Mystery Level Puzzle

1783 Another Mystery Puzzle

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Today’s Puzzle:

Can you find the factors for this mystery-level puzzle? There is only one solution.

Factors of 1783:

  • 1783 is a prime number.
  • Prime factorization: 1783 is prime.
  • 1783 has no exponents greater than 1 in its prime factorization, so √1783 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1783 has exactly 2 factors.
  • The factors of 1783 are outlined with their factor pair partners in the graphic below.

How do we know that 1783 is a prime number? If 1783 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1783. Since 1783 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1783 is a prime number.

More About the Number 1783:

1783 is palindrome 1L1 in base 33 because
1(33²) + 21(33) + 1(1) = 1783.

1783 is the sum of two consecutive numbers:
891 + 892 = 1783.

1783 is the difference of the squares of those same two consecutive numbers:
892² – 891² = 1783.
Of course, every other odd number can make a similar claim.

1781 A Mystery Puzzle for You to Solve

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Today’s Puzzle:

Is this mystery-level puzzle difficult or easy to solve? I’m not telling. You’ll have to try it for yourself to find out. As always, there is only one solution.

Factors of 1781:

1781 ÷ 4 leaves a remainder of 1, and 41² + 10² = 1781. Could 1781 be a prime number? It will be unless it has a prime number hypotenuse less than √1781 as a divisor. In other words, is it divisible by 5, 13, 17, 29, 37, or 41?

1781 obviously isn’t divisible by 5, and since it’s 41² + 10², it isn’t divisible by 41 either. That means we only have to check if it is divisible by 13, 17, 29, and 37.

So is it prime or composite?

  • 1781 is a composite number.
  • Prime factorization: 1781 = 13 × 137.
  • 1781 has no exponents greater than 1 in its prime factorization, so √1781 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 1781 has exactly 4 factors.
  • The factors of 1781 are outlined with their factor pair partners in the graphic below.

More About the Number 1781:

Not only does 41² + 10² = 1781, but
34² + 25² = 1781.

That 34² lets us know right away that 1781 is not divisible by 17, but any number that is the sum of two squares in more than one way is never a prime number.

1781 is the hypotenuse of FOUR Pythagorean triples:

531-1700-1781, calculated from 34² – 25², 2(34)(25), 34² + 25²,
685-1644-1781, which is (5-12-13) times 137,
820-1581-1781, calculated from 2(41)(10), 41² – 10², 41² + 10², and
1144-1365-1781, which is 13 times (88-105-137).

1781 is also the difference of two squares in two different ways:
891² – 890² = 1781, and
75² – 62² = 1781.

1765 On This Memorial Day

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Today’s Puzzle:

This weekend I laid a bouquet of red and white flowers on my husband’s grave and decided to make a red rose Memorial Day puzzle for the blog as well. It is a mystery-level puzzle.

Write the number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. There is only one solution.

Factors of 1765:

  • 1765 is a composite number.
  • Prime factorization: 1765 = 5 × 353.
  • 1765 has no exponents greater than 1 in its prime factorization, so √1765 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 1765 has exactly 4 factors.
  • The factors of 1765 are outlined with their factor pair partners in the graphic below.

More About the Number 1765:

1765 is the sum of two squares in two different ways:
42² + 1² = 1765, and
33² + 26² = 1765.

1765 is the hypotenuse of FOUR Pythagorean triples:
84 1763 1765, calculated from 2(42)(1), 42² – 1², 42² + 1²,
413 1716 1765, calculated from 33² – 26², 2(33)(26), 33² + 26²,
1059-1412-1765, which is (3-4-5) times 353, and
1125-1360-1765, which is 5 times (225-272-353).

1765 is a digitally powerful number:
1⁴ + 7³ + 6⁴ + 5³ = 1765.

1765 is a palindrome in a couple of different bases:
It’s A5A base 13 because 10(13²) + 5(13) + 10(1) = 1765, and
it’s 1D1 base 36 because 1(36²) + 13(36) + 1(1) = 1765.

1763 Daffodil Puzzle

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Today’s Puzzle:

Spring has sprung and perhaps flowers are blooming in your area. I think my favorite flowers are daffodils. I love the way they are shaped and their vibrant colors.

This daffodil puzzle is a great way to welcome spring. It may be a little bit tricky, but I think if you carefully use logic you will succeed! Just write each of the numbers 1 to 12 in the first column and again in the top row so that those numbers are the factors of the given clues. As always there is only one solution.

Here’s the same puzzle if you’d like to print it using less ink:

Factors of 1763:

  • 1763 is a composite number and the product of twin primes.
  • Prime factorization: 1763 = 41 × 43.
  • 1763 has no exponents greater than 1 in its prime factorization, so √1763 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 1763 has exactly 4 factors.
  • The factors of 1763 are outlined with their factor pair partners in the graphic below.

