A Multiplication Based Logic Puzzle

Archive for the ‘Mystery Level Puzzle’ Category

984 Way Too Big Christmas Factor Tree

Some Christmas trees are so big they are difficult to take home in the car. They might even be too big to set up in the house. This puzzle is the biggest one I have ever made. It looks like a very big Christmas tree waiting to be set up. Is it too big to bring fun this Christmastime?

The table below may be helpful in solving the puzzle. There are 400 places to write products in a 20 × 20 multiplication table, but not all the numbers from 1 to 400 appear in such a table. Some numbers don’t appear at all while other numbers appear more than one times. The chart below is color-coded to show how many times a product appears in the 20 × 20 multiplication table. Clues in the puzzle that appear only once (yellow) or twice (green) in the multiplication table won’t cause much trouble when solving the puzzle. Any other clues might stump you. Notice that the number 60 appears twice in the puzzle but eight times (black) in the 20 × 20 multiplication table!

I’d like to share some information about the number 984.

It is the hypotenuse of a Pythagorean triple:
216-960-984 which is 24 times (9-40-41)

Stetson.edu informs us that 8 + 88 + 888 = 984.

984 is 1313 in BASE 9 because 1(9³) + 3(9²) + 1(9¹) + 3(9⁰) = 984.

Here are a couple of the many possible factor trees for 984:

  • 984 is a composite number.
  • Prime factorization: 984 = 2 × 2 × 2 × 3 × 41, which can be written 984 = 2³ × 3 × 41
  • 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 984 has exactly 16 factors.
  • Factors of 984: 1, 2, 3, 4, 6, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984
  • Factor pairs: 984 = 1 × 984, 2 × 492, 3 × 328, 4 × 246, 6 × 164, 8 × 123, 12 × 82, or 24 × 41
  • Taking the factor pair with the largest square number factor, we get √984 = (√4)(√246) = 2√246 ≈ 31.36877

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972 Happy Birthday, Andy!

Today is my brother Andy’s birthday. I know Andy can solve these puzzles so I’ve made him a puzzle cake with factors from 1 to 16. Adding extra factor possibilities complicates the puzzle and makes it a little more difficult to read as a multiplication table, but it is still solvable. Since these puzzles have only one solution and are solved by logic and not by guessing and checking, I added a clue right in the center of the cake to ensure a unique solution. Happy birthday, Andy!

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

Now I’ll share a little about the number 972 which is the 13th Achilles number.  All of the exponents in its prime factorization are greater than 1, yet the greatest common factor of those exponents is still 1. The previous Achilles number, 968, and 972 are the closest two Achilles numbers so far.

I think 972 has some interesting representations when written in some other bases:

It’s 33030 in BASE 4 because 3(4⁴) + 2(4³) + 0(4²) + 3(4) + 0(1) = 3(256 + 64 + 4) = 3(324) = 972
363 in BASE 17 because 3(17²) + 6(17) + 3(1) = 972
300 in BASE 18 because 3(18²) = 3(324) = 972
RR in BASE 35 (R is 27 base 10) because 27(35) + 27(1) = 27(36) = 972
R0 in BASE 36 because 27(36) + 0(1) = 27(36) = 972

  • 972 is a composite number.
  • Prime factorization: 972 = 2 × 2 × 3 × 3 × 3 × 3 × 3, which can be written 972 = 2²× 3⁵
  • The exponents in the prime factorization are 2 and 5. Adding one to each and multiplying we get (2 + 1)(5 + 1) = 3 × 6 = 18. Therefore 972 has exactly 18 factors.
  • Factors of 972: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972
  • Factor pairs: 972 = 1 × 972, 2 × 486, 3 × 324, 4 × 243, 6 × 162, 9 × 108, 12 × 81, 18 × 54 or 27 × 36
  • Taking the factor pair with the largest square number factor, we get √972 = (√324)(√3) = 18√3 ≈ 31.1769

Here are a few of the MANY possible factor trees for 972:

967 Black Friday Shopping Advantage

If your shopping cart were a go-kart, you would have an advantage getting all the shopping bargains Black Friday offers. Not only would you be able to move much faster than the average shopping cart, but you would also be able to do wheelies to get through the crowds, around corners, or tight spaces. After you complete the shopping spree of your dreams, you can lie down exhausted, but ecstatic and work on a puzzle, like this one.

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

I realize I’m really pushing it to make this puzzle have a Thanksgiving week theme. I love that Black Friday has turned into Black November because it means bargains without all the crowds.

You can also imagine the puzzle is a toy on a child’ wishlist. Whatever you think, I hope you enjoy solving the puzzle.

