48 and What’s wrong with this?

/ ivasallay / 2 Comments

Here is a four piece puzzle arranged two different ways. There are 24 blue squares and 24 yellow squares. 24 + 24 = 48. Read on to see why the factors of 48 are significant to this puzzle.

Those triangles are some fabulous mathematical slight of hand.  Something is wrong with the puzzle, but what?

Something is fishy with these two triangle puzzles, too:

And with these two triangles made with six puzzle pieces!

If I had time, I could make an infinite number of these types of puzzles. Why do we get a solid triangle sometimes, but one with a hole in it the other times?

Believe it or not, the answer to the triangle puzzles has something to do with factoring.

Here’s that first puzzle again, but smaller:

The first triangle doesn’t have a hole it while the second one does because

Who would have thought that comparing fractions could turn into a type of magic trick?!!

Both triangles have 48 blue or yellow squares, but the second triangle wants you to think it should have 49.

(Psst….6/7 is also the slope of the green triangle and 7/8 is the slope of the red triangle. Since the two triangles don’t have identical slopes, the big “triangles” formed in the puzzles aren’t really triangles at all.)

I made all the puzzles in this post by comparing fractions. In each case, I simply used two consecutive composite numbers and then factored them and came up with the fractions I wanted to use. The numerators and the denominators of the fractions became the sides of the triangles.

Now, what do you think of this video of a never-ending chocolate bar?

Let me tell you a little bit about the number 48:

48 is a composite number. 48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, or 6 x 8. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Prime factorization: 48 = 2 x 2 x 2 x 2 x 3, which can also be written 48 = 2⁴ x 3.

Sometimes 48 is a clue in the FIND THE FACTORS puzzles. Even though it has many other factors, we use only 6 x 8 for the FIND THE FACTORS 1-10 puzzles and 6 x 8 or 4 x 12 for the FIND THE FACTORS 1-12 puzzles.

 

factor tree

 

47 and Understanding the Equal Sign with Candy

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Understanding the Equal Sign Using Peanut Butter Cups. What a wonderful lesson! Teachers or parents can easily follow Jen’s lesson plan. I’m certain children will always remember what an equal sign means forever after.

  • 47 is a prime number.
  • Prime factorization: 47 is prime.
  • The exponent of prime number 47 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 47 has exactly 2 factors.
  • Factors of 47: 1, 47
  • Factor pairs: 47 = 1 x 47
  • 47 has no square factors that allow its square root to be simplified. √47 ≈ 6.85565

How do we know that 47 is a prime number? If 47 were not a prime number, then it would be divisible by at least one prime number less than or equal to √47 ≈ 6.9. Since 47 cannot be divided evenly by 2, 3, or 5, we know that 47 is a prime number.

47 is never a clue in the FIND THE FACTORS puzzles.

 

46 and South East Arrow Puzzle

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46 is a composite number. 46 = 1 x 46 or 2 x 23. Factors of 46: 1, 2, 23, 46. Prime factorization: 46 = 2 x 23.

46 is never a clue in the FIND THE FACTORS puzzles.

Wikipedia has a chart of 116 different directional arrow symbols. Each one of them has a name. Today’s puzzle represents the South East Arrow.

2014-04 Level 2

To solve the puzzle write the numbers 1 -12 in the top row and again in the first column so that the numbers you write and the clues in the puzzle work together to make a multiplication table.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors of last week’s hook-shaped level 2 puzzle:

2014-03 Level 2 Answer

45 and A Change in Order

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45 is a composite number. 45 = 1 x 45, 3 x 15, or 5 x 9. Factors of 45: 1, 3, 5, 9, 15, 45. Prime factorization: 45 = 3 x 3 x 5, which can also be written 45 = 3² x 5.

When 45 is a clue in the FIND THE FACTORS puzzles, always use 5 and 9 as the factors.

2014-04 Level 1

When I introduce the FIND THE FACTORS puzzles to elementary school students, I only print copies of level 1 and level 2 puzzles. After students have completed those puzzles, I will demonstrate how a level 3 puzzle is completed.

A few students have downloaded the puzzles at home. One very bright sixth grade student was able to complete a level 6 puzzle with the help of his dad.

Kids can be very determined. They may die several times fighting dragons, but they will eventually find the way to save the princess. They can be just as determined to find the way to write the numbers from 1 to 12 in the top row and the first column to make the puzzle work as a multiplication table, especially when I compare it to winning a computer or video game.

Many students tend to guess and check and will get frustrated unless they are shown how to solve the puzzles logically. Frustration can be the enemy of determination.

This week’s puzzles are available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles. If you click on the excel file you may notice that I have changed the order of the puzzles on the main page.

puzzle order

If the puzzles are printed on both sides of a sheet of paper, the paper can be cut down the middle. In this way I can print puzzles for an entire classroom using only half as much paper.

