1697 A Boot in the Window

Today’s Puzzle:

Tonight throughout many parts of the world children will place their polished boots in a window awaiting a visit from St. Nick. In the morning they will find their boots filled with favorite candies.

You can solve this boot puzzle by writing the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will become the start of a multiplication table.

Factors of 1697:

  • 1697 is a prime number.
  • Prime factorization: 1697 is prime.
  • 1697 has no exponents greater than 1 in its prime factorization, so √1697 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1697 has exactly 2 factors.
  • The factors of 1697 are outlined with their factor pair partners in the graphic below.

How do we know that 1697 is a prime number? If 1697 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1697. Since 1697 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1697 is a prime number.

More About the Number 1697:

1697 is the sum of two squares:
41² + 4² = 1697.

1697 is the hypotenuse of a Pythagorean triple:
328-1665-1697, calculated from 2(41)(4), 41² – 4², 41² + 4².

Here’s another way we know that 1697 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 41² + 4² = 1697 with 41 and 4 having no common prime factors, 1697 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1697. Since 1697 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1697 is a prime number.

1696 Inverses

Today’s Puzzle:

Joseph Nebus of Nebusresearch recently wrote an extensive post about inverses at my request. Although inverses are often fascinating, advanced topics in mathematics, they can also be quite simple. For example, solving this puzzle will involve using the inverse of multiplication, division, AND it is the simplest division possible this time. Since it is December, I made this puzzle look like a gift for you. Just write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues could become a multiplication table.

For any gift you receive, you can invoke inverse exploring questions such as “how was this gift made?”

If you want to print the puzzle without color, here it is:

Factors of 1696:

  • 1696 is a composite number.
  • Prime factorization: 1696 = 2 × 2 × 2 × 2 × 2 × 53, which can be written 1696 = 2⁵ × 53.
  • 1696 has at least one exponent greater than 1 in its prime factorization so √1696 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1696 = (√16)(√106) = 4√106.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1696 has exactly 12 factors.
  • The factors of 1696 are outlined with their factor pair partners in the graphic below.

More About the Number 1696:

1696 is the sum of two squares:
36² + 20² = 1696.

1696 is also the hypotenuse of a Pythagorean triple:
896-1440 -1696, which is 32 times (28-45-52).
It can also be calculated from 36² – 20², 2(36)(20), 36² + 20².

1696 is also the difference of two squares in FOUR different ways:
425² – 423² = 1696,
214² – 210² = 1696,
110² – 102² = 1696, and
61² – 45² = 1696.
I found those equations by using the even factor pairs of 1696 and taking the inverse of the fact that a² – b² = (a + b)(a – b).

1695 If You Can Solve 3×3 and 5×5 Magic Squares, Then You Can Solve a 15×15 Magic Square!

Today’s Puzzle:

Completing a 15 × 15 magic square may seem daunting, but I assure you, if you can solve a 3 × 3 magic square and a 5 × 5 magic square, then you can complete a 15 × 15 magic square!

The 15 × 15 magic square below is made with twenty-five 3 × 3 magic squares. See the most famous 3 × 3 square in yellow? Do you see that the first 25 multiples of 9 are along the diagonals I’ve drawn? Do you understand the pattern that was used to make this magic square? Study it, then without looking at this one, can you make your own? Open this Excel file, 12 Factors 1683-1695, enable editing, and the sums of each row, column, and diagonal will automatically populate as you type in the numbers.

This next 15 × 15 magic square is made with nine 5 × 5 magic squares. The one in yellow is one of MANY possible 5 × 5 squares. Do you see the first nine multiples of 25 along the diagonals I’ve drawn? Once you understand this pattern, perhaps you would like to take a turn duplicating it. Try it yourself on that same Excel file, 12 Factors 1683-1695.

Factors of 1695:

  • 1695 is a composite number.
  • Prime factorization: 1695 = 3 × 5 × 113.
  • 1695 has no exponents greater than 1 in its prime factorization, so √1695 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 1695 has exactly 8 factors.
  • The factors of 1695 are outlined with their factor pair partners in the graphic below.

More About the Number 1695:

Since 1695 is the sum of three consecutive numbers, it is the magic sum of a particular 3 × 3 Magic Square. Those three consecutive numbers are in yellow in the square below.

