1735 What Amazing Mathematical Pattern I Noticed Today!

Today’s Puzzle:

Will the common factor of 60 and 10 be 5 or will it be 10? Don’t guess. There is a better place to start this puzzle. After using logic to find some of the other factors, you will know if you should use 5 or 10. Don’t let me trip you up!

Factors of 1735:

  • 1735 is a composite number.
  • Prime factorization: 1735 = 5 × 347.
  • 1735 has no exponents greater than 1 in its prime factorization, so √1735 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 1735 has exactly 4 factors.
  • The factors of 1735 are outlined with their factor pair partners in the graphic below.

More About the Number 1735 and an Amazing Pattern:

1735 is the hypotenuse of a Pythagorean triple:
1041-1388-1735, which is (3-4-5) times 347.

1735 is 567 in base 18 because
5(18²) + 6(18) + 7(1) = 1735.

I know neither of those facts is amazing, but read on:

1735 is the sum of two, five, and ten consecutive positive numbers:
867 + 868 = 1735,
345 + 346 + 347 + 348 + 349 = 1735, and
169 + 170 + 171 + 172 + 173 + 174 + 175 + 176 + 177 + 178 = 1735.

1735 is the difference of two squares in two different ways:
868² – 867² = 1735, and
176² – 171² = 1735.

That first difference of two squares used the two consecutive numbers I listed above in red. Since mathematics is the study of patterns, I wondered if there would also be a relationship of squares with the ten consecutive numbers in blue. I typed into my calculator:
178² – 177² + 176² – 175² + 174² – 173² + 172² – 171² + 170² – 169² =

Guess what the answer was? If you guessed 1735, you would be right! Oh, mathematics is certainly filled with amazing patterns! I’m thrilled that I spotted this one! It appears something similar can be written whenever a number is the sum of an even amount of consecutive numbers! I find that amazing!

I also noticed that the second difference of two squares noted above is contained in the string. I underlined those two squares. Since the difference of those squares by themselves equals the whole string, it follows that the rest of those squares added and subtracted equal zero. That means the sum of the rest of the positive terms must equal the sum of the rest of the negative terms:
178² + 174² + 172² + 170² = 120144 = 177² + 175² + 173² + 169².
I’ve never found a string of four squares equal to another string of four squares before, so naturally, I’m excited! I’m anxious to apply this pattern to other numbers that are the sum of ten consecutive numbers. For example, the first ten numbers add up to 55.
10² – 9² + – 7² + 6² – 5² + 4² – 3² + 2² – 1² =55. And since 8² – 3² = 55, it follows that
10²  + 6² + 4² + 2² = 156 = 9² + 7² + 5²+ 1².
It is no longer “impossible” for me to find these equal sums! Here’s another that connects this year with other recent years:

We can all hope this equation doesn’t predict the future.

1732 Is a Thousand Times More Than…

Today’s Puzzle:

If you know the common factor of 50 and 35 that will put only numbers from 1 to 12 in the first column and the top row of this multiplication table puzzle, then you will have completed the first step in solving the puzzle. Afterward, just work from the top of the puzzle to the bottom filling in factors as you go. That’s how you solve these level-three puzzles.

Factors of 1732:

  • 1732 is a composite number.
  • Prime factorization: 1732 = 2 × 2 × 433, which can be written 1732 = 2² × 433.
  • 1732 has at least one exponent greater than 1 in its prime factorization so √1732 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1732 = (√4)(√433) = 2√433.
  • 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 1732 has exactly 6 factors.
  • The factors of 1732 are outlined with their factor pair partners in the graphic below.

More About the Number 1732:

1732 is the sum of two squares:
34² + 24² = 1732.

1732 is the hypotenuse of a Pythagorean triple:
580-1632-1732, calculated from 34² – 24², 2(34)(24), 34² + 24².
It is also 4 times (145-408-433).

1732 is a palindrome in base 15 because
7(15²) + 10(15) + 7(1) = 1732.

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentine’s day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

That’s how I remember those values, but this tweet reminded me that 1732 is a thousand times more than the square root of three rounded to three decimal places. It also makes a reference to the square root of two.

1731: The Sum of the Squares of Three Consecutive Prime Numbers

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of the numbers you write.

