A Multiplication Based Logic Puzzle

Archive for the ‘Level 4 Puzzle’ Category

1016 and Level 4

If you’ve never tried a level 4 puzzle before, this is a great one to try. Of its twelve clues, eight have only one factor pair in which both factors are from 1 to 12. You should easily be able to place the factors for those eight clues. Since each factor from 1 to 12 must appear exactly one time in the first column and the top row, the factors from those eight clues will eliminate some of the possible factors of the other four clues. Don’t be afraid to give this puzzle a try!

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Here are some reasons why 1016 is an interesting number:

1016 is the sum of seven consecutive powers of two:
2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ = 1016
I know that’s true because 1016 is 1111111000 in BASE 2

1016 is a palindrome in a couple of bases as well:
It’s 13031 in BASE 5 because 1(5⁴) + 3(5³) + 0(5²) + 3(5¹) + 1(5⁰) = 1016
161 in BASE 29 because 1(29²) + 6(29¹) + 1(29⁰) = 1016

1016 is divisible by 2 because 6 is even.
1016 is divisible by 4 because 16 is divisible by 4. (And also because 6 is divisible by 2 but NOT by 4 and 1 is odd.)
1016 is divisible by 8 because 016 is divisible by 8. (And because 16 is divisible by 8 and 0 is an even number.
1016 is NOT divisible by 16 because 016 is divisible by 16 and 1 is an odd number.

  • 1016 is a composite number.
  • Prime factorization: 1016 = 2 × 2 × 2 × 127, which can be written 1016 = 2³ × 127
  • 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 1016 has exactly 8 factors.
  • Factors of 1016: 1, 2, 4, 8, 127, 254, 508, 1016
  • Factor pairs: 1016 = 1 × 1016, 2 × 508, 4 × 254, or 8 × 127
  • Taking the factor pair with the largest square number factor, we get √1016 = (√4)(√254) = 2√254 ≈ 31.87475

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1005 and Level 4

At first, this puzzle is fairly easy to solve, but before long you will probably get stuck. To get unstuck, pick a number. See if there is only one place in the first column or the top row where that number can go. I had to use that strategy over and over again to solve this particular puzzle. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

998 and Level 4

I could easily turn this level 4 puzzle into a level 3 puzzle by rearranging the clues so that you would know an ideal order to work on the clues. Since I’m not doing that, you will have to think more to solve this or any other level 4 puzzle. It still won’t be that hard to do. You will just have to look at all the clues and think about which one should be used next. Remember to always use logic when you consider each clue. Guessing and checking will only frustrate you!

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here’s a little bit about the number 998:

998 is palindrome 828 in BASE 11 because 8(121) + 2(11) + 8(1) = 998

  • 998 is a composite number.
  • Prime factorization: 998 = 2 × 499
  • 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 998 has exactly 4 factors.
  • Factors of 998: 1, 2, 499, 998
  • Factor pairs: 998 = 1 × 998 or 2 × 499
  • 998 has no square factors that allow its square root to be simplified. √998 ≈ 31.5911

989 Christmas Bells at Eventide

I made this puzzle with silver clues to look like a bell. I thought dark blue looked best as a background color with the silver numbers. The puzzle reminds me of a bell in the evening. For the fun of it, I googled Christmas Bells in the evening to see if any poems or songs came up.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I was very surprised that Google found a very old song called Christmas Bells at Eventide. I did not know that such a song existed. Eventide means the same thing as evening. You can listen to the song below.

This is my 989th post.

The number 989 is a palindrome in base 10. What about any other bases?
It’s 373 in BASE 17 because 3(17²) + 7(17) + 3(1) = 989,
252 in BASE 21 because 2(21²) + 5(21) + 2(1) = 989, and
1C1 in BASE 26 (C is 12 base 10) because 1(26²) + 12(26) + 1(1) = 989

  • 989 is a composite number.
  • Prime factorization: 989 = 23 × 43
  • 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 989 has exactly 4 factors.
  • Factors of 989: 1, 23, 43, 989
  • Factor pairs: 989 = 1 × 989 or 23 × 43
  • 989 has no square factors that allow its square root to be simplified. √989 ≈ 31.44837

980 Christmas Factor Trees

This level 4 puzzle has 12 clues in it that are products of factor pairs in which both factors are numbers from 1 to 12. The clues make an evergreen tree, the symbol of everlasting life which is so fitting for Christmas. Can you find the factors for the given clues and put them in the right places?

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now I’ll tell you a little about the number 980:

It has eighteen factors and many possible factor trees. Here are just three of them:

28² + 14² = 980, so 980 is the hypotenuse of a Pythagorean triple:
588-784-980 which is (3-4-5) times 196, but can also be calculated from
28² – 14², 2(28)(14), 28² + 14²

I like the way 980 looks in some other bases:
It is 5A5 in BASE 13 (A is 10 base 10) because 5(13) + 10(13) + 5(1) = 980,
500 in BASE 14 because 5(14²) = 980,
SS in BASE 34 (S is 28 base 10) because 28(34) + 28(1) = 28(35) = 980
S0 in BASE 35 because 28(35) = 980

  • 980 is a composite number.
  • Prime factorization: 980 = 2 × 2 × 5 × 7 × 7, which can be written 980 = 2² × 5 × 7²
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 980 has exactly 18 factors.
  • Factors of 980: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
  • Factor pairs: 980 = 1 × 980, 2 × 490, 4 × 245, 5 × 196, 7 × 140, 10 × 98, 14 × 70, 20 × 49 or 28 × 35
  • Taking the factor pair with the largest square number factor, we get √980 = (√196)(√5) = 14√5 ≈ 31.30495.

974 and Level 4

This puzzle has ten clues and, like always, it has only one solution. If you can figure out where to put the factors 1 to 10 in the first column as well as the top row, then you will have found that solution.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

This is my 974th post.

974 is the sum of three consecutive square numbers:
17² + 18² + 19² = 974

  • 974 is a composite number.
  • Prime factorization: 974 = 2 × 487
  • 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 974 has exactly 4 factors.
  • Factors of 974: 1, 2, 487, 974
  • Factor pairs: 974 = 1 × 974 or 2 × 487
  • 974 has no square factors that allow its square root to be simplified. √974 ≈ 31.20897

 

965 Run, Turkey, Run!

Happy Thanksgiving, everyone!

I didn’t mean to make any Thanksgiving puzzles but after I created the puzzles this week, I could see some Thanksgiving-related pictures in the designs I had already made. This one is my favorite.

Run, Turkey, Run! For millions of turkeys today, it’s already too late.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Here is a little about the number 965:

965 is the sum of two squares two different ways:
26² + 17² = 965
31² + 2² = 965

So it is also the hypotenuse of FOUR Pythagorean triples, two of them primitives:
124-957-965, calculated from 2(31)(2), 31² – 2², 31² + 2²
387-884-965, calculated from 26² – 17², 2(26)(17), 26² + 17²
475-840-965 which is 5 times (95-168-193)
579-772-965 which is (3-4-5) times 193

  • 965 is a composite number.
  • Prime factorization: 965 = 5 × 193
  • 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 965 has exactly 4 factors.
  • Factors of 965: 1, 5, 193, 965
  • Factor pairs: 965 = 1 × 965 or 5 × 193
  • 965 has no square factors that allow its square root to be simplified. √965 ≈ 31.064449

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