Puzzles

1728 Is a Perfect Cube

/ ivasallay / Leave a comment

Today’s Puzzle:

I think this puzzle is a little bit harder than the previous two puzzles, so you might want to solve those two first. You will need to write all the numbers from 1 to 24 where the factors go, but keep the numbers from 1 to 12 together and the numbers from 13 to 24 together. There is only one solution. The possible factors are written to the right of the puzzle. Good luck!

You can solve the puzzle by eliminating factors that must be used for other numbers. For example, 198 cannot be 9×22 because 189 must be 9×21, and the puzzle can only have one nine. Don’t write the factors on the multiplication table until after you know if the top row has the numbers 1 – 12 or if it has the numbers 13 – 24. The first column will then have the other set of numbers.

Factors of 1728:

Let’s find out the factoring information of the puzzle number:

  • 1728 is a composite number and a perfect cube.
  • Prime factorization: 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be written 1728 = 2⁶ × 3³.
  • 1728 has at least one exponent greater than 1 in its prime factorization so √1728 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1728 = (√576)(√3) = 24√3.
  • The exponents in the prime factorization are 6 and 3. Adding one to each exponent and multiplying we get (6 + 1)(3 + 1) = 7 × 4 = 28. Therefore 1728 has exactly 28 factors.
  • The factors of 1728 are outlined with their factor pairs in the graphic below.

More About the Number 1728:

1728 is the 3rd smallest number with exactly 28 factors.

There are 1728 cubic inches in one cubic foot.

12³₁₀ = 10³₁₂ (The cube above is 12 cubed in base 10, and it’s 10 cubed in base 12.)

1728 can be written as the sum of powers in some interesting ways:

2¹⁰ + 2⁹ + 2⁷ + 2⁶ = 1024 + 512 + 128 + 64 = 1728.
1(4⁵) + 2(4⁴) + 3(4³) = 1728.
1(6⁴) + 2(6³) = 1728.
3(8³) + 3(8²) = 3(8³ + 8²) = 1728.

11³ = 1331, and 1(11³) + 3(11²) + 3(11¹) + 1(11⁰) = 1728.

Here’s another connection with the number 11.
3(11²) = 363, and
1728 is palindrome 363 in base 23 because
3(23²) + 6(23¹) + 3(23⁰) = 1728.

1728 is 300 in base 24 because 3(24²) = 1728.

I’ve enjoyed researching the number 1728, and I hope you’ve learned some new and interesting fact about it today.

1726 Find the Factors 1-12 AND 13-24

/ ivasallay / Leave a comment

Today’s Puzzle:

Here’s a puzzle I made to start off 2023. Either all the numbers from 1 to 12 will go in the top row OR they will all go in the first column. All the numbers from 13 to 24 will go in the remaining area. The possible factors are written on the side of the puzzle. Can you find all the factors? There is only one solution.

Factors of 1726:

  • 1726 is a composite number.
  • Prime factorization: 1726 = 2 × 863.
  • 1726 has no exponents greater than 1 in its prime factorization, so √1726 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 1726 has exactly 4 factors.
  • The factors of 1726 are outlined with their factor pair partners in the graphic below.

More About the Number 1726:

1726 is the sum of four consecutive numbers from 430 to 433.

1725 A Toast to You on This Last Day of the Year

/ ivasallay / Leave a comment

Today’s Puzzle:

I haven’t posted as much this past year as I ought, but thank you for bearing with me. Here is a toast to you, faithful reader! Here’s hoping for a great new year for all of us! Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the factors you write.

Factors of 1725:

  • 1725 is a composite number.
  • Prime factorization: 1725 = 3 × 5 × 5 × 23, which can be written 1725 = 3 × 5² × 23.
  • 1725 has at least one exponent greater than 1 in its prime factorization so √1725 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1725 = (√25)(√69) = 5√69.
  • 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 1725 has exactly 12 factors.
  • The factors of 1725 are outlined with their factor pair partners in the graphic below.

More About the Number 1725:

1725 is the difference of two squares in six different ways:

863² – 862² = 1725,
289² – 286² = 1725,
175² – 170² = 1725,
65² – 50² = 1725,
49² – 26² = 1725, and
47² – 22² = 1725.

 

1724 Carol of the Ukrainian Bells

/ ivasallay / Leave a comment

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.

1723 A Little Virgács

/ ivasallay / Leave a comment

Today’s Puzzle:

I haven’t blogged much this year so I guess I deserved a little bit of virgács in my shoes this morning. Mikulás (St. Nick) leaves virgács in the boots of naughty little boys or girls in the wee hours of December 6. Treats are for the good kids. What is virgács? It is small golden spray-painted twigs bound with some pretty red ribbon.  Of course, all children are sometimes naughty and sometimes nice, so they could all expect to get virgács along with their treats in their boots this morning.

You can solve this virgács puzzle by starting with the clues at the top of the grid, finding their factors, and working down the puzzle row by row until you have found all the factors. Each number from 1 to 10 must appear exactly one time in both the first column and the top row.

Factors of 1723:

  • 1723 is a prime number.
  • Prime factorization: 1723 is prime.
  • 1723 has no exponents greater than 1 in its prime factorization, so √1723 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1723 has exactly 2 factors.
  • The factors of 1723 are outlined with their factor pair partners in the graphic below.

How do we know that 1723 is a prime number? If 1723 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1723. Since 1723 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1723 is a prime number.

More About the Number 1723:

1723 and 1721 are twin primes.

1723 is the sum of consecutive numbers:
861 + 862 = 1723.

1723 is the difference of consecutive squares:
862² – 861² = 1723.

