A Multiplication Based Logic Puzzle

Posts tagged ‘Christmas’

721 Merry Christmas, Everybody!

It isn’t difficult to see that 721 is divisible by 7.

  • 721 is a composite number.
  • Prime factorization: 721 = 7 x 103
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 721 has exactly 4 factors.
  • Factors of 721: 1, 7, 103, 721
  • Factor pairs: 721 = 1 x 721 or 7 x 103
  • 721 has no square factors that allow its square root to be simplified. √721 ≈ 26.85144.

Merry Christmas everybody! Today’s puzzle is not too hard and not too easy so enjoy solving it during your leisure hours today.

721 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here is a little more about the number 721:

721 is the sum of the nine prime numbers from 61 to 101.

Stetson.edu informs us that 721 can be expressed as the difference of two cubes two different ways, and is the smallest number that can make that claim. The two ways were fairly easy to find:

  • 9^3 – 2^3 = 721
  • 16^3 – 15^3 = 721

Because it is equal to the difference of the 16th and 15th cubes, 721 is the 16th centered hexagonal number.

And 721 is a palindrome in two bases:

  • 1G1 in base 20 (G = 16 base 10); note that 1(400) + 16(20) + 1(1) = 721.
  • 161 base 24; note that 1(24²) + 6(24) + 1(1) = 721.

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721 Logic

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702 A Couple of Christmas Factor Trees

Since the sum of its digits equals nine, 702 is divisible by nine.

  • 702 is a composite number.
  • Prime factorization: 702 = 2 x 3 x 3 x 3 x 13, which can be written 702 = 2 x (3^3) x 13
  • 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 x 4 x 2 = 16. Therefore 702 has exactly 16 factors.
  • Factors of 702: 1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702
  • Factor pairs: 702 = 1 x 702, 2 x 351, 3 x 234, 6 x 117, 9 x 78, 13 x 54, 18 x 39, or 26 x 27
  • Taking the factor pair with the largest square number factor, we get √702 = (√9)(√78) = 3√78 ≈ 26.49528.

702 is the product of consecutive integers: 26 x 27 = 702. Numbers that can be expressed as such products are known as Pronic numbers.

It seems only natural to make factor trees based on those two multiplication facts:

702 Factor Trees

Today’s Find the Factors puzzle also looks like a couple of small Christmas trees.

702 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are more facts about the number 702:

It is the sum of consecutive prime numbers 349 and 353.

It is also the sum of the seventeen prime numbers from 7 to 73.

And because 13 is one of its factors, 702 is the hypotenuse of Pythagorean triple 270-648-702. Notice that the short leg is a permutation of 702.

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702 Logic

Level 4 Christmas Puzzle #699

  • 699 is a composite number.
  • Prime factorization: 699 = 3 x 233
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 699 has exactly 4 factors.
  • Factors of 699: 1, 3, 233, 699
  • Factor pairs: 699 = 1 x 699 or 3 x 233
  • 699 has no square factors that allow its square root to be simplified. √699 ≈ 26.438608.

Here is a Christmas puzzle for you to solve. It’s numbered 699 to distinguish it from every other puzzle I make:

699 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are a few thoughts I’ve had about the number 699:

699 is the smallest number whose digits add up to 24.

Every odd number greater than 1 is the sum of 2 consecutive numbers. 699 is the sum of 349 and 350.

Every number that is divisible by 3 is the sum of 3 consecutive numbers: 232 + 233 + 234 = 699.

Also 699 is the hypotenuse of Pythagorean triple 315-624-699. Which factor of 699 is the greatest common factor of those three numbers?

699 is palindrome 272 in BASE 17; note that 2(289) + 7(17) + 2(1) = 699

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699 Logic

696 There are lots of goodies in this Christmas Stocking

  • 696 is a composite number.
  • Prime factorization: 696 = 2 x 2 x 2 x 3 x 29, which can be written 696 = (2^3) x 3 x 29
  • 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 x 2 x 2 = 16. Therefore 696 has exactly 16 factors.
  • Factors of 696: 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696
  • Factor pairs: 696 = 1 x 696, 2 x 348, 3 x 232, 4 x 174, 6 x 116, 8 x 87, 12 x 58, or 24 x 29
  • Taking the factor pair with the largest square number factor, we get √696 = (√4)(√174) = 2√174 ≈ 26.38181.

Today’s puzzle is meant to look like a Christmas stocking or boot that can be filled with lots of little treasures.

696 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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What other facts did I find about the number 696?

696 is the sum of all the prime numbers from 71 to 103. Do you know what those eight prime numbers are?

696 is also the sum of consecutive odd numbers 347 and 349 which just happen to also be consecutive prime numbers.

Because 696 is a multiple of 29, it is the hypotenuse of Pythagorean triple 480-504-696. What is the greatest common factor of those three numbers?

696 is a palindrome in two different bases

  • 696 BASE 10; note that 6(100) + 9(10) + 6(1) = 696
  • OO BASE 28; note that O BASE 28 is equivalent to 24 in BASE 10, and  24(28) + 24(1) = 696

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696 Factors

 

321 I Heard the Bells on Christmas Day

321 is made solely from three consecutive numbers so it can be evenly divided by 3.