More About the Number 1763:

1763 is the difference of two squares in two different ways:
882² – 881² = 1763, and
42² – 1² = 1763. (That means the next number will be a perfect square!)

1763 is the hypotenuse of a Pythagorean triple:
387-1720-1763, which is (9-40-41) times 43.

1763 is palindrome 3E3 in base 22 because
3(22²) + 14(22) + 3(1) = 1763.

Lastly and most significantly: 15, 35, 143, 323, 899, and 1763 begin the list of numbers that are the product of twin primes. 1763 is just the sixth number on that list! If we include the products of two consecutive primes whether they are twin primes or not, the list is still fairly small. How rarely does that happen?

When it was 2021, did you realize how significant that year was?

1761 Irish Harp

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Today’s Puzzle:

This mystery-level puzzle was meant to look a little like an Irish harp. Using logic write the numbers 1 to 12 in the first column and again in the top row so that those numbers and the given clues make a multiplication table. There is only one solution.

Notice that the clues 16 and 24 appear THREE times in the puzzle. In each case, you will need to determine if the common factor is 2, 4, or 8. You will have to get the common factor for each one in the right place or it will cause trouble for another clue. Consider what problems each of the following scenarios bring to other clues. For example, 48 must be either 4 × 12 or 6 × 8, but both possibilities are impossible in at least one of these scenarios:

Once you determine the only scenario that doesn’t present a problem for any other clue, you will be able to begin the puzzle.

Here’s the same puzzle without any added color:

Factors of 1761:

1 + 7 + 6 + 1 = 15, a number divisible by 3, so 1761 is divisible by 3. Since 6 is divisible by 3, we didn’t have to include it in our sum: 1 + 7 + 1 = 9, so 1761 is divisible by 3.

  • 1761 is a composite number.
  • Prime factorization: 1761 = 3 × 587.
  • 1761 has no exponents greater than 1 in its prime factorization, so √1761 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 1761 has exactly 4 factors.
  • The factors of 1761 are outlined with their factor pair partners in the graphic below.

More About the Number 1761:

1761 is the difference of two squares in two different ways:
881² – 880² = 1761, and
295² – 292² = 1761.

1761 is palindrome 1N1 in base 32
because 1(32²) + 23(32) + 1(1) = 1761.

1752 So Many Snowflakes

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Today’s Puzzle:

A winter storm hit much of the United States today including the drought-stricken area where I live. We are so grateful for the snow, but also grateful we could stay home all day. Driving in this much snow is not much fun.

This snowflake mystery-level puzzle might have a couple of tricky clues, but I’m confident that you’ll still find it easier than clearing a couple of feet of snow off a driveway! Just write the numbers 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 without any added color:

Factors of 1752:

  • 1752 is a composite number.
  • Prime factorization: 1752 = 2 × 2 × 2 × 3 × 73, which can be written 1752 = 2³ × 3 × 73.
  • 1752 has at least one exponent greater than 1 in its prime factorization so √1752 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1752 = (√4)(√438) = 2√438.
  • 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 1752 has exactly 16 factors.
  • The factors of 1752 are outlined with their factor pair partners in the graphic below.

More About the Number 1752:

1752 is the hypotenuse of a Pythagorean triple:
1152 1320 1752, which is 24 times (48-55-73).

1752 is the difference of two squares in FOUR different ways:
439² – 437² = 1752,
221² – 217² = 1752,
149² -143² = 1752, and
79² – 67² = 1752.

1752 is a palindrome in some other bases:
It’s A4A in base13 because 10(13²) + 4(13) + 10(1) = 1752,
it’s 4G4 in base19 because 4(19²) + 16(19) + 4(1) = 1752, and
it’s 2K2 in base 25 because 2(25²) + 20(25) + 2(1) = 1752.

1747 Getting a Super Bowl Ring Isn’t Easy

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Today’s Puzzle:

On Wednesday I read an article in my local paper titled From undrafted free agent to the biggest stage: Britain Covey’s ‘unique ride’ to Super Bowl LVII. It got me quite excited for today’s game. Unfortunately, between Wednesday and Sunday, Covey suffered a hamstring injury.  Getting a Super Bowl Ring certainly isn’t easy. Being able to contribute to your team’s winning the game isn’t easy either. UPDATE: Good News! He was able to make at least one play in the second quarter! Further update: Even though his team lost the game in the final minutes, this rookie played well when he was on the field. I think both teams played exceptionally well and gave us all an exciting game to watch.

Before the game, it is a mystery which team will win the game. The difficulty level of this puzzle is a mystery also.

If you look at today’s puzzle just right, I think it looks a little like a super bowl ring, but I forwarn you, it will not be easy to get this puzzle either. You will need to place the numbers from 1 to 12 both in the first column and in the top row so that the given clues are the products of the numbers you place. Logic and practice will get you there. Good Luck!