1721 A Gift With Multiple Treasures

/ ivasallay / Leave a comment

Today’s Puzzle:

‘Tis the season of giving, and here’s a gift with multiple treasures inside. Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write. After you find the factors, you can complete the puzzle by finding all of the products.

Factors of 1721:

  • 1721 is a prime number.
  • Prime factorization: 1721 is prime.
  • 1721 has no exponents greater than 1 in its prime factorization, so √1721 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1721 has exactly 2 factors.
  • The factors of 1721 are outlined with their factor pair partners in the graphic below.

How do we know that 1721 is a prime number? If 1721 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1721. Since 1721 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1721 is a prime number.

More About the Number 1721:

1721 is the sum of two squares:
40² + 11² = 1721.

Here’s another way we know that 1721 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 11² = 1721 with 40 and 11 having no common prime factors, 1721 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1721. Since 1721 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1721 is a prime number.

1721 and 1723 are twin primes.

1720 Transforming Puzzle-A-Day’s Wizard’s Hat

/ ivasallay / Leave a comment

Today’s Puzzle:

I recently saw a fun-to-solve puzzle from Puzzle a Day on Twitter:

The puzzle got me thinking, what would happen if the numbers weren’t from 1 to 10, but were from 0 to 9 instead? Would it change the puzzle a little or a lot?

It actually changes the puzzle quite a bit. There are a lot more solutions than for the Puzzle a Day puzzle! And I don’t mean just switching the red line with the green line and/or switching the purple circles at the bottom. How many solutions can you find?

Factors of 1720:

  • 1720 is a composite number.
  • Prime factorization: 1720 = 2 × 2 × 2 × 5 × 43, which can be written 1720 = 2³ × 5 × 43.
  • 1720 has at least one exponent greater than 1 in its prime factorization so √1720 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1720 = (√4)(√430) = 2√430.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1720 has exactly 16 factors.
  • The factors of 1720 are outlined with their factor pair partners in the graphic below.

More About the Number 1720:

1720 is the difference of two squares in FOUR different ways:
431² – 429² = 1720,
217² – 213² = 1720,
91² – 81² = 1720, and
53² – 33² = 1720.

 

1717 Factor Fits for Your Valentine

/ ivasallay / Leave a comment

Today’s Puzzle:

Solve this Valentine-themed Factor Fits puzzle with both logic and heart!

Factors of 1717:

  • 1717 is a composite number.
  • Prime factorization: 1717 = 17 × 101.
  • 1717 has no exponents greater than 1 in its prime factorization, so √1717 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 1717 has exactly 4 factors.
  • The factors of 1717 are outlined with their factor pair partners in the graphic below.

More About the Number 1717:

1717 is the sum of two squares two different ways:
41² + 6² = 1717, and
39² + 14² = 1717.

1717 is the hypotenuse of FOUR Pythagorean triples:
340 1683 1717, which is 17 times (20-99-101),
492 1645 1717, calculated from 2(41)(6), 41² – 6², 41² + 6²,
808 1515 1717, which is (8-15-17) times 101, and
1092 1325 1717, calculated from 2(39)(14), 39² – 14², 39² + 14².

1715 A Lot More of a Subtraction Distraction

/ ivasallay / Leave a comment

Today’s Puzzle:

Last time I published a puzzle with the last clue missing. Leaving out the first or the last clue only makes the puzzle slightly more difficult. What if I left out a clue more in the middle of the puzzle. I gave that some thought and designed today’s puzzle. I soon realized that I had to let you know that the 12 is one of the last eight boxes. There is only one solution. Can you find it?

I posted a solution video for it on Twitter:

Factors of 1715:

  • 1715 is a composite number.
  • Prime factorization: 1715 = 5 × 7 × 7 × 7, which can be written 1715 = 5 × 7³.
  • 1715 has at least one exponent greater than 1 in its prime factorization so √1715 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1715 = (√49)(√35) = 7√35.
  • The exponents in the prime factorization are 1 and 3. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1715 has exactly 8 factors.
  • The factors of 1715 are outlined with their factor pairs in the graphic below.

More About the number 1715:

1715 = 1·7³·1·5. Thank you, OEIS.org for that fun fact!

1715 is the hypotenuse of a Pythagorean triple:
1029-1372-1715, which is (3-4-5) times 343.

1715 = 5·7³.
5·7º = 2² + 1².
5·7¹ cannot be written as the sum of two squares.
5·7² = 14² + 7².
5·7³ cannot be written as the sum of two squares.
5·7⁴ = 98² + 49².

What do you notice? What do you wonder?

1714 Will the Factors in This Puzzle Give You Fits?

/ ivasallay / Leave a comment

Today’s Puzzle:

12 and 24 have several common factors, but only one of them works in this puzzle. Will it be 2, 3, 4, 6, or 12?

What about 40 and 60’s common factors?

Don’t guess which factor to use. Start elsewhere in the puzzle where there’s only one possible common factor. Then use logic to eliminate some of the factor possibilities for 12, 24 and 40, 60. You will have to think, but it won’t be too difficult.

Factors of 1714:

  • 1714 is a composite number.
  • Prime factorization: 1714 = 2 × 857.
  • 1714 has no exponents greater than 1 in its prime factorization, so √1714 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 1714 has exactly 4 factors.
  • The factors of 1714 are outlined with their factor pair partners in the graphic below.

More About the Number 1714:

1714 is the sum of two squares:
33² + 25² = 1714.

1714 is the hypotenuse of a Pythagorean triple:
464-1650-1714, calculated from 33² – 25², 2(33)(25), 33² + 25².
It is also 2 times (232-825-857).