Level 4 Puzzle

1758 Two-Shillelagh O’Sullivan

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Today’s Puzzle:

When I was looking for the song about the shillelagh for my previous post, I found another one called Two-Shillelagh O’Sullivan also by Bing Crosby. It wasn’t a song from my childhood, but it inspired me to make a puzzle with two shillelaghs anyway. In the song, O’Sullivan wears these walking sticks in a holster and can draw them quicker than anyone can draw a gun. He was impossible to beat.

This two-shillelagh puzzle is also a bit difficult to beat. You’re not going to let that stop you from trying, are you? Just use logic and your knowledge of the multiplication table.

Write the numbers from 1 to 12 in the first column and again in the top row so that those numbers are the factors of the given clues.

Here’s the same puzzle in black and white:

Factors of 1758:

  • 1758 is a composite number.
  • Prime factorization: 1758 = 2 × 3 × 293.
  • 1758 has no exponents greater than 1 in its prime factorization, so √1758 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 1758 has exactly 8 factors.
  • The factors of 1758 are outlined with their factor pair partners in the graphic below.

More About the Number 1758:

1758 is the hypotenuse of a Pythagorean triple:
408-1710-1758, which is 6 times (68-285-293).

1758 is palindrome 8E8 in base 14
because 8(14²) + 13(14) + 8(1) = 1758.

 

1744 XOXO Kisses or Multiplication? XOXO

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Today’s Puzzle:

When I was a little girl, my mother told me that OXOXOX stands for Hugs and Kisses. That would mean that O means “hug” and X means “kiss”, but I think X also means multiplication. It’s almost Valentine’s Day, so in today’s puzzle, it can mean either one.

This is a Level-4 puzzle so the clues don’t come in any particular order. Use logic to place all the numbers 1 to 10 in the first column and in the top row.

Factors of 1744:

  • 1744 is a composite number.
  • Prime factorization: 1744 = 2 × 2 × 2 × 2 × 109, which can be written 1744 = 2⁴ × 109.
  • 1744 has at least one exponent greater than 1 in its prime factorization so √1744 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1744 = (√16)(√109) = 4√109.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1744 has exactly 10 factors.
  • The factors of 1744 are outlined with their factor pair partners in the graphic below.

More About the Number 1744:

1744 is the difference of two squares in three ways:
437² – 435² =1744,
220² – 216² =1744, and
113² – 105² =1744.

1744 is also the sum of two squares:
40² + 12² = 1744.

1744 is the hypotenuse of a Pythagorean triple:
960-1456-1744, calculated from 2(40)(12), 40² – 12², 40² + 12².
It is also 16 times (60-91-109).

1724 Carol of the Ukrainian Bells

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Today’s Puzzle:

As I sit in my warm, peaceful house, I often think about the people of Ukraine whose country has been ravaged by war. Many of them, including children, are also facing a winter with no heat. My heart goes out to them.

My childhood was so unlike theirs. I was able to attend school classes and learn about many different topics without fear of dying. In Junior High School choir class, one of the songs I learned was called Carol of the Bells. I just recently learned from Slate magazine of the song’s Ukrainian roots:

A little over a hundred years ago Mykola Leontovych, a Ukrainian composer, arranged several of his country’s folk songs together in a piece he titled Shchedryk. Tragically, he was murdered by a Russian assassin on January 23, 1921, in the Red Terror, when the Bolsheviks were intent on eliminating Ukrainian leaders, intellectuals, and clergy.

During this time of great unrest, the Ukrainian National Chorus performed Shchedryk around the world and in cities large and small in the United States. One performance was even given at the famed Carnegie Hall on October 5, 1922. The haunting melody was heard by Peter Wilhousky who penned alternate words for it: Hark how the bells, Sweet silver bells,…

Now it is one of our most beloved Christmas carols. I am grateful I learned the words and tune in junior high, although I wish I had learned of its Ukrainian history then as well.

These two bells puzzles are reminiscent of Ukraine’s flag. Long may it wave. Write each number 1 to 10 in the yellow columns and rows so that the given clues are the products of the numbers you write.

Here are the same puzzles if you prefer to use less of your printer ink.

Factors of 1724:

  • 1724 is a composite number.
  • Prime factorization: 1724 = 2 × 2 × 431, which can be written 1724 = 2² × 431.
  • 1724 has at least one exponent greater than 1 in its prime factorization so √1724 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1724 = (√4)(√431) = 2√431.
  • 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 1724 has exactly 6 factors.
  • The factors of 1724 are outlined with their factor pair partners in the graphic below.

More About the Number 1724:

1724 is the difference of two squares:
432² – 430² = 1724.

1724₁₀ = 464₂₀, a palindrome, because
4(20²) + 6(20¹) + 4(20°) = 1724.

1699 Sweet Candy Cane

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Today’s Puzzle:

Solving this candy cane puzzle can be a sweet experience. Just 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 make a multiplication table.

