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.

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).

1671 Are Vaccines Really Much of a Mystery?

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

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…

1662 Declare Your Independence!

Today’s Puzzle:

Tomorrow is Independence Day in the United States. Happy Independence Day! Wherever you live, you can have a different kind of independence day, and it can happen any day of the year:

Are you dependent on a calculator, Siri, or someone or something else to give you any of the products or divisors in a multiplication table? Solving these Find the Factors puzzles can help you be more familiar with the table and declare your independence from those outside sources! Use logic to help you find its unique solution. Yes, mystery-level puzzles can be tricky, but I’ll give you a hint under the puzzle if you need it.

The logic to get started: One column has 40, 50 and another column has 10, 60. There are only two numbers that can go at the top of either one of those columns: 5 and 10. We don’t know which column gets which number, however. But it is still enough to tell us that the other 5 and 10 must go in the first column with the 10 being a factor of 70 and the 5 being a factor of 50.

Factors of 1662:

Knowing some divisibility rules can also help you declare your independence!

1662 is even, so it is divisible by 2.
1 + 2 = 3, so 1662 is divisible by 3. (Why wasn’t it necessary to include the 6’s in that calculation?)
Since 1662 is divisible by both 2 and 3, it is divisible by 6, too.

  • 1662 is a composite number.
  • Prime factorization: 1662 = 2 × 3 × 277.
  • 1662 has no exponents greater than 1 in its prime factorization, so √1662 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 1662 has exactly 8 factors.
  • The factors of 1662 are outlined with their factor pair partners in the graphic below.

More About the Number 1662:

1662 is the hypotenuse of a Pythagorean triple:
690-1512-1662, which is 6 times (115-252-277).
Did you notice all the repeating digits in that triple?

 

1660 A 14×14 Mystery Puzzle

Today’s Puzzle:

Adding a few more factors to the multiplication table really complicates this mystery-level puzzle. For example, will the common factor of 28 and 56 be 4, 7, or 14? If it were just a 10 × 10 or a 12 × 12 puzzle, answering that question would be easy. Not so with a 14 × 14 puzzle. Remember to use logic on every step while you find its unique solution.

You can print the puzzle or type the solution on this excel sheet: 10 Factors 1650-1660 with Taxman Scoring Calculator

Factors of 1660:

  • 1660 is a composite number.
  • Prime factorization: 1660 = 2 × 2 × 5 × 83, which can be written 1660 = 2² × 5 × 83.
  • 1660 has at least one exponent greater than 1 in its prime factorization so √1660 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1660 = (√4)(√415) = 2√415.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1660 has exactly 12 factors.
  • The factors of 1660 are outlined with their factor pair partners in the graphic below.

More About the Number 1660:

1660 is the hypotenuse of a Pythagorean triple:
996-1328-1660, which is (3-4-5) times 332.

 

 

1659 Another Mystery

Today’s Puzzle:

Both 20, 10, 30 and 12, 24, 36 have two possible common factors that will only put numbers from 1 to 10 in the first column and the top row of this mystery level puzzle. However, the puzzle has only one solution. Examine all the clues in the puzzle and think logically to determine what those common factors must be.

Factors of 1659:

  • 1659 is a composite number.
  • Prime factorization: 1659 = 3 × 7 × 79.
  • 1659 has no exponents greater than 1 in its prime factorization, so √1659 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 1659 has exactly 8 factors.
  • The factors of 1659 are outlined with their factor pair partners in the graphic below.

More About the Number 1659:

1659 is the difference of two squares in FOUR different ways:
830² – 829² = 1659,
278² – 275² = 1659,
122² – 115² = 1659, and
50² – 29² = 1659.

1658 Mystery Puzzle

Today’s Puzzle:

What’s the mystery?
Will the common factor of 30 and 20 be 5 or 10?
Will the common factor of 36 and 18 be 6 or 9?
Will the common factor of 60 and 30 be 6 or 10? and
Will the common factor of 8 and 16 be 2, 4, or 8?

Don’t guess which common factors to use! Look at all the clues. They work together to help you logically arrive at the puzzle’s unique solution.

Factors of 1658:

  • 1658 is a composite number.
  • Prime factorization: 1658 = 2 × 829.
  • 1658 has no exponents greater than 1 in its prime factorization, so √1658 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 1658 has exactly 4 factors.
  • The factors of 1658 are outlined with their factor pair partners in the graphic below.

More About the Number 1658:

1658 is the sum of two squares:
37² + 17² = 1658.

1658 is the hypotenuse of a Pythagorean triple:
1080-1258-1658, which is 2 times (540-629-829),
and can also be calculated from 37² – 17², 2(37)(17), 37² + 17².

1658 is also a leg in the Pythagorean triple
calculated from 2(829), 829² – 1², 829² + 1².

1647 A Fun Mystery

Today’s Puzzle:

I’ve never made a puzzle quite like this one before, and even though the logic needed is a little tricky, I found it quite enjoyable. Share it with a friend and see if you don’t have some great discussions such as if the common factor of 12 and 36 is 3, 4, 6, or 12.

Factors of 1647:

1 + 6 + 4 + 7 = 18, a multiple of 3 and 9, so 1647 is divisible by both 3 and 9.

  • 1647 is a composite number.
  • Prime factorization: 1647 = 3 × 3 × 3 × 61, which can be written 1647 = 3³ × 61.
  • 1647 has at least one exponent greater than 1 in its prime factorization so √1647 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1647 = (√9)(√183) = 3√183.
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1647 has exactly 8 factors.
  • The factors of 1647 are outlined with their factor pair partners in the graphic below.

More About the Number 1647:

1647 is the hypotenuse of a Pythagorean triple:
297-1620-1647, which is 27 times (11-60-61).

1647 is a repdigit in base 13:
1647₁₀ = 999₁₃ because
9(13² + 13¹ + 13º) =
9(169 + 13 + 1) =
9(183) = 1647.