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.

1656 Seven Ate Nine: Puzzle and a Picture Book

Today’s Puzzle:

Today’s Puzzle is a relatively easy level 6 puzzle with consecutive numbers 7, 8, and 9 prominent among the clues. Write the numbers 1 to 10 in both the 1st column and the top row so that those numbers and all the given clues work together to make a multiplication table. Will the common factor of 24 and 32 be 4 or 8? Will 20 and 12’s common factor be 2 or 4? Don’t guess! Look at the other clues. They all work together to help you find a logical way to solve the puzzle.

The Book Seven Ate Nine:

I ordered several books from my granddaughter’s book order. One of those books was Seven Ate Nine, a delightful tale whose characters are numbers and letters. The back cover summaries the story, “6 has a problem. Everyone knows that 7 is always after him. Word on the street is that 7 ate 9. If that’s true, 6’s days are numbered. Lucky for him, Private I is on the case. But the facts just don’t add up. It’s odd. Will Private I put two and two together and solve the problem . . . or is 6 next in line to be subtracted?”

My preschool grandchildren loved listening to this story. It is filled with math puns and surprising twists and turns. Other than familiarity with the concept of counting, mathematical understanding is not a prerequisite to following the story. Older kids and even adults will enjoy references to several mathematical concepts including odd, even, addition, subtraction, multiplication, division, doubling, measurement, positives, negatives, and pi.

Factors of 1656:

  • 1656 is a composite number.
  • Prime factorization: 1656 = 2 × 2 × 2 × 3 × 3 × 23, which can be written 1656 = 2³ × 3² × 23.
  • 1656 has at least one exponent greater than 1 in its prime factorization so √1656 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1656 = (√36)(√46) = 6√46.
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1656 has exactly 24 factors.
  • The factors of 1656 are outlined with their factor pair partners in the graphic below.

More About the Number 1656:

1656 is the difference of two squares SIX different ways:
415² – 413² = 1656,
209² – 205² = 1656,
141² – 135² = 1656,
75² – 63² = 1656,
55² – 37² = 1656, and
41² – 5² – = 1656.
That last one means we are only 25 numbers away from the next perfect square, 1681.

1655 and Level 5

Today’s Puzzle:

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

Factors of 1655:

  • 1655 is a composite number.
  • Prime factorization: 1655 = 5 × 331.
  • 1655 has no exponents greater than 1 in its prime factorization, so √1655 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 1655 has exactly 4 factors.
  • The factors of 1655 are outlined with their factor pair partners in the graphic below.

More About the Number 1655:

1655 is the hypotenuse of a Pythagorean triple:
993-1324-1655, which is (3-4-5) times 331.

1654 and Level 4

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.

1652 Start at the Top and Work Your Way Down to the Bottom

Today’s Puzzle:

This is a level 3 puzzle so the clues are given in a logical order starting from the top of the puzzle. Begin by writing the factors of 20 and 32 in the appropriate cells. Then write the rest of the numbers so that both the first column and the top row have all the numbers from 1 to 10, and the written numbers are the factors of the given clues.

Factors of 1652:

  • 1652 is a composite number.
  • Prime factorization: 1652 = 2 × 2 × 7 × 59, which can be written 1652 = 2² × 7 × 59.
  • 1652 has at least one exponent greater than 1 in its prime factorization so √1652 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1652 = (√4)(√413) = 2√413.
  • 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 1652 has exactly 12 factors.
  • The factors of 1652 are outlined with their factor pair partners in the graphic below.

More About the Number 1652:

1652 is the difference of two squares two different ways:
414² – 412² = 1652 and
66² – 52² = 1652.

1651 Multiplication Fun

Today’s Puzzle:

Look how much fun these kids are having doing multiplication!

A game like that can help kids get ready to solve a fun puzzle based on the multiplication table.

Write each number from 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table.

Factors of 1651:

  • 1651 is a composite number.
  • Prime factorization: 1651 = 13 × 127.
  • 1651 has no exponents greater than 1 in its prime factorization, so √1651 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 1651 has exactly 4 factors.
  • The factors of 1651 are outlined with their factor pair partners in the graphic below.

