25 and Level 3

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25 is a composite number, and it is 5 squared. 25 = 1 x 25 or 5 x 5. Factors of 25: 1, 5, 25. Prime factorization: 25 = 5 x 5, which can also be written 25 = 5².

Since √25 = 5, a whole number, 25 is a perfect square. 

When 25 is a clue in the FIND THE FACTORS puzzles, write 5 in both the corresponding factor row and the factor column.

2014-02 Level 3

Write the numbers 1-12 in the top row and again in the first column so that those numbers are the factors of the given clues. Use logic, not guessing and checking, to find its unique solution. Level 3 puzzles are designed to be solved starting from a row at the top of the puzzle with 2 clues. First find the common factor of those two clues that will allow you to write only numbers from 1 to 12 in the factor row at the top of the puzzle. Then factor row by row to the bottom of the puzzle making sure each number from 1 to 12 is written only once in both the factor row and the factor column. You will notice a rhythm for the answers as you work. Good luck!

This week’s puzzles are also available in an excel file here if you have a spreadsheet program on your computer. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here is the solution to last week’s level 3 puzzle:

2014-01 Level 3 Answer

24 Think You Have a Snow Problem? Try This:

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Snow Related Puzzles:

It’s January, and some of us are a little tired of the snow while others are longing to hit the ski slopes. I’m writing today to suggest a few ways to deal with snow when it seems like more of an obstacle than an opportunity.

I walked into a second-grade classroom last week and saw books and books about snow, snowflakes, snowmen, etc. Celebrating winter is one way to deal with the snow. Here are a few snow problems for you to solve while you review multiplication facts: I’ve created a nightmare of a snowstorm with huge snowflakes and snowballs and only an itsy bitsy snow shovel to deal with it all. Be grateful this isn’t what you are really facing! How do you solve one of these snow-related puzzles? Write the numbers 1 – 12 in the top row and again in the first column so that those numbers are the factors of the given clues. Each puzzle has only one solution.  Click here for some tips to help you solve the puzzles. Click 12 Factors 2014-01-13 for printable versions of these and a couple other puzzles as well as last week’s answers.

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Tools to Deal with Actual Snow:

Puzzles are fun, but they can’t help you get the car out of the garage after a snowstorm.

If you wake up to find just under a foot of snow (or even much more) on your driveway (or your roof), a good snow shovel can still be the answer. I recommend a shovel with a crooked handle. It makes lifting snow so much easier. This excellent article has everything you need to know to shovel snow.

I have not used cooking oil on my snow shovels, but I have sprayed it with non-stick cooking spray, and that did help when the snow was sticky. (Snow isn’t always sticky.)

Two places I have lived face winter storms on occasion. Those places were Oklahoma County, Oklahoma and Salt Lake County, Utah.

It didn’t snow very often when we lived near Oklahoma City, but when it did, a crooked-handled shovel allowed me to clear my driveway faster than any of my neighbors who had only a straight-handled shovel. Freezing rain, thunderstorms, and tornadoes were a much bigger concern in Oklahoma than snow.

Compared to Oklahoma, we get much more snow near Salt Lake City where we have lived for the last 20 years.   Now my family has two of these crooked handled shovels. For seventeen years we relied solely on those shovels. We found that shoveling was good exercise, but it also took too much time, and time seems to be getting in shorter supply.

Three years ago we bought a good quality snow blower, and it is definitely a time saver. If you buy a snow blower, make sure it is a model that allows the operator to change the direction where the snow is blown. Also if the snowblower gets clogged with snow, always turn it off before attempting to unclog it. We have a friend who lost a finger because he didn’t turn it off first.

We also know people who own tractor snowplows. They easily clear their driveways and sometimes the sidewalks for their entire block. There is a snowplow on the Sears website. The blade on it is currently for sale for just under $250.00. Ouch.

We dream of someday getting an automatic snowmelt system that will turn itself on and melt any snow that falls on our driveway. One such system is described here.

Yes, there are many places that regularly get pounded with much more snow than Salt Lake City does. I’m not sure how helpful any of the above information will be for people who live in such places, so for anyone suffering in those locations I offer a few diversions:

A Few Diversions from Dealing with the Snow:

My daughter-in-law endured the polar vortex that hit the Eastern half of the United States last week. She wrote, “Public service announcement: 37+ weeks pregnant with a head cold and two small sick kids is no fun. I do not recommend it. Today I am grateful for PBS kid shows, giant boxes of tissues, soft blankets, and easy slow cooker meals.” She also survived by finding humor in the situation. She shared a hilarious Sunny Street cartoon panel featuring a very cold pregnant woman whose water broke and turned immediately into ice cubes.

