Author: ivasallay

1690 Today Is My 8th Blogiversary!

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Today’s Puzzle:

I made this Crazy-8 puzzle to commemorate the 8th anniversary of my blog.

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work together like a multiplication table. Some of it might be a little tricky, so make sure you use logic on every step!

My eighth year of blogging has been amazing for me:

  1. Denise Gaskins has a Kickstarter going for her latest book, 312 Things to Do with a Math Journal. One of those 312 things will be journaling about some of my puzzles.
  2. I’ve also hosted her fabulous Math Education Blog Carnival and been featured when other bloggers hosted it.
  3. Bill Davidson interviewed me for his podcast, Centering the Pendulum. Although I’m not one of the many “Eureka Math Giants” he knows, my interview was included in the mix.

  4. In the spring, THREE different types of puzzles I’ve made were published in the Austin Chronicle.

  5. Also a BIG thank you to YOU, reading this right now. I really appreciate you and others who have taken the time to read my thoughts and solve my puzzles.

It’s been a wonderful year. NONE of those things would have happened if I didn’t write a blog. I feel quite fortunate and humbled by it all. I think I’ll go on for another eight years!

Factors of 1690:

  • 1690 is a composite number.
  • Prime factorization: 1690 = 2 × 5 × 13 × 13, which can be written 1690 = 2 × 5 × 13².
  • 1690 has at least one exponent greater than 1 in its prime factorization so √1690 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1690 = (√169)(√10) = 13√10.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1690 has exactly 12 factors.
  • The factors of 1690 are outlined with their factor pair partners in the graphic below.

More About the Number 1690:

1690 is the sum of two squares in THREE different ways:
41² + 3² = 1690,
39² + 13² = 1690, and
31² + 27² = 1690.

1690 is the hypotenuse of SEVEN Pythagorean triples:
232 1674 1690, calculated from 31² – 27², 2(31)(27), 31² + 27²,
246 1672 1690, calculated from 2(41)(3), 41² – 3², 41² + 3²,
416 1638 1690, which is 26 times (16-63-65),
650 1560 1690, which is (5-12-13) times 130.
858 1456 1690, which is 26 times (33-56-65),
1014 1352 1690, calculated from 2(39)(13), 39² – 13², 39² + 13², but it is also (3-4-5) times 338, and
1190 1200 1690, which is 10 times (119-120-169).

1689 Candy Corn Again?!

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Today’s Puzzle:

Here’s yet one more candy corn puzzle. You would think it was everyone’s favorite kind of candy the way it is represented on this blog! This puzzle is level 6, so you will probably find it more difficult to solve. Using logic, write the numbers from 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 1689:

  • 1689 is a composite number.
  • Prime factorization: 1689 = 3 × 563.
  • 1689 has no exponents greater than 1 in its prime factorization, so √1689 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 1689 has exactly 4 factors.
  • The factors of 1689 are outlined with their factor pair partners in the graphic below.

More About the Number 1689:

1689 is the difference of two squares in two different ways:
845² – 844² = 1689, and
283² – 280² = 1689.

 

1688 What Can Grow from a Little Seed?

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Today’s Puzzle:

Pumpkins are often harvested this time of year. It is amazing how one little seed properly planted and tended can grow into a vine that produces pumpkin after pumpkin after pumpkin.  Indeed great things can come from little things.

This level 5 puzzle starts out fairly easy. It doesn’t get very complicated until you’re about halfway through. Don’t give up! Your mind will grow as you use logic to find a way to work it out. Write the numbers from 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 1688:

  • 1688 is a composite number.
  • Prime factorization: 1688 = 2 × 2 × 2 × 211, which can be written 1688 = 2³ × 211.
  • 1688 has at least one exponent greater than 1 in its prime factorization so √1688 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1688 = (√4)(√422) = 2√422.
  • 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 1688 has exactly 8 factors.
  • The factors of 1688 are outlined with their factor pairs in the graphic below.

More About the Number 1688:

1688 is the difference of two squares in two different ways:
423² – 421² = 1688, and
213² – 209² = 1688.

1688₁₀ = 888₁₄ because 8(14² + 14¹ + 14º) = 1688.

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).

1686 Some Candy Corn for You to Chew on

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Today’s Puzzle:

Candy corn probably isn’t your favorite Halloween treat, but this candy corn puzzle could give you something satisfying to chew on. Give it a try!

Find the common factor of 33 and 66, write the factors in the appropriate cells. Since this is a level 3 puzzle, you can then work from the top of the puzzle row by row until you have found all the factors. The numbers from 1 to 12 must appear once in both the first column and the top row.