Also, since 1695 is the sum of the five consecutive numbers shown in yellow below, it is the magic sum of this 5 × 5 magic square:

And lastly, 1695 is the sum of the 15 consecutive numbers shown in yellow below, so here is yet another 15 × 15 magic square with 1695 as the magic sum. This magic square uses the same pattern that works for all odd number magic squares. It is so satisfying to complete it yourself. Study this one and then give it a try! Open that same Excel file, 12 Factors 1683-1695, to make Excel to all the adding for you.

1695 is also the hypotenuse in FOUR Pythagorean triples:
225-1680-1695, which is 15 times (15-112-113),
828-1479-1695, which is 3 times (276-493-565)
1017-1356-1695, which is (3-4-5) times 339, and
1188-1209-1695, which is 3 times (396-403-565).

Did you notice that 565, 339, or 113 was a center number in every magic square in this post?

I hope you enjoy completing these magic squares on your own as you explore the number 1695.

1694 Football Game Day

Today’s Puzzle:

Today all over the United States family and friends will gather to watch or play a game of football. If you would like to change things up a little, here’s a game ball for you to practice multiplication and division facts. Just write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues form a multiplication table. Some clues might be tricky, but enough of them aren’t that I am confident you can score with this football!

Here’s the same puzzle without any distracting color:

Factors of 1694:

  • 1694 is a composite number.
  • Prime factorization: 1694 = 2 × 7 × 11 × 11, which can be written 1694 = 2 × 7 × 11².
  • 1694 has at least one exponent greater than 1 in its prime factorization so √1694 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1694 = (√121)(√14) = 11√14.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1694 has exactly 12 factors.
  • The factors of 1694 are outlined with their factor pair partners in the graphic below.

More About the Number 1694:

From OEIS.org we learn that 1694³ = 4,861,163,384, a number that uses each of the digits 1, 3, 4, 5, and 8 exactly twice.

1693 Tricky Turkey

Today’s Puzzle:

You cannot gobble this turkey up unless you can find all of its factors!

Use logic and multiplication facts. It won’t be easy, but write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Here’s the same puzzle without the added color:

Factors of 1693:

  • 1693 is a prime number.
  • Prime factorization: 1693 is prime.
  • 1693 has no exponents greater than 1 in its prime factorization, so √1693 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1693 has exactly 2 factors.
  • The factors of 1693 are outlined with their factor pair partners in the graphic below.

How do we know that 1693 is a prime number? If 1693 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1693. Since 1693 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1693 is a prime number.

More About the Number 1693:

1693 is the sum of two squares:
37² + 18² = 1693.

1693 is the hypotenuse of a Pythagorean triple:
1045-1332-1693, calculated from 37² – 18², 2(37)(18), 37² + 18².

Here’s another way we know that 1693 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 18² = 1693 with 37 and 18 having no common prime factors, 1693 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1693. Since 1693 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1693 is a prime number.

1692 A Pilgrim’s Belt to Unbuckle

Today’s Puzzle:

The logic needed to unbuckle this Pilgrim’s belt puzzle has several interesting twists and turns in it. Even adults will find it a challenge. Guessing and checking will only frustrate you. Use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work like a multiplication table.

Here’s the same puzzle if you want to print it using less ink.

Factors of 1692:

  • 1692 is a composite number.
  • Prime factorization: 1692 = 2 × 2 × 3 × 3 × 47, which can be written 1692 = 2² × 3² × 47.
  • 1692 has at least one exponent greater than 1 in its prime factorization so √1692 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1692 = (√36)(√47) = 6√47.
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1692 has exactly 18 factors.
  • The factors of 1692 are outlined with their factor pair partners in the graphic below.
  • 1568 is a composite number.

More About the Number 1692:

1692 is the difference of two squares in three different ways:
424² – 422² = 1692,
144² – 138² = 1692, and
56² – 38² = 1692.

The square of 1692 looks a little interesting:
1692² = 2862864.

Do These Records Refer to One Person, Two People, or More?

Some of my husband’s ancestors lived in Dévaványa, Hungary in the 1700s. I came across a death record for an individual who died in Dévaványa but was born in Gyoma, the place where two of my husband’s grandparents were born.

Dévaványa civil death record, number 323, 1899 October 24, Csáki Gergely, age 73 (born about 1827) in Gyoma, married to Karádi Eszter, his parents were the late Csáki István and the late Kis Rebeka.

