A Multiplication Based Logic Puzzle

Posts tagged ‘8’

A Forest of 240 Factor Trees

  • 240 is a composite number.
  • Prime factorization: 240 = 2 x 2 x 2 x 2 x 3 x 5, which can be written (2^4) x 3 x 5
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 240 has 20 factors.
  • Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
  • Factor pairs: 240 = 1 x 240, 2 x 120, 3 x 80, 4 x 60, 5 x 48, 6 x 40, 8 x 30, 10 x 24, 12 x 20, or 15 x 16
  • Taking the factor pair with the largest square number factor, we get √240 = (√16)(√15) = 4√15 ≈ 15.492.

Because 240 has so many factors, it is possible to make MANY different factor trees that create a forest of 240 factor trees. This post only contains eleven of those many possibilities. The two trees below demonstrate different permutations that can be made from the same basic tree. The mirror images of both, as well as mirror images of parts of either tree, would be other permutations.

240 Factor Trees


A good way to make a factor tree for a composite number is to begin with one of its factor pairs and then make factor trees for the composite numbers in that factor pair.

240 Factor Trees 1 - 3

In this first set of three factor trees we can also see the factor trees for 120, 80, 4, & 60.

240 Factor Trees 4 - 6

These three factor trees also include factor trees for 48, 6, 40, 8, and 30.

240 Factor Trees 7 - 9

Finally, these three factor trees also include factor trees for 10, 24, 12, 20, 15, and 16.

This forest of 240 factor trees is dedicated to Joseph Nebus. Read the comments to his post, You might also like, because, I don’t know why, to discover why I was inspired to create images of parts of this forest.


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.




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