Here’s the same puzzle without any color if you prefer:

Factors of 1686:

  • 1686 is a composite number.
  • Prime factorization: 1686 = 2 × 3 × 281.
  • 1686 has no exponents greater than 1 in its prime factorization, so √1686 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 1686 has exactly 8 factors.
  • The factors of 1686 are outlined with their factor pair partners in the graphic below.

More about the number 1686:

1686 is the hypotenuse of a Pythagorean triple:
960-1386-1686, which is 6 times (160-231-281).

1686 is also a leg in these two Pythagorean triples:
1686-710648-710650, calculated from 2(843)(1), 843² – 1², 843² + 1² and
1686-78952-78970, calculated from 2(281)(3), 281² – 3², 281² – 3².

1685 Oh, No! I’ve Created a Monster!

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Today’s puzzle:

You may see some Frankenstein monsters walking about this time of year, but there’s no reason to be afraid of them or of this monster puzzle I’ve created. Simply write the numbers 1 to 12 in both factor areas so that the puzzle functions like a multiplication table.

Here’s the same puzzle without any added color, if that’s what you prefer:

Factors of 1685:

  • 1685 is a composite number.
  • Prime factorization: 1685 = 5 × 337.
  • 1685 has no exponents greater than 1 in its prime factorization, so √1685 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 1685 has exactly 4 factors.
  • The factors of 1685 are outlined with their factor pair partners in the graphic below.

More About the Number 1685:

1685 is the sum of two squares in two different ways:
41² + 2² = 1685, and
34² + 23² = 1685.

1685 is the hypotenuse of FOUR Pythagorean triples:
164-1677-1685, calculated from 2(41)(2), 41² – 2², 41² + 2²,
627-1564-1685, calculated from 34² – 23², 2(34)(23), 34² + 23²,
875-1440-1685, which is 5 times (175-288-337), and
1011-1348-1685, which is (3-4-5) times 337.

 

1684 Triangular Candy Corn

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Today’s Puzzle:

Candy corn is a triangular piece of Halloween candy. 1684 is a centered triangular number formed from the sum of the 32nd, the 33rd, and the 34th triangular numbers. Label the boxes next to the representations of each of those triangular numbers.

 

Factors of 1684:

  • 1684 is a composite number.
  • Prime factorization: 1684 = 2 × 2 × 421, which can be written 1684 = 2² × 421.
  • 1684 has at least one exponent greater than 1 in its prime factorization so √1684 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1684 = (√4)(√421) = 2√421.
  • 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 1684 has exactly 6 factors.
  • The factors of 1684 are outlined with their factor pair partners in the graphic below.

More About the Number 1684:

1684 is the sum of two squares:
30² + 28² = 1684.

1684 is the hypotenuse of a Pythagorean triple:
116-1680-1684, calculated from 30² – 28², 2(30)(28), 30² + 28².
It is also 4 times (29-420-421).

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

1684/2 = 842,  which is the third number in the smallest set of FIVE consecutive numbers whose square roots can be simplified.

1683 Grave Marker

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Today’s Puzzle:

It’s almost Halloween. I hope you enjoy this grave-marker puzzle. Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Here’s the same puzzle, but it won’t use up all your printer ink.

Factors of 1683:

  • 1683 is a composite number.
  • Prime factorization: 1683 = 3 × 3 × 11 × 17, which can be written 1683 = 3² × 11 × 17.
  • 1683 has at least one exponent greater than 1 in its prime factorization so √1683 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1683 = (√9)(√187) = 3√187.
  • 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 1683 has exactly 12 factors.
  • The factors of 1683 are outlined with their factor pair partners in the graphic below.

More About the Number 1683:

1683 is the hypotenuse of a Pythagorean triple:
792-1485-1683, which is (8-15-17) times 99.

1683 is the difference of two squares in SIX different ways:
842² – 841² = 1683,
282² – 279² = 1683,
98² – 89² = 1683,
82² – 71² = 1683,
58² – 41² = 1683, and
42² – 9² = 1683.
That last one means we are 81 numbers away from the next perfect square. I also highlighted a cool-looking difference.

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

1682 This Puzzle Is Not as Difficult as It Looks

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Today’s Puzzle:

Three months ago I was inspired by a puzzle I saw on Twitter:

I enjoyed solving this complicated-looking system of equations, but let me tell you, Looks Can Be Deceiving! The puzzle is not as difficult as it looks.

I decided to make a similar puzzle, and I’ve waited for my 1682nd post to share it with you. If you can solve the Twitter puzzle, then you can solve my puzzle, too!

Why did I wait until my 1682nd post to share this puzzle? Because if you add the three equations together you get:
(x + y + y + z + x + z)(x + y + z) = 1682,
(2x + 2y + 2z)(x + y + z) = 1682,
2(x + y + z)(x + y + z) = 1682,
2(x + y + z)² = 1682.
The factors of 1682 will be quite helpful at this point. What is the greatest common factor of the numbers after the equal signs?