If you like making greeting cards, you could make a snowflake card using an early iteration of Koch’s snowflake. If you completed a few more iterations, you would discover that this snowflake is a lovely six-sided shape that happens to have a finite area, but an infinite perimeter. Here are the directions: http://renegadegeek.wordpress.com/2013/12/31/koch-snowflake-card/

If none of these tips or diversions help you solve or forget your snow problem, I have one more thing for you to try as a last resort:

I am absolutely amazed at how much more snow the Northern Hemisphere gets compared to the Southern Hemisphere during their respective winters. So as a last resort, maybe it is time to move.

I like this animation showing where it snows on earth each month of the year.

If you stay where you are, you may take comfort in knowing there are places that have much more snow. Check out the Top 12 Biggest Snowfall Events in Recorded History!

Twitter informs us that Alaska can deal with a lot of snow with some unusual snowplows:

Factors of 24:

  1. 24 is a composite number.
  2. Prime factorization: 24 = 2 x 2 x 2 x 3, which can be written 24 = 2³ x 3
  3. The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 24 has exactly 8        factors.
  4. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  5. Factor pairs: 24 = 1 x 24, 2 x 12, 3 x 8, or 4 x 6
  6. Taking the factor pair with the largest square number factor, we get √24 = (√4)(√6) = 2√6 ≈ 4.898979.

Sum-Difference Puzzle:

6 has two factor pairs. One of those factor pairs adds up to 5, and the another one subtracts to 5. Those factor pairs are what you need to solve the first puzzle.

24 has four factor pairs. One of those factor pairs adds up to 10, and another one subtracts to 10. If you know what those factor pairs are, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More About the Number 24:

There are 4 counting numbers less than √24. Did you notice that all 4 of those numbers less than √24 are in 24’s factor pairs? OEIS.org informs us that 24 is the LARGEST number that can make that claim.

Another fact: 4! = 1 × 2 × 3 × 4 = 24.

But that’s not all! 24 is a repdigit palindrome in 4 bases!

  • 44 BASE 5 because 4(5) + 4(1) = 24.
  • 33 BASE 7 because 3(7) + 3(1) = 24.
  • 22 BASE 11 because 2(11) + 2(1) = 24.
  • 11 BASE 23 because 1(23) + 1(1) = 24.

24 is also the sum of consecutive prime numbers: 11 + 13 = 24.

When 24 is a clue in the FIND THE FACTORS 1-10 Puzzles, the factors will be either 3 × 8 or 4 × 6. When 24 is a clue in the FIND THE FACTORS 1 -12 puzzles, the factors could be 2 × 12, 3 × 8, or 4 × 6. In each case, only one set of factors will be used for each clue in any particular puzzle.

23 You Call That a Multiplication Table? THIS is a Multiplication Table

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A Discussion About the Multiplication Table:

Almost everyone is familiar with the traditional multiplication table:

Is this traditional way the best way for children to learn multiplication facts? Maybe not!

Beyondtraditionalmath published a post showing the Griddle’s creation of a different kind of multiplication table. This table can really help anyone visualize what multiplication means by relating any multiplication fact to its area. Every child learning to multiply should be exposed to this wonderful table. Even students who already know how to multiply would benefit from looking at it.

On the other hand, I hope it doesn’t completely replace the traditional multiplication table because that old, familiar table is the basis for my Find the Factor puzzles.

Factors of the Number 23:

  • 23 is a prime number.
  • Prime factorization: 23 is prime.
  • The exponent of prime number 23 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 23 has exactly 2 factors.
  • Factors of 23: 1, 23
  • Factor pairs: 23 = 1 x 23
  • 23 has no square factors that allow its square root to be simplified. √23 ≈ 4.79583

How do we know that 23 is a prime number? If 23 were not a prime number, then it would be divisible by at least one prime number less than or equal to √23 ≈ 4.8. Since 23 cannot be divided evenly by 2 or 3, we know that 23 is a prime number.

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

More About the Number 23:

23 is the sum of three consecutive prime numbers:
5 + 7 + 11 = 23.

22 Factor Trees

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22 is a composite number. 22 = 1 x 22 or 2 x 11. Factors of 22: 1, 2, 11, 22. Prime factorization: 22 = 2 x 11.

When 22 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 2 and 11 as the factors.

‘Tis the season to use factor trees to find all the factors of a number. Here is a factor tree that shows all the prime factors of 560. Next to it is a factor tree for 324. Do you see all of their prime factors clearly?