  • 321 is a composite number.
  • Prime factorization: 321 = 3 x 107
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 321 has exactly 4 factors.
  • Factors of 321: 1, 3, 107, 321
  • Factor pairs: 321 = 1 x 321 or 3 x 107
  • 321 has no square factors that allow its square root to be simplified. √321 ≈ 17.916
I heard the bells
Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08
  1. I heard the bells on Christmas day
    Their old familiar carols play,
    And wild and sweet the words repeat
    Of peace on earth, good will to men.
  2. I thought how, as the day had come,
    The belfries of all Christendom
    Had rolled along th’unbroken song
    Of peace on earth, good will to men.
  3. And in despair I bowed my head:
    “There is no peace on earth,” I said,
    “For hate is strong and mocks the song
    Of peace on earth, good will to men.”
  4. Then pealed the bells more loud and deep:
    “God is not dead, nor doth he sleep;
    The wrong shall fail, the right prevail,
    With peace on earth, good will to men.”
  5. Till, ringing, singing, on its way,
    The world revolved from night to day,
    A voice, a chime, a chant sublime,
    Of peace on earth, good will to men!
Text: Henry Wadsworth Longfellow, 1807-1882
I heard the bells factors

19 Last-Minute Gift

19 is a prime number. 19 = 1 x 19. Its only factors are 1 and 19. Prime factorization: none.

How do we know that 19 is a prime number? The square root of 19 is an irrational number approximately equal to 4.36. If 19 were not a prime number, then it would be divisible by at least one prime number less than or equal to 4.36. Since 19 is not divisible by 2 or 3, it is a prime number.

19 is never a clue in the FIND THE FACTORS puzzles.

It’s Christmas Eve or even Christmas day, and maybe all of your shopping didn’t get done. Maybe you didn’t want to drive anyplace because of bad weather, or your favorite stores were closed early for the holiday. Well, if someone on your list likes number placing puzzles (like Sudoku or Kakuro), then I have a last-minute gift idea for you, and it’s free. I design a number placing logic puzzle based on the multiplication table called FIND THE FACTORS. If you have a computer, the internet, and a printer, you can print a little holiday booklet filled with these puzzles and give it as a gift. If the person on your gift list is many miles away, you can even send the booklet electronically. This last minute gift is good for the brain and can be good for the memory. The level 1 and level 2 puzzles can be solved by children 3rd grade and up, but most of the higher level puzzles will be challenging for everyone regardless of age. 

Here is puzzle created to look a little like an angel just for the holidays:

2nd angel

To solve the puzzle above simply write the numbers 1 – 12 in the top row and also in the first column so that those numbers are the factors of the given clues. Okay, maybe it isn’t quite that simple. You have to know basic multiplication facts and use logic to figure out where the numbers go, and yes, I may try to trick you. But you and the people on your gift list have enough skills and persistence to find the one and only correct solution.

Now glancing at the puzzle above you may think you know all the answers, but…

This is what the solved puzzle looks like. Some of those factors may surprise you. That is why using logic is so important when solving the puzzles. (Once the factors are found, filling out the rest of the table is optional.)

angel factors found

Click 2013 Factor Holiday to download a copy of the puzzle booklet. Some of the puzzles in the booklet are a little easier than the one above because they are a lower level or they only use factors up to 10. Have a very Merry Christmas and a Happy New Year!

Related articles with other ideas for last-minute gifts:

17 Christmas Angels

17 is a prime number. 17 = 1 x 17. Its only factors are 1 and 17. Prime factorization: none.

How do we know that 17 is a prime number? The square root of 17 is an irrational number approximately equal to 4.12. If 17 were not a prime number, then it would be divisible by at least one prime number less than or equal to 4.12. Since 17 is not divisible by 2 or 3, it is a prime number.

17 is never a clue in the FIND THE FACTORS puzzles.

Many Christmas trees in the United States have been up and decorated for weeks. Some of them have a beautiful angel on the top to remind us of the angel that visited the shepherds. In Hungary, the angel is remembered in a different way. There the Christmas tree is put up on Christmas Eve. Tradition says that angels are the ones who decorate the tree with the delicious candies called szaloncukor. The candies are wrapped in specially prepared white tissue and fastened to the tree with white yarn. See the related articles at the end of the post for more information about this fascinating tradition.

The angel puzzles that I’ve made for this post have a few extra clues so they will be easier to solve. The first level 5 puzzle even has many of the same clues as the level 4 puzzle. Nevertheless, be careful because each level 5 angel has a few tricks up her sleeve. Still if you can write the numbers 1 to 12 in both the top row and the first column so that those numbers are the factors of the given clues, then you’ve solved the puzzle. There is only one solution to each puzzle. Click 12 Factors 2013-12-19 for a printable version of these and a few other puzzles.

2013-12-19.42013-12-19.5

2013-12-19.72013-12-19.9

Hungary:

United States:

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