A Multiplication Based Logic Puzzle

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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