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…

1669 and Level 6

Today’s Puzzle:

Some sets of clues in this puzzle have two possible common factors, and another set has three possible common factors. Don’t guess which ones to use, but use logic instead! Write the numbers 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 1669:

  • 1669 is a prime number.
  • Prime factorization: 1669 is prime.
  • 1669 has no exponents greater than 1 in its prime factorization, so √1669 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1669 has exactly 2 factors.
  • The factors of 1669 are outlined with their factor pair partners in the graphic below.

How do we know that 1669 is a prime number? If 1669 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1669. Since 1669 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1669 is a prime number.

More About the Number 1669:

1669 is the sum of two squares:
38² + 15² = 1669.

1669 is the hypotenuse of a primitive Pythagorean triple:
1140-1219-1669, calculated from 2(38)(15), 38² – 15², 38² + 15².

Here’s another way we know that 1669 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 15² = 1669 with 38 and 15 having no common prime factors, 1669 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1669. Since 1669 is not divisible by 5, 13, 17, 29, or 37, we know that 1669 is a prime number.

1668 and Level 5

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues create a multiplication table. This is a level 5 puzzle, so some of the clues may be tricky. Remember to use logic and consider all possibilities for factors to avoid being tricked.

Factors of 1668:

  • 1668 is a composite number.
  • Prime factorization: 1668 = 2 × 2 × 3 × 139, which can be written 1668 = 2² × 3 × 139.
  • 1668 has at least one exponent greater than 1 in its prime factorization so √1668 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1668 = (√4)(√417) = 2√417.
  • 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 1668 has exactly 12 factors.
  • The factors of 1668 are outlined with their factor pair partners in the graphic below.

More About the Number 1668:

1668 is the difference of two squares in two different ways:
418² – 416² = 1668, and
142² – 136² = 1668.

 

1667 and Level 4

Today’s Puzzle:

Use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1667:

  • 1667 is a prime number.
  • Prime factorization: 1667 is prime.
  • 1667 has no exponents greater than 1 in its prime factorization, so √1667 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1667 has exactly 2 factors.
  • The factors of 1667 are outlined with their factor pair partners in the graphic below.

How do we know that 1667 is a prime number? If 1667 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1667. Since 1667 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1667 is a prime number.

More About the Number 1667:

Look at these consecutive number facts about the number 1667:
833 + 834 = 1667.
834² – 833² = 1667.

As the chart below shows, 1667 was ALMOST the fourth consecutive prime number ending in 7. Too bad prime number 1663 got in the way of that happening.

 

1666 Demystifying a Tricky Puzzle

Today’s Puzzle:

This puzzle isn’t as tricky as it could be simply because I arranged its clues into a level 3 puzzle. That means that after you write the factors of 99 and 18 in the appropriate boxes in the first column and top row, you work your way down the puzzle clue by clue in the order they appear. Still, you will have to think about what to do with the 70, but I think you can handle it!

Factors of 1666:

  • 1666 is a composite number.
  • Prime factorization: 1666 = 2 × 7 × 7 × 17, which can be written 1666 = 2 × 7² × 17.
  • 1666 has at least one exponent greater than 1 in its prime factorization so √1666 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1666 = (√49)(√34) = 7√34.
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1666 has exactly 12 factors.
  • The factors of 1666 are outlined with their factor pair partners in the graphic below.

More About the Number 1666:

1666 is the sum of two squares because ALL of its odd prime factors either leave a remainder of 1 when divided by 4 OR have an even exponent:
1666 = 2 × 7² × 17,
17÷4 = 4 R1, The exponent on 7² is even.
What are the two squares?
35² +  21² = 1666.

1666 is the hypotenuse of a Pythagorean triple :
784-1470-1666 which is (8-15-17) times 98 and
can also be calculated from 35² –  21², 2(35)(21), 35² +  21².

 

1665 Why I Am Rooting for England over Denmark Today

Today’s Puzzle:

England is playing Denmark in the Euro 2020 Semi-Finals today. Why am I rooting for England when I don’t really follow soccer, as we call it in America? It ISN’T because 5 of my great-grandparents claim English ancestry, and just only one of my great-grandparents was born in Denmark. England wins 5 to 1. No, that isn’t the reason why I’m excited today and created this soccer/football puzzle by freehand in paint:

How many pentagon and hexagon transformations are there on the game ball?

The reason I am rooting for England is that an ingenious mathematics teacher, AC@eymaths, created and shared an exciting and even better transformation puzzle on Twitter: pdf of a transformation puzzle worksheet.

The transformation looked like a wonderful idea, still, I wasn’t sure what all the fuss was about or the meaning of “Miss, is it coming home?” The next day I asked about it:


I googled and found this music video from Three Lions.

I love the enthusiasm shown even while expressing these lyrics:

“Everyone seems to know the score, they’ve seen it all before
They just know, they’re so sure
That England’s gonna throw it away, gonna blow it away
But I know they can play.”

I know that exact feeling! I’ve had high hopes for a team that only disappointed me. I love how everyone in England is in the moment and feeling enthusiastic no matter what! I have watched the music video over and over again. It’s also wonderful that so many mathematics teachers at several different levels are embracing their students’ excitement:

Factors of 1665:

Since this is my 1665th post, I’ll share factoring information about the number 1665.

