A Multiplication Based Logic Puzzle

Archive for the ‘Level 1 Puzzle’ Category

959 and Level 1

Whether you are young, old, or in between, if you can do some simple division, then you can solve this level 1 puzzle. There is a column of clues and a row of clues. Both of them have the same common factor. Write that common factor in the first column to the left of the row of clues and again in the top row above the column of clues. Then simply divide. You will be done in no time at all.

Here are some facts about the number 957:

959 is the hypotenuse of a Pythagorean triple:
616-735-959 which is 7 times (88-105-137)

959 is a palindrome in base 10.

And it is a cool-looking 1110111111 in BASE 2
because (2¹⁰ – 1) – 2⁶ = 959.
In base 2 we would write (if we use commas)
1,111,111,111 – 1,000,000 = 1,110,111,111

 

Advertisements

952 and Level 1

If you’ve never solved a Find the Factors puzzle before, this one will be perfect for you to do. It only has nine clues, but that is enough to find all the factors and fill in the entire multiplication table. You will feel quite smart when you’re done, too.

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

This is my 952nd post so I will mention a few facts about that number.

Stetson.edu informs us that 93 + 53 + 23 + 9 × 5 × 2 = 952

952 is the hypotenuse of a Pythagorean triple:
448-840-952 which is (8-15-17) times 56

952 looks interesting in some other bases:
4224 in BASE 6 because 4(6³) + 2(6²) + 2(6¹) + 4(6⁰) = 952
2C2 in BASE 19 (C is 12 Base 10) because 2(19²) + 12(19¹) +2(19⁰) = 952
SS BASE 33 (S is 28) because 28(33¹) + 28(33⁰) = 28(33 + 1) = 28(34) = 952
S0 BASE 34 because 28(34¹) + 0(34⁰) = 28(34) = 952

  • 952 is a composite number.
  • Prime factorization: 952 = 2 × 2 × 2 × 7 × 17, which can be written 952 = 2³ × 7 × 17
  • 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 952 has exactly 16 factors.
  • Factors of 952: 1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952
  • Factor pairs: 952 = 1 × 952, 2 × 476, 4 × 238, 7 × 136, 8 × 119, 14 × 68, 17 × 56, or 28 × 34
  • Taking the factor pair with the largest square number factor, we get √952 = (√4)(√238) = 2√238 ≈ 29.854497

942 and Level 1

This puzzle is probably as tough as a level 1 puzzle can get, but don’t let that prevent you from giving it a try! Can you figure out where the factors from 1 to 12 go in both the first column and the top row?

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

Now let me tell you a little about the number 942:

It is the sum of four consecutive prime numbers:
229 + 233 + 239 + 241 = 942

It is the hypotenuse of a Pythagorean triple:
510-792-942 which is 6 times (85-132-157)

942 is a palindrome in two other bases and a repdigit in another:
787 in BASE 11, because 7(11²) + 8(11¹) + 7(11⁰) = 942
272 in BASE 20, because 2(20²) + 7(20¹) + 2(20⁰) = 942
666 in BASE 12, because 6(12²) + 6(12¹) + 6(12⁰) = 6(144+12+1) = 6(157) = 942

942³ is 835,896,888. Stetson.edu tells us that 942³ is the smallest perfect cube that contains five 8‘s.

  • 942 is a composite number.
  • Prime factorization: 942 = 2 × 3 × 157
  • 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 942 has exactly 8 factors.
  • Factors of 942: 1, 2, 3, 6, 157, 314, 471, 942
  • Factor pairs: 942 = 1 × 942, 2 × 471, 3 × 314, or 6 × 157
  • 942 has no square factors that allow its square root to be simplified. √942 ≈ 30.6920185

932 and Level 1

These sixteen clues are all you need to solve this puzzle. First, figure out where the factors from 1-10 go in the first column and the top row so that clues in the puzzle and the factors will be like a multiplication table. After you write in all the factors, you can decide if you’d like to fill in the rest of the table. It’s not difficult.