560 green                           324 green

Because sometimes one can’t “see the factors for the trees,” I recommend circling the prime factors or doing something else to make them more distinct.  Here are the trees again with every prime factor clearly visible:

560 red                                    324 red

I make logic puzzles based on the multiplication table. The puzzles for today’s post could also be called factor trees because you have to factor the clues to find the solution, and the puzzles are shaped like evergreen trees in whole or in part. I’ve even included a tree hanging from the ceiling, of all places! It may seem like a modern novelty, but people have been hanging Christmas trees upside down in Eastern Europe for centuries. To solve the puzzles either cut and paste the puzzles into a document to print or click 12 Factors 2013-12-12.

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21 Factors of the Year 2013 and 2014

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21 is a composite number. 21 = 1 x 21 or 3 x 7. Factors of 21: 1, 3, 7, 21. Prime factorization: 21 = 3 x 7.

When 21 is a clue in the FIND THE FACTORS puzzles, use 3 x 7.

Scroll down the page to find factoring information about 2013 and 2014.

2013 year

Near the end of each year movie critics make lists of the ten best movies and the ten worse movies of the year. News agencies list the ten most significant news stories. Time magazine lists the ten most influential people of the year. The music industry lists the top ten songs of the year. As 2013 draws to a close, it is most appropriate for me to review the factors of the year.

2013 had exactly 8 positive factors. These factors were 1, 3, 11, 33, 61, 183, 671, and 2013.

There is no room for argument. I am absolutely certain this list is complete. No one will make any comments disagreeing with me, calling me names, or asking how I could have left Two or Five or Seven off the list. Also no one will wonder why I would include forgettable 671 on the list. Do the Math. 671 was clearly a factor in 2013. Three of the factors of 2013 were also prime factors. They were 3, 11, and 61. This graphic clearly shows those prime factors.

2013 tree

2013 also had 8 negative factors. The first negative factor on the list is no surprise: Minus One. Year in and year out we can count on Minus One being a negative factor. Some other factors were just as negative in 2013, namely -3, -11, -33, – 61, -183, -671, and -2013. Of course, many of those factors were so obscure that most people never gave them a second thought all year long. Again I expect no arguments or negative comments on these selections. Anyone who knows anything about factors will have to agree with this list.

Even though 2014 hasn’t even started, I am going to predict the factors of 2014, and I am absolutely positive that my predictions will be 100% correct. You will not even have to wait until the end of 2014 to verify my accuracy.

The positive factors of 2014 will be (drum roll) 1, 2, 19, 38, 53, 106,1007, and 2014.

Most people expect the number One to be a positive factor every single year, and it will not let us down in 2014. The number Two has a reputation of being a factor only about half the time. Since she was not a factor at all in 2013, I am confident that she will get her act together again in 2014 and become a factor once more. All the other factors I’ve listed have not been factors for a very long time, and each one of them is due to make a difference over and over again in 2014 until they have nothing leftover. I predict that 2014 will have three prime factors, namely 2, 19, and 53, as illustrated in the following graphic.

2014 tree

How can I make such accurate assessments and spot on predictions? I will tell you: I work with factors almost every single day, and I’ve spent years observing them. Every time I have been given an assignment to become acquainted with them, I have approached that assignment with enthusiasm and determination.

Regardless of my astounding record, YOU can become just as much an expert as I am with just a little bit of knowledge and effort. You may discover, as I have, that factoring can be great fun. Here are a couple of logic puzzles that require factoring to solve: 

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All you have to do to solve one of the puzzles is write the numbers 1 – 12 in the top row and again in the first column so that those numbers are the factors of the given clues. Each puzzle has only one solution.

At the top of this post is a page titled How to Find the Factors, and it gives hints to solve the puzzles.   Click 12 Factors 2013-12-30 to find a printable version of these and a few other puzzles as well as the solutions for last week’s puzzles. Excel or comparable spreadsheet program is needed to open the file.

Have a great 2014 and happy factoring!

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20 Underestimation

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20 is a composite number. 20 = 1 x 20, 2 x 10, or 4 x 5. Factors of 20: 1, 2, 4, 5, 10, 20. Prime factorization: 20 = 2 x 2 x 5, which can also be written 20 = 2² x 5.

When 20 is a clue in the FIND THE FACTORS puzzles, use either 4 x 5 or 2 x 10. Only one set of those factors will work for any particular puzzle.

Estimating is a very important skill in mathematics, and I have always done well on any homework or test question involving that skill. However, I have never been very good at estimating real life – like how much time it will take me to do something.