Obviously, 1665 ends with a 5, so it is also divisible by 5.
6, 6, and 1 + 5 use up all the digits and give us three 6’s (three of the same multiple of 3), so 1665 is divisible by 9 and, of course, by 3.

The prime factors of 1665 work together to give us several repdigits as factors, too: 111, 333, and 555.

  • 1665 is a composite number.
  • Prime factorization: 1665 = 3 × 3 × 5 × 37, which can be written 1665 = 3² × 5 × 37.
  • 1665 has at least one exponent greater than 1 in its prime factorization so √1665 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1665 = (√9)(√185) = 3√185.
  • 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 1665 has exactly 12 factors.
  • The factors of 1665 are outlined with their factor pair partners in the graphic below.

More About the Number 1665:

1665 is the sum of two squares in TWO different ways:
39² + 12² = 1665, and
33² + 24² = 1665.

1665 is the hypotenuse of FOUR Pythagorean triples:
513-1584-1665, calculated from 33² – 24², 2(33)(24), 33² + 24²,
but is also 9 times (57-176-185),
540-1575-1665, which is (12-35-37) times 45.
936-1377-1665, calculated from 2(39)(12), 39² – 12², 39² + 12²,
but is also 9 times (104-153-185), and
999-1332-1665, which is (3-4-5) times 333.

1664 and Level 2

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factor Cake for 1664:

We can make a factor cake for 1664 by doing some successive divisions. Divide 1664 by 2, divide that answer by 2, and so forth until you make a factor cake that looks like this:

Factors of 1664:

  • 1664 is a composite number.
  • Prime factorization: 1664 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 13, which can be written 1664 = 2⁷ × 13.
  • 1664 has at least one exponent greater than 1 in its prime factorization so √1664 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1664 = (√64)(√26) = 8√26.
  • 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 1664 has exactly 16 factors.
  • The factors of 1664 are outlined with their factor pair partners in the graphic below.

More About the Number 1664:

1664 is the sum of two squares:
40² + 8² = 1664.
That happened because it has a prime factor that leaves a remainder of 1 when divided by 4 AND all of its other prime factors are powers of 2 or perfect squares:

But that’s not all that cool about 1664. What patterns do you notice below?
2(24² + 16²) = 1664,
4(20² + 4²) = 1664,
8(12² + 8²) = 1664,
16(10² + 2²) = 1664,
32(6² + 4²) = 1664,
64(5² + 1²) = 1664, and
128(3² + 2²) = 1664.

1664 is the hypotenuse of a Pythagorean triple:
640-1536-1664, calculated from 2(40)(8), 40² – 8², 40² + 8².
That triple is also (5-12-13) times 128.

1663 and Level 1

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1663:

  • 1663 is a prime number.
  • Prime factorization: 1663 is prime.
  • 1663 has no exponents greater than 1 in its prime factorization, so √1663 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1663 has exactly 2 factors.
  • The factors of 1663 are outlined with their factor pair partners in the graphic below.

How do we know that 1663 is a prime number? If 1663 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1663. Since 1663 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1663 is a prime number.

More About the Number 1663:

1663 is the sum of consecutive numbers in only one way:
831 + 832 = 1663.

1663 is the difference of two squares in only one way:
832² – 831² = 1663.

What do you notice about those two number facts?

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?

 

1661 A Little About Palindromes

Today’s Puzzle:

A number is a palindrome if it reads the same backward as it does forward. For example, 1661 is a 4-digit palindrome. Today’s puzzle asks you to explore number palindromes and their factors.

Half of the time when a 4-digit palindrome is divided by 11, we get a 3-digit palindrome. Why does that happen?

Are there 3-digit palindromes that were NOT included in the Divided by Eleven column in the table above?

All 2-digit palindromes are divisible by 11. They are 11, 22, 33, 44, 55, 66, 77, 88, 99.

There are only eight 3-digit palindromes that are divisible by 11. They are 121, 242, 363, 484, 616, 737, 858, 979. Some 3-digit palindromes are prime numbers. Others are divisible by 101 or 111. Still, there are plenty of 3-digit palindromes that are composite numbers but not divisible by 11, 101, or 111.

Most 5-digit palindromes are NOT divisible by 11. I was able to construct one that is, 76967, because the red digits minus the blue digits are 2312 = 11, a number divisible by 11.

Here’s another: 81818. It works because 242 = 22, a number divisible by 11.

What 5-digit palindrome can you construct that is divisible by 11?

Will an N-digit palindrome be divisible by 11? What difference does it make if N is an even number or if N is an odd number?

Factors of 1661:

1661 is divisible by 11 because the red digits minus the blue digits equal 0, a number divisible by 11.

  • 1661 is a composite number.
  • Prime factorization: 1661 = 11 × 151.
  • 1661 has no exponents greater than 1 in its prime factorization, so √1661 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 1661 has exactly 4 factors.
  • The factors of 1661 are outlined with their factor pair partners in the graphic below.

More about the Number 1661:

1661 is also a palindrome in base 18:
1661₁₀ = 525₁₈ because
5(18²) + 2(18¹) + 5(18º) =
5(324) + 2(18) + 5(1) =
1620 + 36 + 5 =
1661.