994 and Level 1

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All you need is these eleven clues and the multiplication facts in a normal 12 × 12 multiplication table to completely fill in every square of this abnormal multiplication table, I mean puzzle. Don’t worry about how fast you can solve the puzzle. The more puzzles you solve the better you will get at doing them. Relax and enjoy yourself!

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

When is the number 994 a palindrome?
It is 4334 in BASE 6 because 4(6³) + 3(6²) + 3(6¹) + 4(6⁰) = 994, and
it’s 464 in BASE 15 because 4(15²) + 6(15¹) + 4(15⁰) = 994

  • 994 is a composite number.
  • Prime factorization: 994 = 2 × 7 × 71
  • 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 994 has exactly 8 factors.
  • Factors of 994: 1, 2, 7, 14, 71, 142, 497, 994
  • Factor pairs: 994 = 1 × 994, 2 × 497, 7 × 142, or 14 × 71
  • 994 has no square factors that allow its square root to be simplified. √994 ≈ 31.52777

993 Christmas Angel

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There are 21 clues in this Christmas Angel puzzle. Will it be easy or difficult for you to solve? That is part of the mystery. As always, there is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

When is 993 a palindrome?
It is 5B5 in BASE 13 (B is 11 base 10) because 5(169) + 11(13) + 5(1) = 993,
313 in BASE 18 because 3(18²) + 1(18) + 3(1) = 993,
and it is repdigit 111 in BASE 31 because 31² + 31¹ + 31⁰ = 993

  • 993 is a composite number.
  • Prime factorization: 993 = 3 × 331
  • 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 993 has exactly 4 factors.
  • Factors of 993: 1, 3, 331, 993
  • Factor pairs: 993 = 1 × 993 or 3 × 331
  • 993 has no square factors that allow its square root to be simplified. √993 ≈ 31.5119

992 Christmas Factor Tree

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Artificial Christmas trees have to be assembled. Sometimes the assembly is easy, and sometimes it is frustrating.

This Christmas tree puzzle can be solved using LOGIC and an ordinary multiplication table, but there’s a good chance it will frustrate you. Go ahead and try to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-986-992

The number 992 also can make a nice looking, well-balanced factor tree:

992 is the product of two consecutive numbers: 31 × 32 = 992.
Because of that fact, 992 is the sum of the first 31 EVEN numbers:
2 + 4 + 6 + 8 + 10 + . . . + 54 + 56 + 58 + 60 + 62 = 992

992 is palindrome 212 in BASE 22 because 2(22²) + 1(22) + 2(1) = 922. That was a lot of 2’s and 1’s in that fun fact!

  • 992 is a composite number.
  • Prime factorization: 992 = 2 × 2 × 2 × 2 × 2 × 31, which can be written 992 = 2⁵ × 31
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 992 has exactly 12 factors.
  • Factors of 992: 1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992
  • Factor pairs: 992 = 1 × 992, 2 × 496, 4 × 248, 8 × 124, 16 × 62, or 31 × 32
  • Taking the factor pair with the largest square number factor, we get √992 = (√16)(√62) = 4√62 ≈ 31.49603

991 Carry Your Own Weather

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This snowflake puzzle isn’t for beginners, but making snowflakes goes very well with the idea of carrying your own weather.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I like when I can sneak a little bit of mathematics into a completely unrelated lesson. This lesson is about being proactive and is called “Carry Your Own Weather”.

Carry Your Own Weather (Be Proactive) discussion:

  1. If you could choose the weather, what kind of weather would you choose? (Sunny weather and snowy weather seem to be chosen the most.)
  2. If the weather outside was ALWAYS sunny, would you like that? Would that be a good thing? Would you appreciate the sunny days if every day was sunny? (Variety is good. Without rain and snow, how could food grow?)
  3. Does your mood depend on the weather?
  4. How do you feel when it’s sunny outside?
  5. How do you feel when it’s gloomy outside?
  6. The author of the 7 Habits of Highly Effective People, Stephen R. Covey, talked about carrying your own weather. What do you think that might mean? (Decide for yourself how you will feel. Don’t let the outside weather or other people make that decision for you.)
  7. Let’s listen as Stephen R. Covey talks about Carrying Your Own Weather:

7Habits – CARRY YOUR OWN WEATHER VIDEO.flv.

Have you ever made a snowflake before? Did you know that you can decide how the snowflake will look before you make a single cut? Choosing how the snowflake will look ahead of time is like deciding what kind of weather you will carry with you. You will not leave it up to chance. You will begin with the end in mind. You will decide ahead of time how your snowflake will look. In real life, a snowflake has 6 sides, but you can choose to make your snowflake have four sides, six sides or eight sides. Several layers of a napkin are much easier to cut than the same number of layers of regular paper so you will use white paper napkins to make your snowflakes. Afterwards, you can glue your snowflake onto a sheet of dark blue construction paper. Then you can use a white crayon to sign your name and decorate the dark blue construction paper around your snowflake.