I instruct students to treat the sheet of puzzles like a computer game and see what level they can achieve. They know upfront that I will be extremely surprised if any of them find all the factors to all six puzzles. That statement motivates some students while it reduces stress in others. Not all students like solving the puzzles, but it is a joy to observe the ones who do.

front & back sides

Here is solution for last week’s level 1 puzzle:

2014-03 Level 1 Answer

                                     

44 and Games again…

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Mathematicsgames.wordpress.com that I’ve written about previously is part of the Mathematics, Learning, and Technology blog. This blog is a great resource for teachers, parents, and lovers of mathematics. I really appreciate Find the Factors being included in the post Games, again!

44 is a composite number. 44 = 1 x 44, 2 x 22, or 4 x 11. Factors of 44: 1, 2, 4, 11, 22, 44. Prime factorization: 44 = 2 x 2 x 11, which can also be written 2² x 11.

When 44 is a clue in the FIND THE FACTORS 1 -12 puzzles, always use 4 and 11 as the factors.

42 and A Topological Puzzle

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42 is a composite number. 42 = 1 x 42, 2 x 21, 3 x 14, or 6 x 7. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Prime factorization: 42 = 2 x 3 x 7.

When 42 is a clue in the FIND THE FACTORS puzzles always use 6 and 7 as the factors.

I didn’t take Topology in college, but I knew people who did. It looked pretty fun.

Math and Science with My Kids shared a topological puzzle. It was interesting to read of his child’s frustration and eventual triumph. I learned something reading these, and you can too!

 

 

41 and Life Is Like A Sudoku Puzzle

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  • 41 is a prime number.
  • Prime factorization: 41 is prime.
  • The exponent of prime number 41 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 41 has exactly 2 factors.
  • Factors of 41: 1, 41
  • Factor pairs: 41 = 1 x 41
  • 41 has no square factors that allow its square root to be simplified. √41 ≈ 6.40312

How do we know that 41 is a prime number? If 41 were not a prime number, then it would be divisible by at least one prime number less than or equal to √41 ≈ 6.4. Since 41 cannot be divided evenly by 2, 3, or 5, we know that 41 is a prime number.

41 is never a clue in the FIND THE FACTORS puzzles.

I read an essay titled Life Is Like a Sudoku Puzzle. Typically the only Sudoku puzzles I do are in the newspaper, and it has been weeks since I’ve started one. If I mess up on one, I don’t try to figure out what I did wrong. I simply put a big x on it and throw the paper in the recycling bin. I accept defeat, but I really don’t learn anything from my mistake. But if I would look at those puzzles a little differently, I could learn a lot about life from them, just as that essay suggests.

 

40 and M.C. Escher

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40 is a composite number. 40 = 1 x 40, 2 x 20, 4 x 10, or 5 x 8. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. Prime factorization: 40 = 2 x 2 x 2 x 5, which can also be written 2³ x 5.

When 40 is a clue in the FIND THE FACTORS puzzles always use either 5 x 8 or 4 x 10. Only one of those combinations will work for any particular puzzle.

Today I share a blog post that was part of the 6-word challenge. I encourage other bloggers to accept the challenge on that post and send your finished product to the Figments of a DuTchess blog. Marion Driessen really will visit your blog if you do. She completed this beautiful jigsaw puzzle featuring the mathematically inspired artwork of M. C. Escher and relayed the challenge of putting it together using only six words!

39 and Seriously Hooked on Factoring

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39 is a composite number. 39 = 1 x 39 or 3 x 13. Factors of 39: 1, 3, 13, 39. Prime factorization: 39 = 3 x 13.

39 is never a clue in the FIND THE FACTORS 1-10 or 1-12 puzzles.

Last Tuesday I published a hook-shaped level 2 puzzle and invited readers to get hooked on factoring. Today I present a level 6 hook-shaped puzzle. Admittedly, one would have to be seriously hooked on factoring to attempt and eventually complete the more difficult level 6 puzzle. I assure you that it can be solved using logic with absolutely no guessing and checking.

2014-03 Level 6

This week’s puzzles are also available in an excel file here. If you have a spreadsheet program on your computer, you can access it. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.

Here are the factors from last week’s level 6 snowball puzzle:

2014-02 Level 6 Answer

Below is a table showing one possible way to solve the snowball puzzle using logic alone.

2014-02 L6 steps

Seriously, study the clues for this week’s puzzle and see if you can figure out how to solve it using logic alone. Guessing and checking will most likely only frustrate you. Good luck!

38 and Factor Finding

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38 is a composite number. 38 = 1 x 38 or 2 x 19. Factors of 38: 1, 2, 19, 38. Prime factorization: 38 = 2 x 19.

38 is never a clue in the FIND THE FACTORS puzzles.

One of my favorite bloggers wrote a post about the FIND THE FACTORS puzzles. I am thrilled to share what he wrote in his post Factor Finding!