You also might enjoy this next Super Bowl puzzle I saw on Twitter:

Factors of 1747:

  • 1747 is a prime number.
  • Prime factorization: 1747 is prime.
  • 1747 has no exponents greater than 1 in its prime factorization, so √1747 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1747 has exactly 2 factors.
  • The factors of 1747 are outlined with their factor pair partners in the graphic below.

How do we know that 1747 is a prime number? If 1747 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1747. Since 1747 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1747 is a prime number.

More About the Number 1747:

1747 is a palindrome in base 15.
It’s 7B7₁₅ because 7(15²) + 11(15) + 7(1) = 1747.

1738 Little Surprises Around Every Turn

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Today’s Puzzle:

Is this mystery-level puzzle easy or difficult? I’m not saying. Place 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. If you use logic to solve the puzzle, you will probably find some surprises around every turn.

Factors of 1738:

It may surprise you that 1738 is divisible by 11. Why is it? Because
1 – 7 + 3 – 8 = -11, a multiple of 11, which makes 1738 divisible by 11.

  • 1738 is a composite number.
  • Prime factorization: 1738 = 2 × 11 × 79.
  • 1738 has no exponents greater than 1 in its prime factorization, so √1738 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 1738 has exactly 8 factors.
  • The factors of 1738 are outlined with their factor pair partners in the graphic below.

More About the Number 1738:

Maybe you will be surprised by some of these number facts about the number 1738.

As mentioned in the table, 1738 = 2 × 11 × 79. That means 1738 is the short leg in the Pythagorean triple calculated from 2(79)(11), 79² – 11², 79² + 11².

1738 is the sum of eleven consecutive numbers:
153 + 154 + 155 + 156 +157 + 158 + 159 + 160 + 161 + 162 + 163 = 1738.

1738 is the sum of four consecutive numbers:
433 + 434 + 435 + 436 = 1738.

And because four is an even number:
436² – 435² + 434² – 433² = 1738.
Surprised?

1738 is also the sum of the 44 consecutive numbers from 18 to 61.

Consequently,
61² – 60² + 59² – 68² + 57² – 56² + 55² – 54² + 53² – 52² + 51² – 50² + 49² – 48² + 47² – 46² + 45² – 44² + 43² – 42² + 41² – 40² + 39² – 38² + 37² – 36² + 35² – 34² + 33² – 32² + 31² – 30² + 29² – 28² + 27² – 26² + 25² – 24² + 23² – 22² + 21² – 20² + 19² – 18² = 1738. I bet you weren’t expecting that!

But the biggest surprise about 1738 is something I learned from OEIS.org: 1738 is in an equation that uses every digit from 1 to 9 exactly one time:
9 Different Digit Equation

make science GIFs like this at MakeaGif

 

1694 Football Game Day

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Today’s Puzzle:

Today all over the United States family and friends will gather to watch or play a game of football. If you would like to change things up a little, here’s a game ball for you to practice multiplication and division facts. Just 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. Some clues might be tricky, but enough of them aren’t that I am confident you can score with this football!

Here’s the same puzzle without any distracting color:

Factors of 1694:

  • 1694 is a composite number.
  • Prime factorization: 1694 = 2 × 7 × 11 × 11, which can be written 1694 = 2 × 7 × 11².
  • 1694 has at least one exponent greater than 1 in its prime factorization so √1694 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1694 = (√121)(√14) = 11√14.
  • 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 1694 has exactly 12 factors.
  • The factors of 1694 are outlined with their factor pair partners in the graphic below.

More About the Number 1694:

From OEIS.org we learn that 1694³ = 4,861,163,384, a number that uses each of the digits 1, 3, 4, 5, and 8 exactly twice.

1693 Tricky Turkey

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Today’s Puzzle:

You cannot gobble this turkey up unless you can find all of its factors!

Use logic and multiplication facts. It won’t be easy, but write the numbers from 1 to 10 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 without the added color:

Factors of 1693:

  • 1693 is a prime number.
  • Prime factorization: 1693 is prime.
  • 1693 has no exponents greater than 1 in its prime factorization, so √1693 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1693 has exactly 2 factors.
  • The factors of 1693 are outlined with their factor pair partners in the graphic below.

How do we know that 1693 is a prime number? If 1693 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1693. Since 1693 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1693 is a prime number.

More About the Number 1693:

1693 is the sum of two squares:
37² + 18² = 1693.

1693 is the hypotenuse of a Pythagorean triple:
1045-1332-1693, calculated from 37² – 18², 2(37)(18), 37² + 18².

Here’s another way we know that 1693 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 18² = 1693 with 37 and 18 having no common prime factors, 1693 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1693. Since 1693 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1693 is a prime number.