factorization so √1674 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1674 = (√9)(√186) = 3√186.
  • The exponents in the prime factorization are 1,3 and 1. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1674 has exactly 16 factors.
  • The factors of 1674 are outlined with their factor pair partners in the graphic below.

More about the number 1674:

From OEIS.org we learn that
1/1 + 1/2 + 1/3 + 1/4 + . . . + 1/1672 + 1/1673 ≈ 7.999888, but if you add the next tiny fraction, 1/1674, the sum will be a tiny bit more than 8 or approximately 8.000486.

That’s adding a whole lot of unit fractions just to get a sum over 8.

 

1673 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues become a not-in-the-usual-order multiplication table.

Factors of 1673:

  • 1673 is a composite number.
  • Prime factorization: 1673 = 7 × 239.
  • 1673 has no exponents greater than 1 in its prime factorization, so √1673 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 1673 has exactly 4 factors.
  • The factors of 1673 are outlined with their factor pair partners in the graphic below.

More About the Number 1673:

The factors of 1673 are 1, 7, 239, and 1673. OEIS.org tells us something cool about the sum of the squares of those factors:
1² + 7² + 239² + 1673² = 1690².

1672 A Pythagorean Triple Puzzle with Only Two Clues

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Today’s Puzzle:

If you would like to try a Pythagorean Triple Puzzle that isn’t too difficult, I’ve got you covered with today’s puzzle:

PUZZLE DIRECTIONS: This puzzle is NOT drawn to scale. Although all of the angles may look like 60-degree angles, none of them are. The marked angles are 90 degrees. Lines that look parallel are NOT parallel. Although side lengths look equal, they are NOT equal. Most rules of geometry do not apply here. When drawn to scale, none of the triangles in this particular puzzle overlap, but that is not always the case for some of my other Pythagorean triple puzzles.

No geometry is needed to solve this puzzle. All that is needed is logic and the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

Sorted Triples

You can also print and cut out the triangle cards below to solve the puzzle.

Factors of 1672:

  • Prime factorization: 1672 = 2 × 2 × 2 × 11 × 19, which can be written 1672 = 2³ × 11 × 19.
  • 1672 has at least one exponent greater than 1 in its prime factorization so √1672 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1672 = (√4)(√418) = 2√418.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1672 has exactly 16 factors.
  • The factors of 1672 are outlined with their factor pair partners in the graphic below.

More About the Number 1672:

1672 is the difference of two squares in FOUR different ways:
419² – 417² = 1672,
211² – 207² = 1672,
49² – 27² = 1672, and
41² – 3² = 1672.
That last one means we are only 9 numbers away from the next perfect square, 1681.

From NumbersaPlenty, we learn that 1672 is a digitally powerful number:
1672 = 1¹ + 6⁴ + 7³ + 2⁵.

1671 Are Vaccines Really Much of a Mystery?

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Today’s Puzzle:

Why is this puzzle a mystery-level puzzle? As long as you don’t make any assumptions, it really isn’t all that difficult. Study all the clues. Some of them are tricky, but there is ONE logical place to start that won’t make you guess, check, erase and try again. Don’t let any other clues fool you into thinking you should start with them. Find that one logical place to start, then write the numbers from 1 to 14 in both the first column and the top row so that those numbers and the given clues create a multiplication table. As always, there is only one solution.

Getting vaccinated against COVID is also not much of a mystery as long as you don’t make any dangerous assumptions. I am grateful that I received my doses of the Pfizer vaccine on 22 February and 15 March 2021.

My daughter’s brother-in-law was a healthy police officer who started having difficulty breathing. When it got worse, he was hospitalized. Before long he had to be put into a coma. He had significant lung fibrosis due to COVID complications. Even if he could recover, his life would never again be what it used to be. A few days after he was hospitalized, his wife and children also came down with COVID, so they were no longer allowed to visit him in the hospital.  His coworkers asked his wife what they could do for her. Without reservation, she answered, “Get vaccinated!” The doctor video-chatted with his family and told them to start thinking about preparing themselves for his death. On a Sunday afternoon, they made a video call as his mother held up the phone for him in the hospital. They said goodbye, but how much does a person in a coma hear? How satisfying is a video call for a final goodbye?

On Tuesday, there seemed to be hope. A new doctor felt that while he wasn’t getting any better, he wasn’t getting any worse either. His heart was still strong and his other organs were working. There was still hope. If he got better, he could get a lung transplant. A miracle could still happen.

By Thursday, other organs began to fail, and all hope was gone. His family had recovered enough to visit him in the hospital to say their final goodbyes. It was better than a video call but still terribly heartbreaking.

Although I had probably only met him once, seven years ago when my daughter married into his wife’s family, I spent a lot of time crying when I got the news.

His funeral will be tomorrow. He was the first active-duty police officer in my state to die from COVID 19. Here his wife describes what a great loss his death is to his community and to their family. She pleads with everyone to take the virus seriously and get vaccinated.

Please, do everything you can to protect yourself and others from COVID 19. Get vaccinated. Social distance as best you can. Wear a mask.

Factors of 1671:

This is my 1671st post, so I’ll share the factors of 1671:

  • 1671 is a composite number.
  • Prime factorization: 1671 = 3 × 557.
  • 1671 has no exponents greater than 1 in its prime factorization, so √1671 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 1671 has exactly 4 factors.
  • The factors of 1671 are outlined with their factor pair partners in the graphic below.

More About the Number 1671:

1671 is the hypotenuse of a Pythagorean triple:
495-1596-1671 which is 3 times (165-532-557).

1670 Mystery Puzzle

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Today’s Puzzle:

The logic needed to solve this puzzle is a bit complicated. If you would like a hint, I’ll give one later in the post that will remove most of the mystery in the puzzle. Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues become a multiplication table.

Logic Hint:
The 33 means that the 9 cannot be 3 × 3, so the 9 and the 54 must use both 9’s.
The 54 also must use a 6.
Thus the 36 near the bottom of the puzzle cannot be 9 × 4 or 6 × 6 and must be 3 × 12.
84 and that bottom 36 must use both 12’s, so the 48 must be 6 × 8 and not 12 × 4.
The rest of the puzzle should be rather straightforward.

Factors of 1670:

  • 1670 is a composite number.
  • Prime factorization: 1670 = 2 × 5 × 167.
  • 1670 has no exponents greater than 1 in its prime factorization, so √1670 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 1670 has exactly 8 factors.
  • The factors of 1670 are outlined with their factor pair partners in the graphic below.

More About the Number 1670:

1670 is the hypotenuse of a Pythagorean triple:
1002-1336-1670, which is (3-4-5) times 334.

Although it is very much irrational, OEIS.org informs us that the first few digits of
1670^(1/6) is 3.44444624848…