Here’s a little about prime number 967:

It is 595 in BASE 13 because 5(13²) + 9(13¹) + 5(13⁰) = 967
It is also 1J1 in BASE 23 (J is 19 in base 10) because 1(23²) + 19(23¹) + 1(23⁰) = 967

  • 967 is a prime number.
  • Prime factorization: 967 is prime.
  • The exponent of prime number 967 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 967 has exactly 2 factors.
  • Factors of 967: 1, 967
  • Factor pairs: 967 = 1 × 967
  • 967 has no square factors that allow its square root to be simplified. √967 ≈ 31.09662

How do we know that 967 is a prime number? If 967 were not a prime number, then it would be divisible by at least one prime number less than or equal to √967 ≈ 31.1. Since 967 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, or 31, we know that 967 is a prime number.

957 Mystery Pentagon Puzzle

Pentagons have been on my mind this week so here is another puzzle with a pentagon in it. This time the pentagon is small. How difficult is this Mystery Level puzzle?  That depends on if you recognize one very important piece of logic needed to solve it. If you see that logic, it’s not too bad. If you don’t, it might do you in.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

957 is the hypotenuse of a Pythagorean triple:
660-693-957 which is (20-21-29) times 33

957 is repdigit TT in BASE 32 (T is 29 base 10)
because 29(32) + 29(1) = 29(32 + 1) = 29(33) = 957
957 is also T0 in BASE 33 because 29(33) + 0(1) = 957

  • 957 is a composite number.
  • Prime factorization: 957 = 3 × 11 × 29
  • 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 957 has exactly 8 factors.
  • Factors of 957: 1, 3, 11, 29, 33, 87, 319, 957
  • Factor pairs: 957 = 1 × 957, 3 × 319, 11 × 87, or 29 × 33
  • 957 has no square factors that allow its square root to be simplified. √957 ≈ 30.9354

949 Mystery Level

The difficulty level of a mystery level puzzle is a mystery. You won’t know how hard or how easy it is until you give it a try. You can solve it by using logic and your knowledge of the multiplication table. Can you figure it out?

Print the puzzles or type the solution in this excel file: 12 factors 942-950

Now here is a little about the number 949:

948 and 949 are a Ruth-Aaron pair.

949 is not only a palindrome in base 10, but it is also
434 in BASE 15 because 4(15²) + 3(15¹) + 4(15⁰) = 949

25² + 18² = 949 and 30² + 7² = 949 so 949 is the hypotenuse of four Pythagorean triples:

301-900-949 calculated from 25² – 18², 2(25)(18), 25² + 18²
365-876-949 which is (5-12-13) times 73
420-851-949 calculated from 2(30)(7), 30² – 7², 30² + 7²
624-715-949 which is 13 times (48-55-73)

  • 949 is a composite number.
  • Prime factorization: 949 = 13 × 73
  • 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 949 has exactly 4 factors.
  • Factors of 949: 1, 13, 73, 949
  • Factor pairs: 949 = 1 × 949 or 13 × 73
  • 949 has no square factors that allow its square root to be simplified. √949 ≈ 30.80584

940 Mystery Level

Today’s puzzle reminds me of a gumball machine. I would invite you to stick to solving this puzzle until you find success. I assure you that the factors from 1 to 10 can be placed in the first column and the top row solely by using logic.

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Now let me tell you something about the number 940.

940 is the hypotenuse of a Pythagorean triple:
564-752-940 which is (3-4-5) times 188

  • 940 is a composite number.
  • Prime factorization: 940 = 2 × 2 × 5 × 47, which can be written 940 = 2² × 5 × 47
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 940 has exactly 12 factors.
  • Factors of 940: 1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470, 940
  • Factor pairs: 940 = 1 × 940, 2 × 470, 4 × 235, 5 × 188, 10 × 94, or 20 × 47,
  • Taking the factor pair with the largest square number factor, we get √940 = (√4)(√235) = 2√235 ≈ 30.659419

931 Candy Bar Mystery

What kind of candy bars might these be? That is the mystery. What are your favorite candy bars? My favorites are Snickers, 100 Grand, and Baby Ruth.

What level is this puzzle? I’m not telling. Figuring that out is part of the fun. Give this mystery level puzzle a try. If you can’t solve it, it will be a trick, but if you can, it will be a treat.

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

 

931 is the sum of consecutive prime numbers three different ways:
It is the sum of the 15 prime numbers from 31 to 97,
the sum of the 11 prime numbers from 61 to 107, and
these three consecutive primes, 307 + 311 + 313 = 931.

Here’s 931 in a few different bases:
It’s repdigit 777 in BASE 11, because 7(121) + 7(11) + 7(1) = 7(133) = 931, and
repdigit 111 in BASE 30, because 1(900) + 1(30) + 1(1) = 931.
931 is palindrome 3A3 in BASE 16 (A is 10 base 10), because 3(16²) + 10(16) + 3(1) = 931

  • 931 is a composite number.
  • Prime factorization: 931 = 7 × 7 × 19, which can be written 931 = 7² × 19
  • 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 931 has exactly 6 factors.
  • Factors of 931: 1, 7, 19, 49, 133, 931
  • Factor pairs: 931 = 1 × 931, 7 × 133, or 19 × 49
  • Taking the factor pair with the largest square number factor, we get √931 = (√49)(√19) = 7√19 ≈ 30.51229

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