Factors of 1731:

  • 1731 is a composite number.
  • Prime factorization: 1731 = 3 × 577.
  • 1731 has no exponents greater than 1 in its prime factorization, so √1731 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 1731 has exactly 4 factors.
  • The factors of 1731 are outlined with their factor pair partners in the graphic below.

More About the Number 1731:

1731 is the hypotenuse of a Pythagorean triple:
144-1725-1731 which is 3 times (48-575-577).

OEIS.org informs us that 1731 is the sum of the squares of three consecutive prime numbers. Let’s find those three prime numbers. Since √(1731/3) rounds to 24, I’m guessing the middle prime number is 23. The prime numbers occurring before and after it are 19 and 29.

Is 19² + 23² + 29² = 1731? Yes, it is!

Here are some other ways that 1731 is the sum of three squares:
41² + 7² + 1² = 1731,
41² + 5² + 5² = 1731,
37² + 19² + 1² = 1731, and
29² + 29² + 7² = 1731.

How Is 1730 the Sum of Consecutive Squares?

Today’s Puzzle:

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

Factors of 1730:

  • 1730 is a composite number.
  • Prime factorization: 1730 = 2 × 5 × 173.
  • 1730 has no exponents greater than 1 in its prime factorization, so √1730 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 1730 has exactly 8 factors.
  • The factors of 1730 are outlined with their factor pair partners in the graphic below.

More about the number 1730:

1730 is the sum of two squares in two different ways:
41² + 7² = 1730, and
37² + 19² = 1730.

1730 is the hypotenuse of FOUR Pythagorean triples:
520 1650 1730 which is 10 times (52-165-173)
574 1632 1730 calculated from 2(41)( 7), 41² – 7², 41² + 7²,
1008 1406 1730 calculated from 37² – 19², 2(37)(19), 37² + 19², and
1038 1384 1730 which is 346 times (3-4-5).

Finally, OEIS.org informs us that 1730 is the sum of consecutive squares in two different ways. What are those two ways? I figured it out. Can you?

Here’s a hint: It is the sum of three consecutive squares as well as twelve consecutive squares. That means √(1730/3) rounded is included in one sum and √(1730/12) rounded is included in the other. The solution can be found in the comments. Have fun finding them yourself though!

1729 Not a Dull Number

Today’s Puzzle:

A little more than a hundred years ago near Cambridge University G. H. Hardy took a taxi to visit his young friend and fellow mathematician, Srinivasa Ramanujan, in the hospital. Hardy couldn’t think of anything interesting about his taxi number, 1729, and remarked to Ramanujan that it appeared to be a rather dull number. But even the reason for his hospitalization could not prevent Ramanujan’s genius from shining through. He immediately recognized 1729’s unique and very interesting attribute: it is the SMALLEST number that can be written as the sum of two cubes in two different ways! Indeed,
12³ + 1³ = 1729, and
10³ + 9³ = 1729.

Today’s puzzle looks a little bit like a modern-day American taxi cab with the clues 17 and 29 at the top of the cab. The table below the puzzle contains all the Pythagorean triples with hypotenuses less than 100 sorted by legs and by hypotenuses. Use the table and logic to write the missing sides of the triangles in the puzzle. The right angle on each triangle is the only one that is marked. Obviously, none of the triangles are drawn to scale.

Sorted TriplesHere’s the same puzzle without all the added color:

Print the puzzles or type the solutions in this excel file: 10 Factors 1721-1729.

What taxi cab might Hardy have tried to catch next? He might have had to wait a long time for it, 4104.
16³ + 2³ = 4104, and
15³ + 9³ = 4104.

Factors of 1729:

  • 1729 is a composite number.
  • Prime factorization: 1729 = 7 × 13 × 19.
  • 1729 has no exponents greater than 1 in its prime factorization, so √1729 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 1729 has exactly 8 factors.
  • The factors of 1729 are outlined with their factor pair partners in the graphic below.

More About the Number 1729:

Did you notice these cool-looking factor pairs?
19 · 91 = 1729.
13 · 133 = 1729.

1729 is the hypotenuse of a Pythagorean triple:
665-1596-1729 which is 133 times (5-12-13).

1729 is the difference of two squares in four different ways:
865² – 864² = 1729,
127² – 120² = 1729,
73² – 60² = 1729, and
55² – 36² = 1729.

1728 Is a Perfect Cube

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

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

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

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

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.