A Multiplication Based Logic Puzzle

Archive for November, 2013

11 and Counting Blessings

11 is a prime number. 11 = 1 x 11. Its only factors are 1 and 11. Prime factorization: none.

How do we know that 11 is a prime number? The square root of 11 is an irrational number approximately equal to 3.32. If 11 were not a prime number, then it would be divisible by at least one prime number less than or equal to 3.32. Since 11 is not divisible by 2 or 3, it is a prime number.

Sometimes 11 is a clue in the FIND THE FACTORS 1 – 12 puzzles, and the factors are always 1 and 11.

I have more blessings than I could ever completely count. This is not the place where I will attempt to name them one by one, but I wonder: is the number of blessings that I or anyone else has finite or infinite? Even being able to ponder that question is a blessing. In the last few years in the United States, much of the gratitude part of Thanksgiving has gotten lost in commercialism. Therefore, for some people the number of blessings may be finite and easily measured by counting things. Some of those blessings may be more imaginary than real. Nevertheless, there are still people who can see the hand of God all around them. For them the number of blessings is infinite. Likewise those who rely on the Savior and His infinite atonement have an infinite number of blessings. As I count blessings, I find that some of them are prime, and some are a composite of several blessings working together. Some blessings are rather odd while others are shared evenly. I am grateful for many positive events in my life, but even negative experiences are blessings because they have helped me to grow.

The following blessings may seem trivial, but I am grateful that WordPress has given me a way to share the Find the Factors puzzles not only as jpg pictures, but also in an excel file.  The puzzles have been a blessing to me, and I want to show my gratitude by sharing them with other people. I am grateful for the blogs I follow. They challenge me, entertain me, and teach me so much. I am also thankful to everyone who has looked at my blog.

Click 12 Factors 2013-11-28 to see the same puzzles in excel.

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10 and Just Like Sudoku?

10 is a composite number. 10 = 1 x 10 or 2 x 5. Factors of 10: 1, 2, 5, 10. Prime factorization: 10 = 2 x 5.

When 10 is a clue in the FIND THE FACTORS puzzles, either 1 x 10 or else 2 x 5 will work for that particular puzzle.

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:

skipoku

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?

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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.

factors for 2013-11-25

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.

9 and 50 Years Ago

9 is a composite number, and it is 3 squared. 9 = 1 x 9 or 3 x 3. Factors of 9: 1, 3, 9. Prime factorization: 9 = 3 x 3.
Since √9 = 3, a whole number, 9 is a perfect square.
When 9 is a clue in the FIND THE FACTORS puzzles, the solution will use either 1 x 9 or 3 x 3, but not both.
50 years ago everything that I knew about politics I learned from reading this book:

The book belonged to my sister who was three years older than I was. Both of us delighted in reading about this little girl whose father was president. Here is one of the comics that I remember:

50 years ago today I was sitting in Mr. Stratton’s fifth grade class at Quannah McCall Elementary School in North Las Vegas, Nevada, when Mr. Stratton tearfully gave us the news that President Kennedy had been assassinated. Most of us didn’t know what that word meant, but we very soon learned its meaning. It was a very sad day that robbed us of our innocence and changed who we were.  A few years ago my husband gave me a tour of the school he attended when he also was in fifth grade at Elsinore Military Academy. He showed me the exact spot he was standing when he heard the news. This is a day burned into the memories of people from all over the world. We are all different, but we are united in our thoughts and feelings of that day.  I wish we would be more united in showing respect to the office of president. We do not have to agree with everything or even anything that a president says or does, but we can still be much more civil and respectful when we do disagree. President Kennedy was killed by someone who thought it was okay to be hateful and to act upon that hate. In this blog I usually write about a puzzle, but a much more important puzzle to be solved is how do we all get along with each other.

 

 

8 and Dimensions

8 is a composite number. 8 = 1 x 8 or 2 x 4. Factors of 8: 1, 2, 4, 8. Prime factorization: 8 = 2 x 2 x 2, which can also be written 8 = 2^3.

When 8 is a clue in the FIND THE FACTORS puzzles, use either 1 x 8 or 2 x 4.

My daughter-in-law, Julayne, is brilliant. She earned a master’s degree in mathematics and teaches at the college level. She is also an expert at solving the Hungarian puzzle known as a Rubik’s cube. Since she can easily solve a 3 x 3 x 3 cube, she decided to conquer 4 x 4 x 4, 5 x 5 x 5, and 6 x 6 x 6 cubes. She has even been able to solve the 7 x 7 x 7 cube. Amazing! If you hand her a physical cube in any of those dimensions, she will be able to solve it. She can even easily solve a virtual Rubik’s cube. Someone gave her this cube for her birthday, and she can solve it, too.

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When I showed her the Find the Factors puzzles, she naturally asked if I could make one that used factors all the way to fifteen.