The numbers in one of 1682’s Pythagorean triples, 580-609-1682, are featured prominently in this puzzle.

I hope you enjoy solving my puzzle, and maybe you will make and solve some puzzles of your own!

Factors of 1682:

  • 1682 is a composite number.
  • Prime factorization: 1682 = 2 × 29 × 29, which can be written 1682 = 2 × 29².
  • 1682 has at least one exponent greater than 1 in its prime factorization so √1682 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1682 = (√841)(√2) = 29√2.
  • The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 1682 has exactly 6 factors.
  • The factors of 1682 are outlined with their factor pair partners in the graphic below.

More About the Number 1682:

1682 is the sum of two squares in two different ways:
29² + 29² = 1682, and
41² + 1² = 1682.

1682 is the hypotenuse of two Pythagorean triples:
82-1680-1682, calculated from 2(41)(1), 41² – 1², 41² + 1², and
1160-1218-1682, which is (20-21-29) times 58.

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

1682/2 = 841, which is the second number in the smallest set of FIVE consecutive numbers whose square roots can be simplified.

What Is 1681’s Claim to Fame?

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Today’s Puzzle:

1681 shares that claim to fame with those other numbers, but is the only one on the list that ______________________________________________________________.

Why Do Pythagorean Triples Do What They Do?

Four years ago I noticed something remarkable about Pythagorean triples and wrote a post that I titled Why Do Pythagorean Triples Do That?

 This graphic shows some of what I found so amazing:

It happens over and over again because (a + b)² = a² +2ab + b² is always true.

That statement can be rearranged: (a + b)² = a² + b² +2ab.

Here are some more examples:

Earlier THIS year, Duncan Fraser found my post and left a series of comments that are just too good to keep to myself. I decided I would share his comments the next time I wrote about an odd perfect prime, which 1681 happens to be. Here is our conversation (I’ve replaced ^ with exponents to make reading it a little easier.):

Duncan Fraser

The sum of the even leg and hypotenuse for all ppts (primitive Pythagorean triples) is the square of an odd number. This is a rule for ppts that has been sadly overlooked. Because of this, you can find several ppts that have the sum of the even leg and hypotenuse with the same numerical value. Example (35,12,37), (21,20,29), (7,24,25). If their sum is the square of an odd number, p, the number of ppts is (p-1)/2 ppts. If the sum of the square of an odd number is not prime, then (p-1)/2 Pythagorean triples are produced.
Some are ppts and the others are scalar multiples. Example 15 will produce 4 ppt and 3 scalar multiples.

Since for ppts, the sum of the even leg and the hypotenuse is an odd number squared, and an odd number is the sum of an even and an odd number. (m+n)squared gives the m, n parts for the hypotenuse and even leg of ppts.

Wow, thanks! So since 15² = 225, the seven Pythagorean triples whose even side added to the hypotenuse equals 225 are (197-28-197), (165-52-173), (135-72-153), (105-88-137), (75-100-125), (45-108-117), and (15-112-113). I just had to find them myself after finding this out!

Duncan Fraser

Glad you were able to get the 7 triples. The rule for odd numbers is an odd number is the sum of an even and an odd. I did show that the even side 2mn and hypotenuse m²+ n² is m²+n²+2mn which is (m+n)², m+n is odd.

M+N =15 is an equation with two variables. However, each has to be an integer. Thus (m,n) are (14,1) (13,2) (12,3) (11,4) (10,5) (9,6) (8,7). Note 3 sets of(m,n) have gcfs and these will be the non ppts.

The last pair of (8,7) has m=n+1. In your output (15, 112,113) you will note that 15 is now your odd leg., and the difference between the hypotenuse and the even leg is also 1.

This leads to a very interesting rule, which is

If the difference between the hypotenuse and even leg of a ppt is1. Then the area divided by the perimeter of the ppt is (odd leg-1)/4. Since your odd leg is 15, the ratio is(15-1)/4= 3.5

Now if the odd leg is a prime number then only one ppt is formed eg( 7,24,25).
All composite or multiples will produce more than one Pythagorean triple, one of which will produce a ppt with the hypotenuse minus the even leg equal 1,
For example 9 will be ( 9, 40,41) with m= n+1 .The other (9,12,15).
We can simply square 9 so (81+1)/2 =41 and(81-1)/2=40 are the hypotenuse and even leg respectively.

Or To get the the m and n ,m =(x+1)/2, n=(x-1)/2 . Thus x= 9 gives m=5 and n=4

So for the odd legs starting with the series of odd numbers 3,5,7,9…….. and
Applying the above rule where the difference between the hypotenuse and even leg is1

The area/ perimeter = (x-1)/4 we will get an arithmetic series with first term 0.5 and a common difference of 0.5

Using the formula, 3 gives 0.5, 5(1),7(1,5), 9 (2) ………
Thus producing the arithmetic series.