First, you need to fold your napkin. Folding into eighths is the easiest. Just find the corner where all the folds in the napkin already meet and fold that corner again, thus making a 45° angle. Fold that corner in half again and you’ll get the 22.5° angle that you see at the bottom of the napkin in the picture below. The other napkin is folded into sixths and then into twelfths. If you don’t cut off the uneven edges at the top of those napkins, your snowflake will only have four sides, but if you do cut off the uneven edges, you will get a six-sided or an eight-sided snowflake. (Depending on that cut, you might also get a twelve-sided or sixteen-sided snowflake. They’ll look great, too!)

Making a perfect six-sided snowflake is a little more difficult than an eight-sided one. I recommend reading this post from Paula Beardell Krieg for complete instructions on six-sided snowflake cutting: ‘Tis the Season to Make Paper Snowflakes (She is the one who told me about using easy-to-cut paper napkins for the snowflakes, too.)

I found these three triangles helpful in making snowflakes with perfect 60° angles.

 

Place the center of the folded napkin at the bottom center of triangles. You can easily see through the napkin to see where the folds need to go.

When each side of the napkin has been folded up, it will look have a 60° angle at the bottom. The red line shows where to cut the top off the napkin to get a perfect hexagon folded into an equilateral triangle.

Fold the napkin in half again so that bottom angle becomes a 30° angle before making your decorative cuts. These next instructions tell how to make those cuts to get the exact snowflakes that you want. These tips were made for six-sided snowflakes, but you can also apply the tips to eight-sided snowflakes:

Paper Snowflake Cutting Tips

Please be aware that these snowflakes are delicate. They can rip easily. I suggest you mount them on sturdy paper as soon as possible.

After you finish making your snowflakes, I’m sure you will enjoy a story that I like very much, even though you have heard it many times before. The main character felt sorry for himself because he was bullied and nobody thought he was good at anything. When he decided to find a way to be helpful, he started to carry his own weather. He became proactive. Doing so not only lifted him but lifted everyone around him, too. Can you guess the name of the story? (Rudolph the Red-Nosed Reindeer)

Rudolph the Red-Nosed Reindeer story and song from youtube

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This is my 991st post, so I’ll tell you a little bit about the number 991:

991 is a prime number that can be written as the sum of consecutive prime numbers two different ways:
127 + 131 + 137 + 139 + 149 + 151 + 157 = 991; that’s seven consecutive primes.
191 + 193 + 197 + 199 + 211 = 991; that’s five consecutive primes.

991 is a palindrome two different ways:
33133 in BASE 4 because 3(4⁴) + 3(4³) + 1(4²) + 3(4¹) + 3(4⁰) = 991
131 in BASE 30 because 1(30²) + 3(30) + 1(1) = 991

  • 991 is a prime number.
  • Prime factorization: 991 is prime.
  • The exponent of prime number 991 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 991 has exactly 2 factors.
  • Factors of 991: 1, 991
  • Factor pairs: 991 = 1 × 991
  • 991 has no square factors that allow its square root to be simplified. √991 ≈ 31.48015

How do we know that 991 is a prime number? If 991 were not a prime number, then it would be divisible by at least one prime number less than or equal to √991 ≈ 31.5. Since 991 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 991 is a prime number.

 

990 Christmas Factor Trees

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Today’s Puzzle:

This puzzle has a couple of small Christmas trees in it. Don’t let their smallness fool you into thinking this is an easy puzzle. Can you solve it? Remember to use logic and not guess and check to find the solution.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

Factors of 990:

  • 990 is a composite number.
  • Prime factorization: 990 = 2 × 3 × 3 × 5 × 11, which can be written 990 = 2 × 3² × 5 × 11
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 990 has exactly 24 factors.
  • Factors of 990: 1, 2, 3, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990
  • Factor pairs: 990 = 1 × 990, 2 × 495, 3 × 330, 5 × 198, 6 × 165, 9 × 110, 10 × 99, 11 × 90, 15 × 66, 18 × 55, 22 × 45, or 30 × 33
  • Taking the factor pair with the largest square number factor, we get √990 = (√9)(√110) = 3√110 ≈ 31.464265

Sum-Difference Puzzle:

990 has twelve factor pairs. One of the factor pairs adds up to 101, and a different one subtracts to 101. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 990:

There are many interesting facts about the number 990:

9 × 10 × 11 = 990

Because 44 × 45/2 = 990, it is the 44th triangular number. That means that the sum of all the numbers from 1 to 44 is 990.

990 is the sum of the twelve prime numbers from 59 to 107.
It is also the sum of six consecutive prime numbers:
151 + 157 + 163 + 167 + 173 + 179 = 990,
and the sum of two consecutive primes:
491 + 499 = 990

990 is the hypotenuse of a Pythagorean triple:
594-792-990 which is (3-4-5) times 198

990 looks interesting in some other bases:
It is 6A6 in BASE 12 (A is 10 base 10) because 6(144) + 10(12) + 6(1) = 990,
2E2 in BASE 19 (E is 14 base 10) because 2(19²) + 14(19) + 2(1) = 990
1K1 in BASE 23 (K is 20 base 10) because 1(23²) + 20(23) + 1(1) = 990
UU in BASE 32 (U is 30 base 10) because 30(32) + 30(1) = 30(33) = 990
U0 in BASE 33 because 30(33) = 990

989 Christmas Bells at Eventide

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I made this puzzle with silver clues to look like a bell. I thought dark blue looked best as a background color with the silver numbers. The puzzle reminds me of a bell in the evening. For the fun of it, I googled Christmas Bells in the evening to see if any poems or songs came up.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I was very surprised that Google found a very old song called Christmas Bells at Eventide. I did not know that such a song existed. Eventide means the same thing as evening. You can listen to the song below.

This is my 989th post.

The number 989 is a palindrome in base 10. What about any other bases?
It’s 373 in BASE 17 because 3(17²) + 7(17) + 3(1) = 989,
252 in BASE 21 because 2(21²) + 5(21) + 2(1) = 989, and
1C1 in BASE 26 (C is 12 base 10) because 1(26²) + 12(26) + 1(1) = 989

  • 989 is a composite number.
  • Prime factorization: 989 = 23 × 43
  • 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 989 has exactly 4 factors.
  • Factors of 989: 1, 23, 43, 989
  • Factor pairs: 989 = 1 × 989 or 23 × 43
  • 989 has no square factors that allow its square root to be simplified. √989 ≈ 31.44837

988 Christmas Factor Tree

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To solve a level 3 puzzle, find two clues in a single row or column. They will help you know what factors to put in the top cell of the first column and two other cells in either the first column or top row. You’ll then be able to work your way down the puzzle by finding the factors of each clue in turn in the most ideal order. The clues below a current clue can still affect the logic of that clue, as you should discover with this puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

Here’s a little about the number 988:

It is divisible by 4 because 88 is divisible by 4. Here are a couple of its factor trees that are nicely well-balanced:

988 is the sum of these four consecutive prime numbers:
239 + 241 + 251 + 257 = 988

988 is also the hypotenuse of a Pythagorean triple:
380-912-988 which is (5-12-13) times 76

  • 988 is a composite number.
  • Prime factorization: 988 = 2 × 2 × 13 × 19, which can be written 988 = 2² × 13 × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 988 has exactly 12 factors.
  • Factors of 988: 1, 2, 4, 13, 19, 26, 38, 52, 76, 247, 494, 988
  • Factor pairs: 988 = 1 × 988, 2 × 494, 4 × 247, 13 × 76, 19 × 52, or 26 × 38
  • Taking the factor pair with the largest square number factor, we get √988 = (√4)(√247) = 2√247 ≈ 31.432467

Now, what do you imagine is the total number of triangles of any size in the graphic below? You guessed it, 988.

987 Christmas Star

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

Print the puzzles or type the solution in this excel file: 10-factors-986-992

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

 

986 and Level 1

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Today’s puzzle looks like a simple holiday napkin, and it’s actually quite simple to solve. Even kids who have learned how to multiply but haven’t even heard of division could solve this puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

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

31² + 5² = 986
25² + 19² = 986

986 is the hypotenuse of FOUR Pythagorean triples:
264-950-986
310-936-986
680-714-986
464-870-986

986 looks interesting in some other bases:
12421 BASE 5
TT in BASE 33 (T is 29 base 10)
T0 in BASE 34

  • 986 is a composite number.
  • Prime factorization: 986 = 2 × 17 × 29
  • 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 986 has exactly 8 factors.
  • Factors of 986: 1, 2, 17, 29, 34, 58, 493, 986
  • Factor pairs: 986 = 1 × 986, 2 × 493, 17 × 58, or 29 × 34
  • 986 has no square factors that allow its square root to be simplified. √986 ≈ 31.4006

985 Make an Origami Santa Star

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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