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

Now here’s a few facts about the number 932:

26² + 16² = 932
That means 420² + 832² = 932²

Why? Because 420-832-932 can be calculated from 26² – 16², 2(26)(16), 26² + 16²

  • 932 is a composite number.
  • Prime factorization: 932 = 2 × 2 × 233, which can be written 932 = 2² × 233
  • 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 932 has exactly 6 factors.
  • Factors of 932: 1, 2, 4, 233, 466, 932
  • Factor pairs: 932 = 1 × 932, 2 × 466, or 4 × 233
  • Taking the factor pair with the largest square number factor, we get √932 = (√4)(√233) = 2√233 ≈ 30.528675

923 Grave Marker

To me graveyards are beautiful places where the dearly departed are laid to rest. Find A Grave and Billiongraves are two genealogical sources that assist individuals in finding grave sites. When my son and I visited graveyards in Hungary and Slovakia a few years ago, we saw many wood and stone grave markers which had been eroded by weather. Some were almost impossible to read. We also suspect some people were too poor when they died to get a headstone of any type. We were very excited when we saw any readable grave markers with our family surnames.

Recently on twitter I saw these paintings of gothic graveyards by M J Forster. I knew immediately I wanted to include them in this post. The paintings are quite stunning.

//platform.twitter.com/widgets.js

Finding departed ancestors can sometimes be difficult, but very rewarding. Finding the factors in today’s puzzle will be very easy:

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

Here’s a fun fact about the number 923:

Stetson.edu informs us that 923(923 + 1) = 852,852. Below are two of the MANY possible factor trees for 852,852. The first one includes factor trees for 923 and 924, the second one shows why their product uses digits that repeat itself in order.

  • 923 is a composite number.
  • Prime factorization: 923 = 13 × 71
  • 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 923 has exactly 4 factors.
  • Factors of 923: 1, 13, 71, 923
  • Factor pairs: 923 = 1 × 923 or 13 × 71
  • 923 has no square factors that allow its square root to be simplified. √923 ≈ 30.380915

915 Traditional Vampire Deterrent

915 is the hypotenuse of four Pythagorean triples:

  • 165-900-915 which is 15 times (11-60-61)
  • 408-819-915 which is 3 times (136-273-305)
  • 621-672-915 which is 3 times (207-224-305)
  • 549-732-915 which is (3-4-5) times 183

 

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

Here’s a little more about the number 915:

915 is repdigit 555 in BASE 13 because 5(13²) + 5(13) + 5(1) = 5(183) = 915.

It is palindrome 393 in BASE 16 because 3(16²) + 9(16) +3(1) = 915.

And as 195 in BASE 26, it uses its base 10 digits in a different order. Note that 1(26²) + 9(26) + 5(1) = 915.

  • 915 is a composite number.
  • Prime factorization: 915 = 3 × 5 × 61
  • 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 915 has exactly 8 factors.
  • Factors of 915: 1, 3, 5, 15, 61, 183, 305, 915
  • Factor pairs: 915 = 1 × 915, 3 × 305, 5 × 183, or 15 × 61
  • 915 has no square factors that allow its square root to be simplified. √915 ≈ 30.2489669.

 

905 and Level 1

905 is the sum of the seventeen prime numbers from 19 to 89.

905 is also the sum of these seven consecutive prime numbers:

  • 109 + 113 + 127 + 131 + 137 + 139 + 149 = 905

Here’s today’s puzzle:

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

29² + 8² = 905, and 28² + 11² = 905, making 905 the hypotenuse of four Pythagorean triples:

  • 95-900-905, which is 5 times (19-180-181)
  • 464-777-905, computed from 2(29)(8), 29² – 8², 29² + 8²
  • 543-724-905, which is (3-4-5) times 181
  • 616-663-905, computed from 2(28)(11), 28² – 11², 28² + 11²

The numbers in red are factors of 905.

  • 905 is a composite number.
  • Prime factorization: 905 = 5 × 181
  • 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 905 has exactly 4 factors.
  • Factors of 905: 1, 5, 181, 905
  • Factor pairs: 905 = 1 × 905 or 5 × 181
  • 905 has no square factors that allow its square root to be simplified. √905 ≈ 30.0832179

Tag Cloud