Just Like Sudoku?

On numerous occasions when I have demonstrated how to solve a Find the Factors puzzle, someone will remark that the puzzle is just like Sudoku. What are common factors that both a Find the Factors puzzle and a Sudoku puzzle will have? 1) Both will have only one solution. 2) Both require the solver to be able to count, write, and place the numerals 1 to 9, but Find the Factors also requires the number 10 be placed. 3) Both were originally designed to require logic to be solved. 4) Both puzzles utilize a square grid. 5) Both puzzles have several difficulty levels and variations that make the puzzles more challenging.

What factors do the two puzzles NOT have in common? 1) A difficult Sudoku puzzle can take some people almost an hour to solve while Find the Factors  would never take that long. 2) Sudoku has been a wildly popular puzzle while Find the Factors is known only among a small circle of people who have had some kind of contact with me. 3) Some more recent Sudoku puzzles require the solvers to guess and check which is getting away from its logic puzzle roots and is making it less popular for some people.  4) Sudoku could just as easily be made with letters of an alphabet, colors, or the names of the planets (if you include Pluto), while Find the Factors has to be made with numbers. 5) Sudoku requires only counting, while Find the Factors also requires the solver to factor and multiply. So really, if Find the Factors were just like Sudoku, it would look like this:

Requiring skip counting to solve the Skipoku puzzle does make it more challenging, but I became annoyed with the skip counting by the time I finished the puzzle.

One complaint about some advanced Sudoku puzzles is the need to guess and check to find a solution. Is it necessary to guess and check all the possibilities to solve this level SIX Find the Factors puzzle?

Click 10 Factors 2013-11-25 for more puzzles.

No. Guessing and checking is not necessary even though the clues 12 and 24 have several common factors.  We easily eliminate 1 and 2 because both of them would require a partner greater than 10. We also eliminate 12 because it is greater than 10. What about 3, 4, and 6? Do we have to try each of those possibilities? When we examine the puzzle we notice that it has 10 clues with only two of the clues paired together. We also notice there is one column that contains no clue, so supposedly any factor could fit there. Here is a chart of all the possible factors and the clues each could satisfy.

Remember that each factor must be written twice, once in the factor row and one in the factor column. Notice that the number 9 is a factor of only one of the clues. That means that 9 has to be put over the column with no clues. From there it is easy to know where the other 9 goes and both 8's and so forth until it is completed. Not all level SIX puzzles can be completed that easily, but using logic instead of guessing and checking is the key to solving these puzzles.

Related articles

• Playing Sudoku On Mobile Phones (tekan6games12.wordpress.com)
• The Technology Behind the Sudoku Questions (sarahwilson256mnaruaez517.wordpress.com)
• Sudoku Game (mbalac2.wordpress.com)
• http://voices.yahoo.com/sudoku-math-they-related-51807.html?cat=2
• Taking Sudoku Seriously - The Math Behind the World's Most Popular Pencil Puzzle (topoldmovies.wordpress.com)
• A new Epoch in Sudoku (mbalac2.wordpress.com)
• A Gamification Scenario (thoughtsandme.wordpress.com)

The Doorbell Rang

The Doorbell Rang by Pat Hutchins is about cookies and sharing. It takes less than five minutes for an adult to read every delightful word aloud to a child.  It is also a good book for beginning readers because it is filled with reliable repetition, and it is also sprinkled with a few interesting multi-syllabic words. Some words that do NOT appear in the text are mathematics, multiplication, division, or factoring. Still the book very cleverly helps children recognize all the factors of 12. Chiix Moses wrote in a review, "Something I firmly believe is that learning is best when it doesn't feel like learning, and that is precisely what this books accomplishes." This book almost effortlessly teaches students to think win-win, so it is also an excellent choice for reinforcing the Seven Habits. The puzzle below will requires knowledge of the factors of 12 as well as thirteen other numbers. It is a level five puzzle, meant to be completed by adults or very bright children.   Click 12 Factors 2013-11-21 for more puzzles. Related articles Using Children's Books to Enhance Mathematics (pulaskischoolslearningservices.wordpress.com)

Spaghetti and Meatballs for All!

