1427 Mysterious Cat

This mysterious Halloween cat shares twelve clues that can help you solve its puzzle. Each clue is the products of two factors from 1 to 12 multiplied together. Will you be able to solve its mystery?

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you a little bit about the puzzle number, 1427:

  • 1427 is a prime number.
  • Prime factorization: 1427 is prime.
  • 1427 has no exponents greater than 1 in its prime factorization, so √1427 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1427 has exactly 2 factors.
  • The factors of 1427 are outlined with their factor pair partners in the graphic below.

How do we know that 1427 is a prime number? If 1427 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1427. Since 1427 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1427 is a prime number.

1427 is 272 in BASE 25 because 2(25²) + 7(25) + 2(1) = 1427

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1425 Jack O’Lantern

Most of my puzzles don’t have as many clues as this Jack O’lantern puzzle has. Those extra clues could make it easier to solve. On the other hand, some of the clues might still be tricky.

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Here are some facts about the puzzle number, 1425:

  • 1425 is a composite number.
  • Prime factorization: 1425 = 3 × 5 × 5 × 19, which can be written 1425 = 3 × 5² × 19
  • 1425 has at least one exponent greater than 1 in its prime factorization so √1425 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1425 = (√25)(√57) = 5√57
  • 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 1425 has exactly 12 factors.
  • The factors of 1425 are outlined with their factor pair partners in the graphic below.

1425 is the hypotenuse of TWO Pythagorean triples:
399-1368-1425 which is (7-24-25) times 57
855-1140-1425 which is (3-4-5) times 285

 

1269 Tangram Witch Puzzle

If you have the seven tangram pieces, then you can create this Halloween witch riding across the moon (or a paper plate). Best Witches creating all kinds of things with those fabulous tiles!

In case you would like to know some facts about the number 1269, here’s what I’ve learned:

  • 1269 is a composite number.
  • Prime factorization: 1269 = 3 × 3 × 3 × 47, which can be written 1269 = 3³ × 47
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1269 has exactly 8 factors.
  • Factors of 1269: 1, 3, 9, 27, 47, 141, 423, 1269
  • Factor pairs: 1269 = 1 × 1269, 3 × 423, 9 × 141, or 27 × 47
  • Taking the factor pair with the largest square number factor, we get √1269 = (√9)(√141) = 3√141 ≈ 35.62303

1269 is the difference of two squares four different ways:
37² – 10² = 1269
75² – 66² = 1269
213² – 210² = 1269
635² – 634² = 1269

1268 Halloween Cat Mystery

Cats can be quite mysterious. They are a favorite pet for many every day, even though suspicious stories abound about them on Halloween. Can you solve the mystery of this cat-like puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

Now I’ll share a few facts about the number 1268:

  • 1268 is a composite number.
  • Prime factorization: 1268 = 2 × 2 × 317, which can be written 1268 = 2² × 317
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1268 has exactly 6 factors.
  • Factors of 1268: 1, 2, 4, 317, 634, 1268
  • Factor pairs: 1268 = 1 × 1268, 2 × 634, or 4 × 317
  • Taking the factor pair with the largest square number factor, we get √1268 = (√4)(√317) = 2√317 ≈ 35.60899

28² + 22² = 1268

1268 is the hypotenuse of a Pythagorean triple:
300-1232-1268 calculated from 28² – 22², 2(28)(22), 28² + 22².
It is also 4 times (75-308-317)

1267 Frankenstein Mystery

There are legends of Dr. Frankenstein creating a monster years ago. Nowadays Frankenstein’s Monster can often be seen walking through neighborhoods on Halloween night. This puzzle looks a little bit like him.

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

But if you take all the color away, he looks completely different and quite harmless:

Now I’ll share some information about the number 1267:

  • 1267 is a composite number.
  • Prime factorization: 1267 = 7 × 181
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1267 has exactly 4 factors.
  • Factors of 1267: 1, 7, 181, 1267
  • Factor pairs: 1267 = 1 × 1267 or 7 × 181
  • 1267 has no square factors that allow its square root to be simplified. √1267 ≈ 35.59494

1267 is the sum of nine consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1267

1267 is the hypotenuse of a Pythagorean triple:
133-1260-1267 which is 7 times (19-180-181)

 

928 Halloween Cat

This cat has arrived just in time for Halloween. Find the factors that go with the clues in the grid to make this Halloween Cat puzzle a multiplication table:

Print the puzzles or type the solution on this excel file: 12 factors 923-931

Now let me tell you a little about the number 928.

It is the sum of four consecutive prime numbers:
227 + 229 + 233 + 239 = 928
and the sum of eight other consecutive prime numbers:
101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 928

It is the sum of two squares:
28² + 12² = 928
That means 928 is the hypotenuse of a Pythagorean triple:
640-672-928 which is 28² – 12², 2(28)(12), 28² + 12²

Here’s how 928 looks in a few other bases:
It’s 565 in BASE 13, because 5(169) + 6(13) + 5(1) = 928.
It’s 4A4 in BASE 14 (A is 10 base 10), because 4(196) + 10(14) + 4(1) = 928.
It’s TT in BASE 31 (T is 29 base 10), because 29(31) + 29(1) = 29(32) = 928.
It’s T0 in BASE 32, because 29(32) = 928.

  • 928 is a composite number.
  • Prime factorization: 928 = 2 × 2 × 2 × 2 × 2 × 29, which can be written 732 = 2⁵ × 29
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 928 has exactly 12 factors.
  • Factors of 928: 1, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928
  • Factor pairs: 928 = 1 × 928, 2 × 464, 4 × 232, 8 × 116, 16 × 58, or 29 × 32
  • Taking the factor pair with the largest square number factor, we get √928 = (√16)(√58) = 4√58 ≈ 30.463092423.