Having never blogged before writing this one, I seriously underestimated how much time each puzzle would take me to create and test, and how much time each post would take me to write. Even though I have a large number of level 4, 5, and 6 puzzles already created, I still like to test them all again before I publish them, and that takes time.

Like most people, I work to make a living, and I have only a limited amount of free time each day. There are also other things that I ought to do or want to do that I haven’t given sufficient time in the last two months.

Because I underestimated my time and overestimated other things, I’ve decided to downsize and only publish posts on Mondays. Therefore there will not be any new puzzles today. Here is the easiest puzzle and the hardest puzzle from last week:

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To solve them, place the numbers 1- 12 in the top row and again in the first column so that those numbers are the factors of the given clues. Click 12 Factors 2013-12-26 to see the solutions for all of the puzzles from last Thursday. Excel or similar spreadsheet program is needed to open the file. This Monday I will publish more 12 Factor puzzles followed by 10 factor puzzles the following week.

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There are much worse things to underestimate:

19 Last-Minute Gift

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Today’s Puzzle and a Last-minute Gift Idea:

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 a  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!

Factors of the Number 19:

  • 19 is a prime number.
  • Prime factorization: 19 is prime.
  • The exponent of prime number 19 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 19 has exactly 2 factors.
  • Factors of 19: 1, 19
  • Factor pairs: 19 = 1 x 19
  • 19 has no square factors that allow its square root to be simplified. √19 ≈ 4.3588989

How do we know that 19 is a prime number? If 19 were not a prime number, then it would be divisible by at least one prime number less than or equal to √19 ≈ 4.4. Since 19 cannot be divided evenly by 2 or 3, we know that 19 is a prime number.

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

More About the Number 19:

19 is the fourth centered triangular number. There are 19 squares in the graphic below:

Why? Because 1 + 3(1) + 3(2) + 3(3) = 19.

19 is also the third centered hexagonal number.
Why? Because 1 + 6(1) + 6(2) = 19.
Imagine lines forming concentric hexagons in the drawing from the tweet below:

 

Related articles about puzzles and/or the brain:

Related articles with other ideas for last-minute gifts:

18 Up on the Housetop

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18 is a composite number. 18 = 1 x 18, 2 x 9, or 3 x 6. Factors of 18: 1, 2, 3, 6, 9, 18. Prime factorization: 18 = 2 x 3 x 3, which can also be written 18 = 2 x 3².

When 18 is a clue in the FIND THE FACTORS puzzles, use either 2 x 9 or 3 x 6. Only one set of factors will work for any particular puzzle.

“Up on the Housetop” is a famous children’s Christmas carol.  Imagine Santa stopping on each housetop to go down its chimney.

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Up on the housetop – to solve these puzzles will require you to have knowledge of the multiplication facts and just a little imagination to see the housetops (and maybe the ability to tilt your head for most of them.) Place the numbers 1 – 10 in both the top row and the first column so that those numbers are the factors of the given clues. Click 10 Factors 2013-12-23 to print these puzzles or see last week’s solutions. In order to view the file, you need excel or other comparable spreadsheet program.

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17 Christmas Angels

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  • 17 is a prime number.
  • Prime factorization: 17 is prime.
  • The exponent of prime number 17 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 17 has exactly 2 factors.
  • Factors of 17: 1, 17
  • Factor pairs: 17 = 1 x 17
  • 17 has no square factors that allow its square root to be simplified. √17 ≈ 4.123.

How do we know that 17 is a prime number? If 17 were not a prime number, then it would be divisible by at least one prime number less than or equal to √17 ≈ 4.1. Since 17 cannot be divided evenly by 2 or 3, we know that 17 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.

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Hungary:

United States:

16 Silver Bells

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16 is a composite number, and it is 4 squared. 16 = 1 x 16, 2 x 8, or 4 x 4. Factors of 16: 1, 2, 4, 8, 16. Prime factorization: 16 = 2 x 2 x 2 x 2, which can also be written 16 = 2⁴.

Since √16 = 4, a whole number, 16 is a perfect square.

When 16 is a clue in the FIND THE FACTORS puzzles, use either 2 x 8 or 4 x 4. Only one of those sets of factors will work for any particular puzzle.

“Silver bells, silver bells.
It’s Christmas time in the city.”

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Find the Factors is a type of logic puzzle. To solve one of the above puzzles, place the numbers 1 – 10 in both the top row and in the first column so that those numbers are factors of the given clues. For each puzzle, there is only one solution. Click on 10 Factors 2013-12-16 to find these and a few more puzzles, as well as last Monday’s solutions.

Some of these Related articles have the lyrics or soundtrack to Silver Bells: