A Multiplication Based Logic Puzzle

Posts tagged ‘Christmas’

987 Christmas Star

Today’s puzzle is a lovely Christmas star whose golden beams shine throughout the dark night. Solving this puzzle could also enlighten your mind.

987 is made from three consecutive numbers so it is divisible by 3.

It is also the sixteen number in the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, . . .

987 is a palindrome when written in base 11 or base 29:
818 in BASE 11 because 8(121) + 1(11) + 8(1) = 987
151 in BASE 29 because 1(29²) + 5(29) + 1(1) = 987

  • 987 is a composite number.
  • Prime factorization: 987 = 3 × 7 × 47
  • 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 987 has exactly 8 factors.
  • Factors of 987: 1, 3, 7, 21, 47, 141, 329, 987
  • Factor pairs: 987 = 1 × 987, 3 × 329, 7 × 141, or 21 × 47
  • 987 has no square factors that allow its square root to be simplified. √987 ≈ 31.416556

 

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985 Make an Origami Santa Star

Almost immediately when I saw Paula Beardell Krieg’s origami stars, I thought about turning one into a Santa Star. It occurred to me that the white pentagon formed on the back of the star would make a nice beard for Santa. I made a prototype and tweaked it and tweaked it until I got this result:

Why did I want to do this? My daughter-in-law, Michelle, adores Santa Stars. They are her favorite Christmas decoration. When I gave her this Santa Star, she got so excited. She recently took a picture of her collection, and I am thrilled that the one I made for her was included.

If you would like to make this Santa Star, follow these steps.

  1. Click on the pentagon above, then copy and paste it into a document. Make it as big as your printer allows.
  2. Print the pentagon and cut it out.
  3. Follow the directions in the video by Tobias that Paula recommends.

Here are some pictures I took as I folded mine. Click on them if you want to see them better. I’ve also included a few tips to help you in folding the star:

In this picture, you can see that the pentagon was folded in half five different ways in the first set of folds. The second set of folds creates a smaller pentagon in the center of the pentagon as well as a star-like shape.

The third set of folds creates a new crease. I make a flower-like shape by refolding that crease on each side. To me, this “flower” is a very important step to get the paper to form the star.

Those creases will help form the small white pentagon you see in the picture below that will become Santa’s beard.

Turn the paper over to reveal a bigger pentagon. You will fold the vertices of this pentagon to the center of the pentagon. Fold the red and black tips at the same time as you fold the vertices. After you make the first fold, I recommend unfolding it. Fold the other vertices in order so that first fold will eventually become your last fold. The last fold is the most difficult to do. If it has already been folded once, it will be much easier to fold at the end.

Again, here is the finished Santa star.

Now I’ll share some facts about the number 985:

985 is the sum of three consecutive prime numbers:
317 + 331 + 337 = 985

29² + 12² = 985 and 27² + 16² = 985

985 is the hypotenuse of FOUR Pythagorean triples
140-975-985
591-788-985
473-864-985
696-697-985

When is 985 a palindrome?
It’s 505 in BASE 14 because 5(14²) + 5(1) = 5(196 + 1) = 5(197) = 985
It’s 1H1 BASE in 24 (H is 17 base 10) because 1(24²) + 17(24) + 1(1) = 985

  • 985 is a composite number.
  • Prime factorization: 985 = 5 × 197
  • 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 985 has exactly 4 factors.
  • Factors of 985: 1, 5, 197, 985
  • Factor pairs: 985 = 1 × 985 or 5 × 197
  • 985 has no square factors that allow its square root to be simplified. √985 ≈ 31.3847

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

983 Candy Cane

Candy canes have been a part of the Christmas season for ages. Here’s a candy cane puzzle for you to try. It’s a level 6 so it won’t be easy, but you will taste its sweetness once you complete it. Go ahead and get started!

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here’s some information about prime number 983:

983 is the sum of consecutive prime numbers two different ways:
It is the sum of the seventeen prime numbers from 23 to 97.
It is also the sum of the thirteen prime numbers from 47 to 103.

  • 983 is a prime number.
  • Prime factorization: 983 is prime.
  • The exponent of prime number 983 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 983 has exactly 2 factors.
  • Factors of 983: 1, 983
  • Factor pairs: 983 = 1 × 983
  • 983 has no square factors that allow its square root to be simplified. √983 ≈ 31.35283

How do we know that 983 is a prime number? If 983 were not a prime number, then it would be divisible by at least one prime number less than or equal to √983 ≈ 31.4. Since 983 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 983 is a prime number.

980 Christmas Factor Trees

This level 4 puzzle has 12 clues in it that are products of factor pairs in which both factors are numbers from 1 to 12. The clues make an evergreen tree, the symbol of everlasting life which is so fitting for Christmas. Can you find the factors for the given clues and put them in the right places?

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now I’ll tell you a little about the number 980:

It has eighteen factors and many possible factor trees. Here are just three of them:

28² + 14² = 980, so 980 is the hypotenuse of a Pythagorean triple:
588-784-980 which is (3-4-5) times 196, but can also be calculated from
28² – 14², 2(28)(14), 28² + 14²

I like the way 980 looks in some other bases:
It is 5A5 in BASE 13 (A is 10 base 10) because 5(13) + 10(13) + 5(1) = 980,
500 in BASE 14 because 5(14²) = 980,
SS in BASE 34 (S is 28 base 10) because 28(34) + 28(1) = 28(35) = 980
S0 in BASE 35 because 28(35) = 980

  • 980 is a composite number.
  • Prime factorization: 980 = 2 × 2 × 5 × 7 × 7, which can be written 980 = 2² × 5 × 7²
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 980 has exactly 18 factors.
  • Factors of 980: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
  • Factor pairs: 980 = 1 × 980, 2 × 490, 4 × 245, 5 × 196, 7 × 140, 10 × 98, 14 × 70, 20 × 49 or 28 × 35
  • Taking the factor pair with the largest square number factor, we get √980 = (√196)(√5) = 14√5 ≈ 31.30495.

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

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

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