A Multiplication Based Logic Puzzle

Archive for November, 2013

11 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.








10 Just Like Sudoku?









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


factors for 2013-11-25


9 Remembering 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 What Happens When Puzzle Dimensions Change?

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.


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.




7 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.


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. 


6 A Piece of Cake






Birthday cake

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

quarter note



factor tree


more common factor tree


Cake method




5 Easy as 1-2-3

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

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.


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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