1687 Fly Me to the Moon!

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

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!

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

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

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

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?

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.

1680 is a Spectacular Number!

Today’s Puzzle:

1680 is the smallest number with at least 40 factors! That fact alone makes 1680 a pretty spectacular number!

Those 40 factors are:
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, and 1680.

Without looking at a list of triangular numbers, determine how many of 1680’s factors are triangular numbers. Then list which factors are triangular numbers. Scroll down to the Puzzle Hints section if needed.

The prime factorization of 1680 is 2⁴ · 3 · 5 · 7.
How many of 1680’s factors are square numbers? What are they?

How many of 1680’s factors are hexagonal numbers? What are they? Scroll down to the Puzzle Hints if you need another puzzle hint.

Each of those questions could be answered by looking at either its list of factors or its prime factorization. The graphic below will help with the next question:
How many of 1680’s factors are octagonal numbers? What are they? The first 24 octagonal numbers are listed along a line in the center of the octagon below.

Puzzle Hints:

Hint for the triangular numbers: The number of triangular numbers is the same as the number of pairs of consecutive numbers listed as factors. For example, 14 and 15 are both factors of 1680, so (14 · 15)/2 = 105 is a triangular number AND it is a factor of 1680.

Hint for the hexagonal numbers: how many of those pairs of consecutive numbers begin with an odd number? For example, 15 is odd and 16 is the next consecutive number. Both of them are factors of 1680, so (15 · 16)/2 = 120 is both a triangular number AND a hexagonal number.

1680 Factor Trees:

There are MANY possible factor trees for 1680. The possibilities are more than for any previous number. Since it’s October, I’ll take you into a haunted forest to see three of its trees.

If those trees give you the willies, here are three other not-so-scary-looking factor trees:

Notice that the prime factors are the same for all of its trees.

Finding the Factors of 1680:

  • You can divide 1680 by 1 because it’s a whole number.
  • You can divide it by 2 because it’s even.
  • You can divide it by 4 because its last digit is 0, and 8 is even.
  • You can divide it by 8 because 80 is divisible by 8, and 6 is even.
  • You can divide it by 5 or by 10 because the last digit is 0.
  • You can divide it by 3 and/or 7 because these special numbers: 21, 42, 63, 84, 105, 126, 147, 168, and 189 are all divisible by both 3 and 7.
  • For the reasons listed above and their prime factorizations, you can also divide 1680 by these composite numbers: 6, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, and so forth.

Here’s some general factoring information for the number 1680:

  • 1680 is a composite number.
  • Prime factorization: 1680 = 2 × 2 × 2 × 2 × 3 × 5 × 7, which can be written 1680 = 2⁴ × 3 × 5 × 7.
  • 1680 has at least one exponent greater than 1 in its prime factorization so √1680 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1680 = (√16)(√105) = 4√105.
  • The exponents in the prime factorization are 4, 1, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) )(1 + 1) = 5 × 2 × 2 × 2 = 40. Therefore 1680 has exactly 40 factors.
  • The factors of 1680 are outlined with their factor pair partners in the graphic below.

More about the number 1680:

1680 is the hypotenuse of only one Pythagorean triple:
1008-1344-1680, which is (3-4-5) times 336.

1680 is the difference of two squares in 12 different ways! That’s also more than any previous number, another spectacular fact! Here are those 12 ways:

  1.   421² – 419² = 1680,
  2.   212² – 208² = 1680,
  3.   143²  – 137² = 1680,
  4.   109² – 101² = 1680,
  5.   89² – 79² = 1680,
  6.   76² – 64² = 1680,
  7.   67² – 53² = 1680,
  8.   52² – 32² = 1680,
  9.   47² – 23² = 1680,
  10.   44² – 16² = 1680,
  11.   43² -13² = 1680, and
  12.   41² -1² = 1680.

That last one means we are just one number away from the next perfect square,
41² = 1681. We could predict that as soon as we saw that 40 · 42 = 1680.

The first graphic I made shows 1680 tiny squares arranged into an octagon. I thought it might be fun to use the octagon to demonstrate that 1680 can be divided into 42 groups of 40. It wasn’t always as fun as I thought it would be, but here’s my work:

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

1680 square roots

(Remarkably, the first number in the smallest set was 1680/2 = 840.) So cool!

I hope you have enjoyed reading about this very fascinating number!

 

 

1679 and Level 6

Today’s Puzzle:

Can you complete this multiplication table puzzle? It’s a level 6, so some of the clues might be a little tricky. Use logic every step of the way, and everything will work out for you!

Factors of 1679:

  • 1679 is a composite number.
  • Prime factorization: 1679 = 23 × 73.
  • 1679 has no exponents greater than 1 in its prime factorization, so √1679 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 1679 has exactly 4 factors.
  • The factors of 1679 are outlined with their factor pair partners in the graphic below.

More About the Number 1679:

From OEIS.org we learn that
1 + 6 + 7 + 9 = 23, AND 1679 is divisible by 23.
This is cool: 1679 is the smallest multiple of 23 that can make that claim!

1679 is the hypotenuse of a Pythagorean triple:
1104-1265 -1679, which is 23 times (48-55-73).

1678 and Level 5

Today’s Puzzle:

Using logic, write the numbers from 1 to 10 in both the first column and the top row so that this puzzle will function like a multiplication table.

Factors of 1678:

  • 1678 is a composite number.
  • Prime factorization: 1678 = 2 × 839.
  • 1678 has no exponents greater than 1 in its prime factorization, so √1678 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 1678 has exactly 4 factors.
  • The factors of 1678 are outlined with their factor pair partners in the graphic below.

More about the number 1678:

Because 2(839)(1) = 1678, you can calculate the only Pythagorean triple that contains the number 1678:
The smaller leg will be 2(839)(1) = 1678.
The longer leg will be 839² – 1² = 703920.
The hypotenuse will be 839² + 1² = 703922.