More About the Number 1651:

1651 is the hypotenuse of a Pythagorean triple:
635-1524-1651, which is (5-12-13) times 127.

1651 is the 26th heptagonal number because
5(26²)/2 – 3(26)/2 = 1651.

1651 is a nice-looking palindrome in base 2:
1651₁₀ = 11001110011₂.
That just means that
2¹⁰ + 2⁹ + 2⁶ + 2⁵ + 2⁴+ 2¹+ 2⁰ = 1024 + 512 + 64 + 32 + 16 + 2 + 1 = 1651.

 

 

1650 Wrinkles in the Multiplication Table

Today’s Puzzle:

Are you familiar with the book A Wrinkle in Time? Kat of The Lily Cafe’s blog loves books and recently compared Meg in that book to her six-year-old son. She wrote a post titled Am I Raising a Meg? Her six-year-old LOVES math and is very much interested in multiplication and division. When Mom thought he was playing a game on her phone, he was actually playing with the calculator app! I felt so happy inside as I read that!

I wonder if they have discovered the storybooks in the Math Book Magic blog. Such books could combine Mom’s love for reading with her son’s love of math.

Someday her son might like to solve a “wrinkled” multiplication table puzzle like this one that has only nine clues.

Write all the numbers 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table.

Factor Cake for 1650:

This is my 1650th post.
1650 is divisible by 2 and by 5 because it ends with a 0.
1650 is divisible by 3 because 1 + 6 + 5 + 0 = 12, a number divisible by 3.
1650 is divisible by 11 because 1 – 6 + 5 – 0 = 0, a number divisible by 11.

I think we can make a lovely factor cake for 1650:

Factors of 1650:

  • 1650 is a composite number.
  • Prime factorization: 1650 = 2 × 3 × 5 × 5 × 11, which can be written 1650 = 2 × 3 × 5² × 11.
  • 1650 has at least one exponent greater than 1 in its prime factorization so √1650 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1650 = (√25)(√66) = 5√66.
  • The exponents in the prime factorization are 1, 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1)(1 + 1) = 2 × 2 × 3 × 2 = 24. Therefore 1650 has exactly 24 factors.
  • The factors of 1650 are outlined with their factor pair partners in the graphic below.

More About the Number 1650:

1650 is the hypotenuse of TWO Pythagorean triples:
462-1584-1650, which is (7-24-25) times 66, and
990-1320-1650, which is (3-4-5) times 330.

1649 Tweaking a Puzzle Posed by Sunil Singh @Mathgarden

Today’s Puzzle:

A couple of days ago on Twitter, I saw an interesting puzzle posed by Sunil Singh @Mathgarden.

After I found one of several of its solutions, I wondered if I could add a bridge that would use all nine numbers from 1 to 9 in the solution, so I tweaked it. I decided to move the puzzle to the ocean when I added that extra bridge.

I was able to solve this problem using logic and addition facts, rather than algebra. Try solving it yourself. If you want to see any of the steps I used to solve the puzzle, scroll down to the end of the post.

Factors of 1649:

  • 1649 is a composite number.
  • Prime factorization: 1649 = 17 × 97.
  • 1649 has no exponents greater than 1 in its prime factorization, so √1649 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 1649 has exactly 4 factors.
  • The factors of 1649 are outlined with their factor pair partners in the graphic below.

More About the Number 1649:

1649 is the sum of two squares in TWO different ways:
32² + 25² = 1649, and
40² + 7² = 1649.

1649 is the hypotenuse of FOUR Pythagorean triples:
399-1600-1649, calculated from 32² – 25², 2(32)(25), 32² + 25²,
560-1551-1649, calculated from 2(40)(7), 40² – 7², 40² + 7²,
776 1455 1649, which is (8-15-17) times 97, and
1105 1224 1649, which is 17 times (65-72-97).

