A Multiplication Based Logic Puzzle

Posts tagged ‘Puzzle’

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.

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

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

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922 Boo!

Have you ever cut holes in a sheet, put it over your head, and jumped out in front of people as you hollered, “Boo!”? Today’s puzzle is made to look like a ghost. It’s a level 6, but don’t let that spook you! Attack the puzzle using logic, and after you solve it, you can claim to be a ghost-buster!

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

When 922 floats around in a different base, you may think you’re seeing an apparition:

922 becomes 1234 in BASE 9 because 1(9³) + 2(9²) + 3(9¹) + 4(9º) = 922.
922 becomes palindrome 262 in BASE 20

922 is also the sum of the 18 prime numbers from 17 to 89.

922 = 29² + 9², so 922 is the hypotenuse of a Pythagorean triple:
522-760-922, which is the same as 2(29)(9), 29² – 9², 29² + 9².
That Pythagorean triple is also 2 times (261-380-461).

  • 922 is a composite number.
  • Prime factorization: 922 = 2 × 461
  • 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 922 has exactly 4 factors.
  • Factors of 922: 1, 2, 461, 922
  • Factor pairs: 922 = 1 × 922 or 2 × 461
  • 922 has no square factors that allow its square root to be simplified. √922 ≈ 30.3644529

 

 

921 Is This Bug Cute or Creepy?

Some bugs make creepy Halloween decorations. Other bugs, like ladybugs, might make a very cute costume.

Today’s puzzle looks like a bug, but there is no reason to run and hide from this one. Yes, it’s a level 5, so some parts of it may be tricky.

This is what you need to do to solve it: stay calm; don’t guess and check. Figure out where to put each number from 1 to 10 in both the top row and the first column so that the clues make the puzzle work like a multiplication table. Don’t write a number down unless you are absolutely sure it belongs where you’re putting it. Use logic, step by step, and this puzzle will be a treat.

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

When you put on a Halloween costume, you may look completely different.

When a number is written in a different base, it may look completely different. For example,
921 looks like repdigit 333 in BASE 17 because 3(17²) + 3(17¹) + 3(17º) = 3(289 + 17 + 1) = 3(307) = 921
(307 is 111 in BASE 17)

921 looks like palindrome 1H1 in BASE 23 (H is 17 base 10). As you might suspect, 1(23²) + 17(23¹) + 1(23º) = 529 + 391 + 1 = 921

When it’s not written in a different base, 921 looks pretty familiar. You can tell quite quickly that it is divisible by 3:

  • 921 is a composite number.
  • Prime factorization: 921 = 3 × 307
  • 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 921 has exactly 4 factors.
  • Factors of 921: 1, 3, 307, 921
  • Factor pairs: 921 = 1 × 921 or 3 × 307
  • 921 has no square factors that allow its square root to be simplified. √921 ≈ 30.34798181

920 Witches’ Cauldron

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

What besides “eye of newt” goes in witches’ cauldrons? The list includes some horrifying ingredients that you can read here from one scene from Shakeaspeare’s play, MacBeth.

Instead of putting “Eye of newt, and toe of frog, Wool of bat, and tongue of dog” and so forth in today’s Halloween cauldron puzzle, I just put a bunch of asterisks.

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

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

Double 115 is 230.

Double 230 is 460.

Double 460 is 920, today’s post number.

920 is the hypotenuse of a Pythagorean triple:
552-736-920 which is (3-4-5) times 184.

920 is palindrome 767 in BASE 11 because 7(121) + 6(11) + 7(1) = 920

  • 920 is a composite number.
  • Prime factorization: 920 = 2 × 2 × 2 × 5 × 23, which can be written 920 = 2³ × 5 × 23
  • 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 920 has exactly 16 factors.
  • Factors of 920: 1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920
  • Factor pairs: 920 = 1 × 920, 2 × 460, 4 × 230, 5 × 184, 8 × 115, 10 × 92, 20 × 46, or 23 × 40
  • Taking the factor pair with the largest square number factor, we get √920 = (√4)(√230) = 2√230 ≈ 20.331501776.

 

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

916 Witch’s Hat

Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

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

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • 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 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

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.

 

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