Spaghetti and Meatballs for All! by Marilyn Burns is a delightful story, the kind that children enjoy hearing over and over again.

http://www.youtube.com/watch?v=w4p7X3nUAes

I work at a Leader in Me school, where we promote the Seven Habits. I used this book when I taught about habit 4, think win-win. When we think win-win, we do not allow someone to “step on us’ to give them a win. Mrs. Comfort’s relatives stepped on her over and over again, and they didn’t even realize it. Finally she cried, “I give up!” and planted herself on a chair. She definitely felt like she was losing. The class listened to the story intently trying to identify places where the Seven Habits were used or could have been used. We had a great discussion afterwards. Also since the book did not use the words, “area” or “perimeter” at all, the class hardly realized that the story was also about those concepts. When we followed the suggestions at the back of the book, the class was able to learn about perimeter and area as we had a great discussion about those topics as well. 

Click 10 Factors 2013-11-18 for more puzzles

Related articles

• Read ” Spaghetti And Meatballs For All “, Another Excellent Children’s Book Author Marilyn Burns (kidbooksreview.wordpress.com)
• Spaghetti and Meatballs for All! (ngainer.wordpress.com)

A Piece of Cake

Happy birthday, Kathy! I hope your day is wonderful. You have grown into a beautiful, talented, prayerful, intelligent, hard-working, and loving young woman.  I am grateful you are my daughter.  So for your birthday today and for this blog, I’ve created three special puzzles: the first is a birthday cake to celebrate your happy day. To highlight your love of music, the second puzzle is a quarter note. The third puzzle is either a violin, a guitar, or a ukulele, you decide. I love listening as you sing or as you play any of those instruments or the piano. Today for your birthday I will also cut down a tree and make yet another cake with two birthday candles on top in this blog post.  So have a fun birthday, today.  I love you.

Click 12 Factors 2013-11-14 for more puzzles.

What did I mean by cutting down a tree and making yet another cake?  Today I will discuss two methods for finding the prime factors of a whole number.  One method is making a factor tree and the other is the cake method.  To factor a number means to write it as a product of two or more factors. When those two or more factors are all prime factors, it is called the prime factorization of the number. A composite number always has more than two factors. A prime number always has exactly 2 factors, 1 and itself. (ZERO and ONE are neither prime or composite numbers.) Usually to find the prime factors of a number, a person will usually make a factor tree.   The following example shows how this is done:

 

From this example, you can certainly understand why this algorithm is called a factor tree.  It looks exactly like a perfectly-shaped evergreen tree.  The problem is that a factor tree doesn’t always look so neat and trim.  Here is a factor tree that even Charlie Brown wouldn’t choose:

720 isn’t even that big of a number, but gathering all the prime numbers from it and putting them in numerical order would be like picking up a bunch of scattered leaves or doing yard work.  Imagine if you had a number that had many more factors. If one or two of the factors gets lost in the mess, your answer wouldn’t be correct. Chop down that factor tree. Even if you like to do yard work, do you really want to deal with that big of a mess, …. especially when you can have cake instead?  Look, the cake method is so much more pleasing to the eye, and it is simply an extension of the very familiar division algorithm:

With the cake method, the more factors you have, the bigger the cake will be, but it will always be neatly organized with all the factors on the outside of the cake.  And if the largest prime factor of your given number is eleven, you will also have two candles on top of your cake!  I find using the cake method to be much less confusing than using a factor tree.  Yes, finding prime factors can actually be a piece of cake. The only disadvantage to the cake method is that since you work from the bottom up you have to leave enough space for the cake to rise.

In spite of my opinion, it is best to use whichever method you are more comfortable with.

Now if your appetite for cake has not been satisfied, click on the amazing Spider-Man Cake or one of the other links.  Enjoy!

Related articles about cake:

• Splendid Spider-Man Cake (betweenthepagesblog.typepad.com)
• Another Surprise Rainbow Cake (seraphinacakes.wordpress.com)
• The Case for Cake (benmo45.wordpress.com)
• “Birthday cake” (smilingmushrooms.wordpress.com)
• In Search of The Best Chocolate Cake (lavaletteerica.wordpress.com)

Easy as 1-2-3

Being able to identify factors of a whole number is a very important skill in mathematics.

Click 10 Factors 2013-11-11 for more puzzles.

It is a skill that is commonly used in many areas of mathematics ranging from reducing fractions to solving differential equations.  The Find the Factors puzzles can help make that skill second nature, but to solve the puzzles, we are only interested in the limited set of factors that are represented in the following table. (Click on the table to enlarge it.)