To develop the formula, Let x be the length of the odd leg. The hypotenuse plus the even leg is x² with the hypotenuse (x²-1)/2 and the even leg (x²-1)/2. Thus the perimeter is x +x² or x(x+1)
The area is x (x²-1)/4 or x(x+1)(x-1)/4

Area divided by perimeter (x-1)/4. This formula can be used by kids as young as grade 7.

The next formula is area/perimeter = n/2. Where is equal to m minus one

You can develop this formula starting with the hypotenuse m²+n² minus the even leg 2mn which leads to (m-n)²=1 thus (m-n)=1

Since the odd leg is m²-n² which is (m+n)(m-n) =(m+n)

I will leave the rest to any interested person.

ivasallay

Thank you for explaining more. This will be my topic when I write my 1681st post in a couple of months. (I should have said 7 months!) I will share your comments in that post so hopefully, more people will see them. Thanks again!

Duncan Fraser

If the odd leg of a ppt is a prime number where its rightmost digit is 1 or 9 then the even leg is a multiple of 60. As you showed (11, 60, 61). (19, 180, 181). Some of the hypotenuses will also be a prime number. Of course, the even leg is a multiple of 4 (2mn) and either m or n is an even number.

The uniqueness of the A²+B² = C² formula is that it holds for any A and B and C. The Pythagorean triples are a special case where all three are integers. The amazing truth is for all As, Bs and Cs the three sides of a right triangle where A is greater than B. (a+b)² +(a-b)² = 2(c)²

Example 3²+4²=5² , (3+4)² + (4-3)²= 2(5)²

Algebraically (a+b)²+(a-b)² is 2a²+2b² which is 2c² and thus diving by 2 gives the original formula.

Can exponents greater than 2 produce a similar result.?

Let us assume a³+ b³= c³ a>b. And a and b are integers

Then (a+b)³+(a-b)³ is 2a³ + 6ab². Is 6ab² = 2b³?

Thus 6ab²-2b³=0 and 2b²(3a-b)=0. Either b=0 or b = 3a. If b=0 then a³=c³. If b=3a then a³+(3a)³=28a³=c³
The cube root of 28 is not an integer and thus c is not an integer.

Let us consider a⁴+b⁴=c⁴ then

(a+b)⁴ +(a-b)⁴ = 2a⁴+12a²b²+2b⁴. The extra term must be equal to zero to have the equation equal to 2c⁴. Thus either a =0 or b=0. Thus the extra term shows that c cannot be an integer in the original fourth-degree equation.

All higher degrees behave like n equal 3 for odd exponents and n equal 4 for even exponents.

Did Fermat have a simple proof? He did not have the tools available to Prof. Wiles.

That was wonderfully proven!

A Puzzle with Pythagorean Triples:

Whether or not you were able to follow all that Duncan Fraser wrote, fill in the blanks on this next puzzle. Seriously, you should be able to complete it in less than a couple of minutes!

Now sit back, relax, take notice and wonder about the patterns in this table:

Mathematics is truly a thing of beauty!

Factors of 1681:

  • 1681 is a composite number and is a perfect square.
  • Prime factorization: 1681 = 41 × 41 which can be written 1681 = 41².
  • 1681 has at least one exponent greater than 1 in its prime factorization so √1681 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1681 = (√41)(√41) =
  • The exponent in the prime factorization is 2. Adding one to that exponent, we get (2 + 1) = 3. Therefore 1681 has exactly 3 factors.
  • The factors of 1681 are outlined with their factor pair partners in the graphic below.

More About the Number 1681:

1681 is the sum of two squares:
40² + 9² = 1681.

1681 is the hypotenuse of two Pythagorean triples:
369-1640-1681, which is 41 times (9-40-41).
720-1519-1681, calculated from 2(40)(9), 40² – 9², 40² + 9².

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

Puzzle Solution:

All of the numbers are perfect squares AND concatenations of exactly two perfect squares:
7² = 49; 2² = 4, 3² = 9.
13² = 169; 4² = 16, 3² = 9.
19² = 361; 6² = 36, 1² = 1.
35² = 1225; 1² = 1, 15² = 225.
38² = 1444; 12² = 144, 2² = 4.
41² = 1681; 4² = 16, 9² = 81.
57² = 3249; 18² = 324, 3² = 9.
65² = 4225; 2² = 4, 15² = 225.
70² = 4900; 2² = 4, 30² = 900.

1681 shares that claim to fame with those other numbers, but is the only one on the list that is a concatenation of two 2-digit squares.