Some Logical Steps to Solve Today’s Puzzle:

I found four different ways to write the solution but they are all rotations or reflections of each other. Here are the steps to one of those four ways:

1st Step: The biggest number that can be used is 9. Since every island must be included in at least two sums, neither 8 nor 9 can be an addend; they both must be sums. 7 must be an addend in their sums because adding any number to 7 will yield 8, 9, or some larger forbidden number. Thus 8 and 9 are bridges that connect to island 7.

I chose to write 7, 8, and 9 in the top right section, but 8 and 9 could change places with each other. I could have also chosen to write those numbers in the bottom left area.

2nd Step: 1 + 7 = 8, and 7 + 2 = 9.

3rd Step: 1 + 2 = 3.

4th Step: The last island must be the smallest remaining number (4) because the smallest remaining number can’t be the sum of a bigger number and the number on either adjacent island.

Final Step: 1 + 4 = 5, and 4 + 2 = 6.

Did you enjoy this puzzle? How did my steps compare to the steps that you took?

Please, check the comments for another solution.

1648 A Pythagorean Triple Logic Puzzle with a Triangular Card Deck

Today’s Puzzle:

It’s been a few years since I’ve made one of these Pythagorean triple logic puzzles. The triangles in it are shaped a little different than in years past because I also wanted to make a deck of Pythagorean triple cards that are shaped like equilateral triangles or at least as close as I can get to equal sides. This puzzle won’t be easy, but do give it a try!

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: in fact, non-adjacent triangles in the drawing might actually overlap.

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

Print the puzzles or type the solution in this excel file: 12 Factors 1639-1648

Triangular Card Deck of Pythagorean Triples:

You can make this deck of 50 playing cards to help you solve the puzzle or perhaps to play a domino-type game. Print each group of 25 cards on a separate sheet of paper. Cut the cards out along the solid lines and fold the cards on the dotted lines. Use a glue stick to keep the front of each card attached to its back. Laminate the cards, if desired. The right angles are the only angles marked on the cards, and the hypotenuses are all along the folded edges. Note: Some sides will not match with any other side in the deck, and the 57-76-95 triangle does not match sides with ANY triangle with a hypotenuse less than 100.

Factors of the Number 1648:

  • 1648 is a composite number.
  • Prime factorization: 1648 = 2 × 2 × 2 × 2 × 103, which can be written 1648 = 2⁴ × 103.
  • 1648 has at least one exponent greater than 1 in its prime factorization so √1648 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1648 = (√16)(√103) = 4√103.
  • 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 1648 has exactly 10 factors.
  • The factors of 1648 are outlined with their factor pair partners in the graphic below.

More About the Number 1648:

1648 is not the hypotenuse of any Pythagorean triples because none of its prime factors leave a remainder of 1 when divided by 4.

1648 is the difference of two squares three different ways:
413² – 411² = 1648,
208² – 204² = 1648, and
107² – 99² = 1648.

That means 1648 is a leg in THREE Pythagorean triples calculated from
413² – 411², 2(413)(411), 413² + 411²;
208² – 204², 2(208)(204), 208² + 204²; and
107² – 99², 2(107)(99), 107² + 99².

1648 can be expressed as 2(824)(1), 2(412)(2), 2(206)(4), as well as 2(103)(8).

That means 1648 is a leg in FOUR Pythagorean triples calculated from
2(824)(1), 824² – 1², 824² + 1²;
2(412)(2), 412² – 2², 412² + 2²;
2(206)(4), 206² – 4², 206² + 4²; and
2(103)(8), 103² – 8², 103² + 8².

Sometimes those formulas produce duplicate triples, but not this time. 1648 is in SEVEN Pythagorean triples!
1648-339486-339490,
1648-84864-84880,
1648-21186-21250,
1648-678975-678977,
1648-169740-169748,
1648-42420-42452,
1648-10545-10673.

Aren’t you glad that today’s puzzle was limited to triples with hypotenuses less than 100? I certainly wouldn’t want any of those triples with 1648 as a side to be part of any puzzle! But 2-digit sides, 16 and 48, helped to make today’s puzzle, and I do hope you were able to solve it today!

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.