A Multiplication Based Logic Puzzle

Archive for the ‘Mystery Level Puzzle’ Category

914 Jack-O’Lantern Mystery Level

25² + 17² = 914, so 914 is the hypotenuse of a Pythagorean triple:
336-850-914, which is 2 times (168-425-457).
You can also calculate it from 2(25)(17), 25² – 17², 25² + 17².

914 becomes 194 in BASE 26. Notice that both bases use the same digits.

914 is also palindrome 494 in BASE 14.

Today’s puzzle is a jack o’lantern that will kick off two weeks worth of Halloween themed puzzles.

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

  • 914 is a composite number.
  • Prime factorization: 914 = 2 × 457
  • 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 914 has exactly 4 factors.
  • Factors of 914: 1, 2, 457, 914
  • Factor pairs: 914 = 1 × 914 or 2 × 457
  • 914 has no square factors that allow its square root to be simplified. √914 ≈ 30.2324

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911 Mystery Level Puzzle

911 is a prime number that is also the sum of three consecutive primes:

  • 293 + 307 + 311 = 911

911 is 191 in BASE 26 because 1(26²) + 9(26) + 1(1) = 911

911 is palindrome 12121 in BASE 5 because 1(5⁴) + 2(5³) + 1(5²) + 2(5) + 1(1) = 911

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 911 is a prime number.
  • Prime factorization: 911 is prime.
  • The exponent of prime number 911 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 911 has exactly 2 factors.
  • Factors of 911: 1, 911
  • Factor pairs: 911 = 1 × 911
  • 911 has no square factors that allow its square root to be simplified. √911 ≈ 30.1827765

How do we know that 911 is a prime number? If 911 were not a prime number, then it would be divisible by at least one prime number less than or equal to √911 ≈ 30.2. Since 911 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 911 is a prime number.

 

 

 

 

908 Haunted House

I combined most of this week’s puzzles to make a haunted house. None of the levels are listed on it, but I will include the levels as I publish each puzzle separately sometime this week. If you enter this haunted house, will you be able to escape? Warning: This should not be the first Find The Factors puzzle you try. A couple of the puzzles in it may be easy, but the rest could be very scary!

Print the puzzles or type the solution on this excel file: 12 factors 905-913

908 can be written as the sum of two consecutive odd numbers:

  • 453 + 455 = 908

Four consecutive even numbers:

  • 224 + 226 + 228 + 230 = 908

And eight consecutive counting numbers:

  • 110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 = 908

There wasn’t much to say about the number 908, but it’s always possible to find something.

  • 908 is a composite number.
  • Prime factorization: 908 = 2 × 2 × 227, which can be written 908 = 2² × 227
  • 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 908 has exactly 6 factors.
  • Factors of 908: 1, 2, 4, 227, 454, 908
  • Factor pairs: 908 = 1 × 908, 2 × 454, or 4 × 227
  • Taking the factor pair with the largest square number factor, we get √908 = (√4)(√227) = 2√227 ≈ 30.1330383

903 Mystery Level

903 can be written as the sum of consecutive numbers seven different ways:

  • 451 + 452 = 903; that’s 2 consecutive numbers.
  • 300 + 301 + 302 = 903; that’s 3 consecutive numbers.
  • 147 + 148 + 149 + 150 + 151 + 153 = 903; that’s 6 consecutive numbers.
  • 126 + 127 + 128 + 129 + 130 + 131 + 132 = 903; that’s 7 consecutive numbers.
  • 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 = 903; that’s fourteen consecutive numbers.
  • 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 903; that’s 21 consecutive numbers.
  • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 903; that’s 42 consecutive numbers.

That last way means that 903 is the 42nd triangular number. It happened because (42 × 43)/2 = 903.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 903 is a composite number.
  • Prime factorization: 903 = 3 × 7 × 43
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 903 has exactly 8 factors.
  • Factors of 903: 1, 3, 7, 21, 43, 129, 301, 903
  • Factor pairs: 903 = 1 × 903, 3 × 301, 7 × 129, or 21 × 43
  • 903 has no square factors that allow its square root to be simplified. √903 ≈ 30.0499584.

892 Tribute to The Mysteries of Harris Burdick

When I finished making today’s puzzle, I remembered a particular picture from The Mysteries of Harris Burdick. Can you guess which picture that would be?

Print the puzzles or type the solution on this excel file: 12 factors 886-896

This popular children’s book contains only pictures with short captions. Children use their imaginations to write short stories for the curious pictures and captions. The book is available in municipal libraries everywhere and on amazon.com.