My husband had a 4th great-grandmother named Kis Erzsébet who lived in Gyoma. Perhaps this Csáki Gergely would be related to her somehow? I decided to look for him in other records and found that he and his wife had one child born in Gyoma.

Gyoma Reformed Church christening record, page 877, Line 188, born 1859 August 6, christened 1859 August 8, a girl, Ester, born to Csáki Gergely and Karádi Ester.

That matches the information in Csáki Gergely’s death record. Great! Let me see if I can find his christening record. I’m looking for a Gergely who was born about 1827 and his parents were Csáki István and Kis Rebeka. I did find a Gergely christened on 12 September 1827 whose father was Csáki István, but his mother’s name was Fekete Rebeka. Also that Gergely died a half year later on 25 April 1828. The couple had another son, István, christened 23 Jan 1829, and one more they named Gergely christened on 12 January 1830. Could that second Gergely be the same as the one who died in 1899?

Suppose one of your relatives died and you reported the death to the authorities. Would you get his/her birth year and mother’s maiden name right? I’m not ready to declare that these two documents refer to the same person, but I’m also not going to eliminate the possibility either.

Some people in Hungary essentially had two surnames. Perhaps Rebeka sometimes listed her surname as Kis. I searched for István and Rebeka’s marriage record.

Gyoma Reformed Church marriage record, page 1009, last entry for 1826 November 15, the widower young Csáki István, age 25 (born about 1801) and Fekete György’s daughter Rebeka, age 29 (born about 1797).

There is no mention of Kis there, and young Csáki István was already a widower? I looked for Rebeka’s christening record and found she appears to have been christened on 13 Jun 1797, the daughter of Vanyai Fekete György and Szabó Ilona. (Vanyai means he was from Dévaványa.) Still no mention of the surname Kis. I found these parents’ marriage record.  Gyoma Reformed Church marriage record, 2nd entry for 29 January 1794, Dévaványa resident Fekete György, age 25 (born about 1769), and the late Bálint Szabó Márton’s daughter Ilona, age 21 (born about 1773). Again, no mention of Kis on either her father’s side or her mother’s side. It looks like I’m at a dead end.

All this means Csáki Gergely might not be related to my husband’s 4th great-grandmother after all, but my curiosity was still peaked about Csáki István being a widower at age 25. I found three records of interest.

  1. Gyoma Reformed Church marriage record page 1000, 16 November 1825, Csáki Gergely’s son István, age 24, wed B. Szabó János’s daughter, Mária, age 17 (born about 1808). This is one day less than one year before the widower Csáki István married Fekete Rebeka. And her name is B. Szabó Mária? Could she be related to Fekete Rebeka’s mother, Bálint Szabó Ilona?
  2. Gyoma Reformed Church christening record, page 73, 21 September 1826, the child christened, Imre, his parents were Csáki István and Kaptsos Mária.  Mária is the right given name, but Kaptsos and B. Szabó are not the same names.
  3. Gyoma Reformed Church death record, page78, 22 September 1826, Csáki Istvánné (Mrs. István Csáki), age 24 (born about 1802). This is just one day after the christening mentioned in the previous record. I looked up the cause of death, pokolvar, to see if it had anything to do with childbirth, but it did not. Pokol means hell, pokolvar means carbuncles. I do not know if the two words are related. Her death occurred less than two months before Fekete Rebeka’s 15 November 1826 wedding mentioned above. (It wasn’t unusual for a father of a new baby to look for a new mother soon after the death of his wife.)

Do these three records belong to one person, two people, or three? I looked for a possible christening record for Mária, and found several records for children born to János (Kaptsos or Bálint) and Ersebet Czegledi. They may be one couple or they may be two, but I arranged the information from the records in a chart here:

Note that Győri István was the godfather of three of the children. Balás Judit (not Ersébet) might have been the godmother all three times as well.

Discrepancies might have been from the participants, the informants, or even the priests. Most people were illiterate back then. I can imagine a person going to the priest when Mrs. István Csáki died. “How old was she?” asks the priest. “I don’t know,” the informant responds. “What were her parents’ names? I’ll determine her age from the christening book,” says the priest. He finds the birth of Mária in 1802, totally unaware that Mária 1802 died in 1807. “It looks like she was 24 when she died,” and that’s what he records in the book. Another possibility is that all the records are exactly correct and belong to different people so there are no discrepancies.

