A Multiplication Based Logic Puzzle

Archive for the ‘Level 3 Puzzle’ Category

963 Arrow

The numbers 1 to 12 fit someplace in the first column as well as in the top row.  Can you figure out where those places are so that this puzzle can become a multiplication table?

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

Now I’d like to mention a few things about the number 963:

It is the sum of the 24 prime numbers from 2 to 89. Pretty cool!

I like the way 963 looks in base 10 as well as in a few other bases:
33003 in BASE 4 because 3(4⁴) + 3(4³) + 3(1) = 3 × 321 = 963
3C3 in BASE 16 (C is 12 in base 10) because 3(16²) + 12(16) + 3(1) = 963
1B1 in BASE 26 (B is 11 in base 10) because 1(26²) + 11(26) + 1(1) = 963
123 in BASE 30 because 1(30²) + 2(30¹) + 3(30⁰) = 963

  • 963 is a composite number.
  • Prime factorization: 963 = 3 × 3 × 107, which can be written 963 = 3² × 107
  • 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 963 has exactly 6 factors.
  • Factors of 963: 1, 3, 9, 107, 321, 963
  • Factor pairs: 963 = 1 × 963, 3 × 321, or 9 × 107
  • Taking the factor pair with the largest square number factor, we get √963 = (√9)(√107) = 3√107 ≈ 31.03224

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954 and Level 3

The ten clues in this puzzle are all that is needed to solve the puzzle and then make it a complete multiplication table. Seriously, ten clues! Yes, the factors and the products will not be in their usual places, but the completed puzzle will still be a legitimate multiplication table. Can YOU make it work?

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

954 is the sum of ten consecutive prime numbers:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 954

27² +  15² = 954 so 954 is the hypotenuse of a Pythagorean triple:
504-810-954 which is 18 times (28-45-53)

954 is a palindrome in two consecutive bases:
676 in BASE 12 because 6(144) + 7(12) + 6(1) = 954
585 in BASE 13 because 5(169) + 8(13) + 5(1) = 954

  • 954 is a composite number.
  • Prime factorization: 954 = 2 × 3 × 3 × 53, which can be written 954 = 2 × 3² × 53
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 954 has exactly 12 factors.
  • Factors of 954: 1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954
  • Factor pairs: 954 = 1 × 954, 2 × 477, 3 × 318, 6 × 159, 9 × 106, or 18 × 53,
  • Taking the factor pair with the largest square number factor, we get √954 = (√9)(√106) = 3√106 ≈ 30.88689

944 and Level 3

The division facts needed to solve today’s puzzle are not complicated. You can fill in all the cells of this puzzle if you know the multiplication facts from 1 × 1 to 12 × 12.

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

Now here are some facts about the number 944:

944 is divisible by 2 because it is even.
944 is divisible by 4 because the last number is divisible by 4 and the digit before it is even.
944 can be evenly divided by 8 because 44 is divisible by 4, but not by 8, and the digit before 44 is odd.

944 is a funny-looking palindrome, 1I1, in BASE 23 (I is 18 in base 10) because 1(23²) + 18(23¹) + 1(23⁰) = 944

  • 944 is a composite number.
  • Prime factorization: 944 = 2 × 2 × 2 × 2 × 59, which can be written 944 = 2⁴ × 59
  • 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 944 has exactly 10 factors.
  • Factors of 944: 1, 2, 4, 8, 16, 59, 118, 236, 472, 944
  • Factor pairs: 944 = 1 × 944, 2 × 472, 4 × 236, 8 × 118, or 16 × 59
  • Taking the factor pair with the largest square number factor, we get √944 = (√16)(√59) = 4√59 ≈ 30.72458

935 Is the Second Lucas-Carmichael Number

935 = 5 × 11 × 17, and 935 + 1 is evenly divisible by 5 + 1, 11 + 1, and 17 + 1. That makes 935 only the SECOND Lucas-Carmichael number. Thanks to Stetson.edu for that fun fact.

Today’s puzzle is a level 3, a good transition from the easier puzzles to the more difficult ones.

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

Here’s more about the number 935:

935 is the sum of the nineteen prime numbers from 13 to 89.

935 is the hypotenuse of four Pythagorean triples:
143-924-935, which is 11 times (13-84-85)
396-847-935, which is 11 times (36-77-85)
440-825-935, which is (8-15-17) times 55
561-748-935, which is (3-4-5) times 187

  • 935 is a composite number.
  • Prime factorization: 935 = 5 × 11 × 17
  • 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 935 has exactly 8 factors.
  • Factors of 935: 1, 5, 11, 17, 55, 85, 187, 935
  • Factor pairs: 935 = 1 × 935, 5 × 187, 11 × 85, or 17 × 55
  • 935 has no square factors that allow its square root to be simplified. √935 ≈ 30.5777697

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

 

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

907 and Level 3

88 is 8 × 11, and 24 is 2 × 12, 3 × 8, or 4 × 6. Those are all their factor pairs in which both factors are less than or equal to 12. There is only one number that is a common factor for both 88 and 24. Write that number in the top cell of the first column and the rest of 88’s factor pair directly above 88 and the rest of 24’s factor pair directly above 24 in the top row.

Next think of a factor pair for 80 in which both factors are less than or equal to 12. You probably thought of 8 × 10, the only factor pair that qualifies. The first column already has an 8, so this 8 must go in the top row above 80. Write 10 in the first column.

The next row doesn’t have a clue, but you already have enough information to write what number must go in the first column. (Hint: it is a number that is already in the top row and can’t go in any other cell in the first column.) If you cannot figure out what goes in this cell, skip that row until later, and figure out what goes in the next cells continuing from the top cell of the first column to the bottom cell. You will fill out the top row at the same time, but each factor 1- 12 will be written above its appropriate clue instead of in order from left to right. Good luck!

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

907 is the first prime number since 887. We will not have to wait nearly as long for the next prime number. It will be 911.

907 is palindrome 32023 in BASE 4 because 3(4⁴) + 2(4³) + 0(4²) + 2(4¹) + 3(4º) = 907.

  • 907 is a prime number.
  • Prime factorization: 907 is prime.
  • The exponent of prime number 907 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 907 has exactly 2 factors.
  • Factors of 907: 1, 907
  • Factor pairs: 907 = 1 × 907
  • 907 has no square factors that allow its square root to be simplified. √907 ≈ 30.11644

How do we know that 907 is a prime number? If 907 were not a prime number, then it would be divisible by at least one prime number less than or equal to √907 ≈ 30.1. Since 907 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 907 is a prime number.

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