892 looks interesting in a couple of different bases:

  • It is 4044 in BASE 6, because 4(6³) + 4(6) + 4(1) = 4(216 + 6 + 1) = 4(223) = 892
  • It is 161 in BASE 27, because 1(27²) + 6(27) + 1(1) = 892

892 is also the sum of consecutive prime numbers: 443 + 449 = 892

  • 892 is a composite number.
  • Prime factorization: 892 = 2 × 2 × 223, which can be written 892 = 2² × 223
  • 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 892 has exactly 6 factors.
  • Factors of 892: 1, 2, 4, 223, 446, 892
  • Factor pairs: 892 = 1 × 892, 2 × 446, or 4 × 223
  • Taking the factor pair with the largest square number factor, we get √892 = (√4)(√223) = 2√223 ≈ 29.866369

891 Mystery Level Puzzle

Sometimes revealing the puzzle level reveals more than is needed. I think I will periodically publish a Mystery Level puzzle. Can you solve this one?

Print the puzzles or type the solution on this excel file: 12 factors 886-896

8 + 9 + 1 = 18, so 891 is divisible by 3 and by 9.

8 – 9 + 1 = 0, so 891 is divisible by 11.

891 looks interesting in a few different bases:

  • 1(2⁹) + 1(2⁸) + 0(2⁷) + 1(2⁶) + 1(2⁵) + 1(2⁴) + 1(2³) + 0(2²) + 1(2¹) + 1(2⁰) =891, so it’s palindrome 1101111011 in BASE 2.
  • RR in BASE 32 (R is 27 in base 10), because 27(32) + 27(1) = 27(33) = 891
  • R0 in BASE 33, because 27(33) = 891

891 is also the sum of five consecutive prime numbers: 167 + 173 + 179 + 181 + 191 = 891

  • 891 is a composite number.
  • Prime factorization: 891 = 3 × 3 × 3 × 3 × 11, which can be written 891 = 3⁴ × 11
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 891 has exactly 10 factors.
  • Factors of 891: 1, 3, 9, 11, 27, 33, 81, 99, 297, 891
  • Factor pairs: 891 = 1 × 891, 3 × 297, 9 × 99, 11 × 81, or 27 × 33
  • Taking the factor pair with the largest square number factor, we get √891 = (√81)(√11) = 9√11 ≈ 29.849623

 

881 What Level Should This Puzzle Be?

Before I started this blog I shared a sheet of six puzzles with a coworker. The most difficult puzzle on the sheet looked similar to this one.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

He skipped ALL the easier puzzles and went straight for the most difficult one. Even though I advised him to use logic to solve the puzzle, he used guess and check and solved the puzzle within a couple of minutes. He then bragged that he could also solve a difficult Sudoku puzzle in about five minutes. He told me that my puzzles could never be a challenge to him and he wasn’t interested in ever doing another one. Ouch.

After that experience, when I began publishing my puzzles on my blog, I made sure the most difficult puzzle on the sheet were more difficult than his puzzle was.

Still I like this level of puzzle. I’m just not sure where it should be categorized.

It’s more difficult than level 4 because 20 and 16 have more than one possible common factor. However, 20 and 16 are the only set of multiple clues in any row or column, so it’s easier than a level 6. It doesn’t exactly qualify as a level 5 so I’m not assigning it that level.

Logic is still very important in finding the solution, although I suppose some lucky guess-and-checker might find it without logic. I think most people would only get frustrated if they just guessed and checked.

So give this puzzle a try. I’m calling it level ????, and its difficulty level is somewhere between a level 4 and a level 6.

Here are a few facts about the number 881:

25² + 16² = 881, so 881 is the hypotenuse of a Pythagorean triple which happens to be a primitive:

  • 369-800-881, which is calculated from 25² – 16², 2(25)(16), 25² + 16²

881 is the sum of the nine prime numbers from 79 to 113.

  • 881 is a prime number.
  • Prime factorization: 881 is prime.
  • The exponent of prime number 881 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 881 has exactly 2 factors.
  • Factors of 881: 1, 881
  • Factor pairs: 881 = 1 × 881
  • 881 has no square factors that allow its square root to be simplified. √881 ≈ 29.681644

How do we know that 881 is a prime number? If 881 were not a prime number, then it would be divisible by at least one prime number less than or equal to √881 ≈ 29.7. Since 881 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 881 is a prime number.

Here’s another way we know that 881 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 16² = 881 with 25 and 16 having no common prime factors, 881 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √881 ≈ 29.7. Since 881 is not divisible by 5, 13, 17, or 29, we know that 881 is a prime number.

 

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