Do the dimensions of the puzzle matter? The puzzles usually ask you to find the factors from 1 to 10 or 1 to 12. What if we had a Find the Factors 1-5 puzzle?

Factors 1-5

This puzzle is so easy to solve that most people will not think it worth their time. The number combinations are so few that it should  never prove to be very challenging. If you solve this puzzle, you may find it hard to believe that the numbers were chosen at random, but they really were. You probably will agree with me that 5 factors are too few. If we made puzzles with 6 factors, then 7, 8, or 9, they would all be more difficult than a 5 factor puzzle, but 10 factors seems like the best place to start. Most everyone is expected to know the times table up to 10 x 10 = 100. Many people were also taught the multiplication facts up to 12 x 12 = 144.

A coworker once mentioned to me that he noticed that puzzles with 12 factors are much more difficult to solve than ones with 10 factors. Whether the factors are prime or composite numbers is only part of what makes a puzzle easy or difficult. For example, even though 2, 3, and 5 are prime numbers, they don’t make a puzzle easier because they have multiples that can also be factors in the puzzle. Prime numbers 7 and 11 help make a puzzle easier, so a Find the Factors 1-11 would actually be easier than a 10 factor puzzle. Adding 12 complicates the puzzle significantly because the following multiples of 12 have other factors that also could be the correct answer in the puzzle.

multiples of 12

Thirteen, another prime number, also makes the puzzle a little easier. My brother, Andy, will have hiss 65 birthday soon. I made him a Find the Factors 1-13 puzzle because I wanted to include the number 65 which is 13 x 5. Most people only know a few of the 13 multiplication facts like 4 suits x 13 ranks = 52 playing cards, but if you solve this puzzle, you will still probably find it easier than a 1-12 factor puzzle.

Happy 65th Birthday

Adding 14 as a possible factor takes away the advantages of prime number 7, so a 14 factor puzzle would be more difficult. Also most people have not memorized the first 14 multiples of 14. Making or solving a 15 factor puzzle makes the multiples of 3 and 5 become even more complicated clues. Of course, most people don’t recall the first 15 multiples of 15 either. The many possible factors for the clues make it more difficult to create a puzzle that has only one solution. Try to solve this Find the Factors 1-15 puzzle. It expects you to know the two factors of 195 that are both between 1 and 15. Also the common factor of 15 and 30 could be 3, 5, or 15. This puzzle can still be solved using logic only, but it will be more challenging than puzzles of smaller dimensions.

15 puzzle

You can cut and paste the puzzle into a document and make it any size you wish or you can open 10 Factors 2013-12-02 to view it along with the following puzzles.

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7 and Spaghetti and Meatballs for All!

7 is a prime number. 7 = 1 x 7. Its only factors are 1 and 7. Prime factorization: none.

How do we know that 7 is a prime number? The square root of 7 is an irrational number approximately equal to 2.65. If 7 were not a prime number, then it would be divisible by at least one prime number less than or equal to 2.65. Since 7 is not divisible by 2, it is a prime number.

When 7 is a clue in the FIND THE FACTORS puzzles, one factor will be 7 and the other will be 1.

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

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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. 

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6, 48, 720, 1100 and A Piece of Cake

6 is a composite number. 6 = 1 x 6 or 2 x 3. Factors of 6: 1, 2, 3, 6. Prime factorization: 2 x 3.

When 6 is a clue in the FIND THE FACTORS  puzzle, the pair that will work for that particular puzzle might be 1 x 6, or it might be 2 x 3.

Scroll down the page to see the factors of 48, 720, and 1100.

In this post I talk about factor trees, and I describe how to use the cake method to find the factors of a number.

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.

Birthday cake

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

quarter note

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:

 

factor tree

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:

more common factor tree

720 isn’t even that big of a number, but gathering all of the prime numbers from the factor tree and putting them in numerical order would be like picking up a bunch of scattered leaves. It would be like 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. Notice that some of the prime factors of 720 (2,2,2,2,3,3,5) are not as easy to see as others on the factor tree.  That is why I want to chop down that 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:

Cake method

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.

Still, 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!

5, 435, 144 and Easy as 1-2-3

5 is a prime number. 5 = 1 x 5. Its only factors are 1 and 5. Prime factorization: none.

Scroll down the page to see the factors of 435 and 144.

How do we know that 5 is a prime number? The square root of 5 is an irrational number approximately equal to 2.24. If 5 were not a prime number, then it would be divisible by at least one prime number less than or equal to 2.24. Since 5 is not divisible by 2, it is a prime number.

When 5 is a clue in a FIND THE FACTORS puzzle, use 1 for one of the factors and 5 for the other.

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

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

To solve the puzzles, we are only interested in the limited set of factors that are represented in the following table:

Puzzle Clues Chart

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.

Thinking part 1

Thinking part 2

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.

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:

144 table

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.

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