Factors of 1699:

  • 1699 is a prime number.
  • Prime factorization: 1699 is prime.
  • 1699 has no exponents greater than 1 in its prime factorization, so √1699 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1699 has exactly 2 factors.
  • The factors of 1699 are outlined with their factor pair partners in the graphic below.

How do we know that 1699 is a prime number? If 1699 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1699. Since 1699 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1699 is a prime number.

More About the Number 1699:

1699 is the third prime in a prime triple. What are the other two primes in that triple?

1699 is the difference of two squares:
850² – 849² = 1699.

1687 Fly Me to the Moon!

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Today’s Puzzle:

A witch flying on a broomstick in front of a bright full moon is a common Halloween image. Here is a level 4 puzzle shaped like a broom. If you succeed in solving it, you might just feel like you are flying to the moon, too. Just write the numbers from 1 to 12 in both the first column and in the top row so that those numbers and the given clues form a multiplication table. Best Witches!

Factors of 1687:

1687 is divisible by 7 because 16 is the double of 8, and the last digit is 7.
217, 427, 637, 847, 1057, 1267, 1477, 1687, 1897 are all divisible by 7.

  • 1687 is a composite number.
  • Prime factorization: 1687 = 7 × 241.
  • 1687 has no exponents greater than 1 in its prime factorization, so √1687 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 1687 has exactly 4 factors.
  • The factors of 1687 are outlined with their factor pair partners in the graphic below.

More About the Number 1687:

1687 is the hypotenuse of a Pythagorean triple:
840-1463-1687, which is 7 times (120-209-241).

1667 and Level 4

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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.

 

1654 and Level 4

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Today’s Puzzle:

Use logic to write all the numbers 1 to 10 in both the first column and the top row of the puzzle so that those numbers are the factors of the given clues.

Factors of 1654:

  • 1654 is a composite number.
  • Prime factorization: 1654 = 2 × 827.
  • 1654 has no exponents greater than 1 in its prime factorization, so √1654 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 1654 has exactly 4 factors.
  • The factors of 1654 are outlined with their factor pair partners in the graphic below.

More About the Number 1654:

1654 is a leg in one Pythagorean triple:
1654-683928-683930, calculated from 2(827)(1), 827² – 1², 827² + 1².

1642 and Level 4

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Today’s Puzzle:

There are several clues in this puzzle with more than one factor pair, but if you use logic every step of the way, you can still write the factors 1 to 12 in the appropriate places to turn the puzzle into a multiplication table.

Factors of 1642:

  • 1642 is a composite number.
  • Prime factorization: 1642 = 2 × 821.
  • 1642 has no exponents greater than 1 in its prime factorization, so √1642 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 1642 has exactly 4 factors.
  • The factors of 1642 are outlined with their factor pair partners in the graphic below.

More About the Number 1642:

1642 is the sum of two squares:
39² + 11² = 1642.

1642 is the hypotenuse of a Pythagorean triple:
858-1400-1642, calculated from 2(39)(11), 39² – 11², 39² + 11².
That triple is also 2 times (429-700-821).

1631 The Importance of Practice

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Today’s Puzzle:

I did not have the privilege of learning a musical instrument when I was growing up, but I did make sure my children had that opportunity. One of the topics discussed in this next episode of Bill Davidson’s Podcast is the importance that practice plays in both music and mathematics. I thought it was quite good.

I think practicing is best when it is enjoyable. If you solve this musical note puzzle, it will hopefully be an enjoyable way for you to practice a few multiplication and division facts. Just 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 will function like a multiplication table.

Factors of 1631:

  • 1631 is a composite number.
  • Prime factorization: 1631 = 7 × 233.
  • 1631 has no exponents greater than 1 in its prime factorization, so √1631 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 1631 has exactly 4 factors.
  • The factors of 1631 are outlined with their factor pair partners in the graphic below.

More about the Number 1631:

1631 is the hypotenuse of a Pythagorean triple:
735-1456-1631, which is 7 times (105-208-233).

1631 is the difference of two squares in two different ways:
816² – 815² = 1631, and
120² – 113² = 1631.

I found those number facts just from looking at the factors of 1631.

 

1598 See the Logic in This Level 4 Puzzle

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Today’s Puzzle:

Put the numbers from 1 to 10 in both the first column and the top row to turn this level 4 puzzle into a multiplication table. The logic in the ten clues is fairly straight-forward. Go ahead give it a try!

Factors of 1598:

  • 1598 is a composite number.
  • Prime factorization: 1598 = 2 × 17 × 47.
  • 1598 has no exponents greater than 1 in its prime factorization, so √1598 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 1598 has exactly 8 factors.
  • The factors of 1598 are outlined with their factor pair partners in the graphic below.

More about the Number 1598:

1598 is the hypotenuse of a Pythagorean triple:
752-1410-1598, which is (8-15-17) times 94.