What about all the other factors of these numbers?  And what about all the other whole numbers not on the chart?  How do you find ALL of the factors of a given whole number?  For example, suppose you were asked to find all of the factors of 435.  Some people might notice right away that it is divisible by 5 because its last digit is 5.  While that is true, beginning with 5 is not the best place to start because there is an advantage in considering all possible factors in an organized way.  When you are asked to find ALL of the factors of any number, starting at 1 will make finding all of the factors as easy as 1-2-3.

So what are the factors of 435?  Using a calculator, I notice that the square root of 435 is about 20.85.  That means I can find absolutely all of the factors of 435 by considering as divisors just the whole numbers from 1 to 20!  Each factor will have a partner that is greater than 20 but will be found at the same time with these few short calculations. To demonstrate my thinking process, I will put each possible factor from 1 to 20 in a chart and write my thoughts as I consider each one.

As you may notice, once a possible factor is eliminated, it is not necessary to do any actual division by ANY of the multiples of that number. (4, 6, 8, 10, 12, 14, 16, 18, and 20 are all multiples of 2, which was not a factor, so I didn't actually divide 435 by any of those multiples.)

As I carefully consider each possible factor, I only WRITE DOWN a number if it is an actual factor.  Therefore, with only a little bit of effort I would list ALL of the factors of 435 in one tidy list: 1 x 435, 3 x 145, 5 x 87, 15 x 29.

See, it was as easy as 1-2-3!  Now let’s find all of the factors of 144.

Even though 144 is less than 435, it has more factors. One of its factors is paired with itself because the square root of 144  is 12.  That fact is also the signal that we can stop looking for more factors, and we can list all the factors of 144 on the following chart:

  There are 8 multiplication facts that produce 144, but 12 x 12 = 144 is the only fact we consider when solving a Find the Factors 1-12 puzzle with 144 as one of the clues. In every other case one of the pair of numbers in the multiplication fact will be greater than 12 and not eligible to be written in the factor row or factor column. However in solving mathematical problems, any of the factors of a whole number could be the star of the show. Knowing how to find those factors is indeed an important skill and is as easy as 1-2-3.

Related articles

• Divide and conquer (rookiesreminders.wordpress.com)

Rhyme and Rhythm

Click on this link, 12 Factors 2013-11-07,  for more puzzles. Seventeen years ago, when my daughter was learning the multiplication facts, I came across a rhyme that taught one fact: 5, 6, 7, 8……….56 is 7 times 8. Coincidentally, 1, 2, 3, 4……….I know twelve is 3 times 4. 3 x 4 = 12 isn’t as difficult to remember as 7 x 8 = 56. Still I find it fun to notice the relationship between counting to eight and these two multiplication facts. I enjoyed the rhymes mentioned in my 10/31/2013 post, and I have found yet another site with rhymes for learning the multiplication facts. Two rhymes similar to these 12 and 56 counting rhymes were even included!  The site is : http://www.teacherweb.com/NY/Quogue/MrsLevy/MULTIPLICATION-RHYMES.pdf Some children have no problem memorizing number facts, but for some children, a rhyme makes learning the facts more fun and much easier.  Even though I already know all of the basic multiplication facts, I am going to memorize these rhymes simply because I personally enjoy them.  I also know I will have at least one opportunity every week to share them with someone trying to memorize the facts: Already when a student asks me, "what's 7 x 8?" I always answer in rhyme. Besides the two rhymes listed above, my favorites are: Times One: Mirror, mirror look and see, it's the other number, not me. Times Zero:  Zero is always the hero Six times six / Magic tricks / Abracadabra / thirty-six A tree on skates fell on the floor / Three times eight is twenty-four. A 4 by 4 is a big machine, Iʼm going to get one when Iʼm 16. This week I even wrote one myself:  Twelve times twelve / Is a dozen dozen / A gross one forty-four / Just ask my cousin. Robin Liner writes a blog (crazygoodreaders.wordpress.com) that discusses reading and dyslexia.  On October 5, 2013 , she wrote Rhythm and Rhyme: A Phonological Power Tool.  She wrote, "Rhymes provide subconscious clues." That means someone is more likely to get an answer right when that answer rhymes with the question. What a fun and powerful way to learn! Much of what she wrote not only applies to learning to read but also to learning math, science, history, ....... anything. Related articles Rhythm and Rhyme: A Phonological Power Tool (crazygoodreaders.wordpress.com)

Uncluttered Post

Click on this link, Findthefactors WordPress, to get to the puzzles and last Monday’s answers.