A Multiplication Based Logic Puzzle

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.

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.

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

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

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.

12 The Doorbell Rang

12 is a composite number. 12 = 1 x 12, 2 x 6, or 3 x 4. Factors of 12: 1, 2, 3, 4, 6, 12. Prime factorization: 12 = 2 x 2 x 3, which can also be written 12 = (2^2) x 3.

When 12 is a clue in the FIND THE FACTORS 1 – 12 puzzles, any pair of its factors could be the correct  choice. In the 1 – 10 puzzles, only 2 x 6 or 3 x 4 will be the correct choice.

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.

Here is part of an email that my blogging friend, Paula Krieg, sent after reading this post, “I’ve been looking at some Islamic Geometry, learning to draw some of those rosettes, and was struck by how the 12-fold pattern seemed particularly rich. I may be wrong about this, but it got me thinking about 12. 12 makes a dozen. 12 months to a year. 12 inches to a foot. 12 days of Christmas, 12 numbers on a clock,  12 apostles. My cupcake pans makes 12 cup cakes, and I guess Grama’s cookie pan makes 12 cookies in The Doorbell Rang book.”

I should also mention that some people think we should switch from base 10 to base 12 because 12 is divisible by 50% of the numbers less than or equal to it while 10 is only divisible by 40% of the numbers less than or equal to it.

The puzzle below will require 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.