Although I was very interested in looking at these records, and I have an opinion of which records refer to which people, I’m not offering any of those opinions. But if you are related to any of these people, and find my research helpful, go ahead, form your own opinions, and make any connections you feel are appropriate.

 

1691 Jack 0’Lantern Time

Today’s Puzzle:

This smiling Jack O’Lantern actually has a few tricks up his sleeve. Don’t let yourself get fooled by any of them. Using logic, write the numbers from 1 to 12 in the first column as well as in the top row so that those numbers and the given clues can be a multiplication table.

Have a safe and happy Halloween!

Here’s the same puzzle with just the clues:

Factors of 1691:

  • 1691 is a composite number.
  • Prime factorization: 1691 = 19 × 89.
  • 1691 has no exponents greater than 1 in its prime factorization, so √1691 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 1691 has exactly 4 factors.
  • The factors of 1691 are outlined with their factor pair partners in the graphic below.

More About the Number 1691:

1691 is the hypotenuse of a Pythagorean triple:
741-1520-1691, which is 19 times (39-80-89).

1690 Today Is My 8th Blogiversary!

Today’s Puzzle:

I made this Crazy-8 puzzle to commemorate the 8th anniversary of my blog.

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work together like a multiplication table. Some of it might be a little tricky, so make sure you use logic on every step!

My eighth year of blogging has been amazing for me:

  1. Denise Gaskins has a Kickstarter going for her latest book, 312 Things to Do with a Math Journal. One of those 312 things will be journaling about some of my puzzles.
  2. I’ve also hosted her fabulous Math Education Blog Carnival and been featured when other bloggers hosted it.
  3. Bill Davidson interviewed me for his podcast, Centering the Pendulum. Although I’m not one of the many “Eureka Math Giants” he knows, my interview was included in the mix.

  4. In the spring, THREE different types of puzzles I’ve made were published in the Austin Chronicle.

  5. Also a BIG thank you to YOU, reading this right now. I really appreciate you and others who have taken the time to read my thoughts and solve my puzzles.

It’s been a wonderful year. NONE of those things would have happened if I didn’t write a blog. I feel quite fortunate and humbled by it all. I think I’ll go on for another eight years!

Factors of 1690:

  • 1690 is a composite number.
  • Prime factorization: 1690 = 2 × 5 × 13 × 13, which can be written 1690 = 2 × 5 × 13².
  • 1690 has at least one exponent greater than 1 in its prime factorization so √1690 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1690 = (√169)(√10) = 13√10.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1690 has exactly 12 factors.
  • The factors of 1690 are outlined with their factor pair partners in the graphic below.

More About the Number 1690:

1690 is the sum of two squares in THREE different ways:
41² + 3² = 1690,
39² + 13² = 1690, and
31² + 27² = 1690.

1690 is the hypotenuse of SEVEN Pythagorean triples:
232 1674 1690, calculated from 31² – 27², 2(31)(27), 31² + 27²,
246 1672 1690, calculated from 2(41)(3), 41² – 3², 41² + 3²,
416 1638 1690, which is 26 times (16-63-65),
650 1560 1690, which is (5-12-13) times 130.
858 1456 1690, which is 26 times (33-56-65),
1014 1352 1690, calculated from 2(39)(13), 39² – 13², 39² + 13², but it is also (3-4-5) times 338, and
1190 1200 1690, which is 10 times (119-120-169).

1689 Candy Corn Again?!

Today’s Puzzle:

Here’s yet one more candy corn puzzle. You would think it was everyone’s favorite kind of candy the way it is represented on this blog! This puzzle is level 6, so you will probably find it more difficult to solve. Using logic, write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues form a multiplication table.

Factors of 1689:

  • 1689 is a composite number.
  • Prime factorization: 1689 = 3 × 563.
  • 1689 has no exponents greater than 1 in its prime factorization, so √1689 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 1689 has exactly 4 factors.
  • The factors of 1689 are outlined with their factor pair partners in the graphic below.

More About the Number 1689:

1689 is the difference of two squares in two different ways:
845² – 844² = 1689, and
283² – 280² = 1689.