927 Candy Corn

Candy corn is a traditional Halloween candy.

Figure out what number goes in the top cell of the first column of this level three candy corn puzzle, and work your way down the first column, cell by cell, to make this puzzle a treat to complete.

Print the puzzles or type the solution on this excel file: 12 factors 923-931

Fibonacci numbers begin with 1, 1, with the rest of the numbers in the sequence being the sum of the previous two.

Tribonacci numbers begin with 0, 0, 1 with the rest of the numbers in the sequence being the sum of the previous THREE.

The first 15 tribonacci numbers are 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927. Thank you, Stetson.edu, for that fun fact.

  • 927 is a composite number.
  • Prime factorization: 927 = 3 × 3 × 103, which can be written 927 = 3² × 103
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 927 has exactly 6 factors.
  • Factors of 927: 1, 3, 9, 103, 309, 927
  • Factor pairs: 927 = 1 × 927, 3 × 309, or 9 × 103
  • Taking the factor pair with the largest square number factor, we get √927 = (√9)(√103) = 3√103 ≈ 30.44667

 

921 Is This Bug Cute or Creepy?

Some bugs make creepy Halloween decorations. Other bugs, like ladybugs, might make a very cute costume.

Today’s puzzle looks like a bug, but there is no reason to run and hide from this one. Yes, it’s a level 5, so some parts of it may be tricky.

This is what you need to do to solve it: stay calm; don’t guess and check. Figure out where to put each number from 1 to 10 in both the top row and the first column so that the clues make the puzzle work like a multiplication table. Don’t write a number down unless you are absolutely sure it belongs where you’re putting it. Use logic, step by step, and this puzzle will be a treat.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

When you put on a Halloween costume, you may look completely different.

When a number is written in a different base, it may look completely different. For example,
921 looks like repdigit 333 in BASE 17 because 3(17²) + 3(17¹) + 3(17º) = 3(289 + 17 + 1) = 3(307) = 921
(307 is 111 in BASE 17)

921 looks like palindrome 1H1 in BASE 23 (H is 17 base 10). As you might suspect, 1(23²) + 17(23¹) + 1(23º) = 529 + 391 + 1 = 921

When it’s not written in a different base, 921 looks pretty familiar. You can tell quite quickly that it is divisible by 3:

  • 921 is a composite number.
  • Prime factorization: 921 = 3 × 307
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 921 has exactly 4 factors.
  • Factors of 921: 1, 3, 307, 921
  • Factor pairs: 921 = 1 × 921 or 3 × 307
  • 921 has no square factors that allow its square root to be simplified. √921 ≈ 30.34798181

920 Witches’ Cauldron

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

What besides “eye of newt” goes in witches’ cauldrons? The list includes some horrifying ingredients that you can read here from one scene from Shakeaspeare’s play, MacBeth.

Instead of putting “Eye of newt, and toe of frog, Wool of bat, and tongue of dog” and so forth in today’s Halloween cauldron puzzle, I just put a bunch of asterisks.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

Double 115 is 230.

Double 230 is 460.

Double 460 is 920, today’s post number.

920 is the hypotenuse of a Pythagorean triple:
552-736-920 which is (3-4-5) times 184.

920 is palindrome 767 in BASE 11 because 7(121) + 6(11) + 7(1) = 920

  • 920 is a composite number.
  • Prime factorization: 920 = 2 × 2 × 2 × 5 × 23, which can be written 920 = 2³ × 5 × 23
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 920 has exactly 16 factors.
  • Factors of 920: 1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920
  • Factor pairs: 920 = 1 × 920, 2 × 460, 4 × 230, 5 × 184, 8 × 115, 10 × 92, 20 × 46, or 23 × 40
  • Taking the factor pair with the largest square number factor, we get √920 = (√4)(√230) = 2√230 ≈ 20.331501776.

 

918 Grim Reaper’s Scythe

Sometime on Halloween you are likely to see the Grim Reaper carrying a scythe. Together they look pretty scary. This puzzle isn’t that bad though. You should give it a try.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

Scythe, now that is a good word to try when playing hangman. ☺

Let me tell you about the number 918:

It is the sum of consecutive prime numbers: 457 + 461 = 918

It is the hypotenuse of a Pythagorean triple:
432-810-918, which is (8-15-17) times 54

918 looks interesting in a few other bases:

  • 646 in BASE 12, because 9(144) + 4(12) + 6(1) = 918
  • 330 in BASE 17, because 3(289) + 3(17) + 0(1) = 3(289 + 17) = 3(306) = 918
  • 198 in BASE 26, which is the digits of 918 in a different order. Note that 1(26²) + 9(26) + 8(1) = 918
  • RR in BASE 33, (R is 27 in base 10), because 27(33) + 27(1) = 27(33 + 1) = 27(34) = 918
  • R0 in BASE 34, because 27(34) = 918

918 has consecutive numbers, 17 and 18, as two of its factors. That means 918 is a multiple of the 17th triangular number, 153.

  • 918 is a composite number.
  • Prime factorization: 918 = 2 × 3 × 3 × 3 × 17, which can be written 918 = 2 × 3³ × 17
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 918 has exactly 16 factors.
  • Factors of 918: 1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918
  • Factor pairs: 918 = 1 × 918, 2 × 459, 3 × 306, 6 × 153, 9 × 102, 17 × 54, 18 × 51, or 27 × 34
  • Taking the factor pair with the largest square number factor, we get √918 = (√9)(√102) = 3√102 ≈ 30.29851