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.

43 and Find the Factors

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  • 43 is a prime number.
  • Prime factorization: 43 is prime.
  • The exponent of prime number 43 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 43 has exactly 2 factors.
  • Factors of 43: 1, 43
  • Factor pairs: 43 = 1 x 43
  • 43 has no square factors that allow its square root to be simplified. √43 ≈ 6.5574

How do we know that 43 is a prime number? If 43 were not a prime number, then it would be divisible by at least one prime number less than or equal to √43 ≈ 6.6. Since 43 cannot be divided evenly by 2, 3, or 5, we know that 43 is a prime number.

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

So far today FIND THE FACTORS has received nearly four times as many views as its previous high. When I investigated why I discovered one of the reasons was a blog called mathematicsgames.wordpress.com. On that blog was this post recommending the FIND THE FACTORS puzzles. I am excited for the great write-up and the new people who have looked at the puzzles and used the excel file I made! I am humbled that mathematicsgames wrote the post, and I’m also quite proud to tell you about it.

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!

37 and How Many Trapezoids I Can Draw

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A trapezoid is often defined for young students as a four-sided shape with EXACTLY two parallel sides. Once a person studies higher level math, the definition changes: A trapezoid is a four-sided shape with AT LEAST two parallel sides. How many different kinds of trapezoids can a person draw? It depends on which definition you use. If you use the second definition, you can also include parallelograms, rectangles, rhombuses, and squares. Either definition will allow the standard isosceles trapezoid and several others. But how many? Whichever definition you use, figuring out how many different ones can be drawn is a nice puzzle to solve. NebusResearch’s blog post does a nice job explaining the different ones, and it even came up with ones I hadn’t considered!

Now I’ll share some facts about the number 37:

  • 37 is a prime number.
  • Prime factorization: 37 is prime.
  • The exponent of prime number 37 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 37 has exactly 2 factors.
  • Factors of 37: 1, 37
  • Factor pairs: 37 = 1 x 37
  • 37 has no square factors that allow its square root to be simplified. √37 ≈ 6.08276

How do we know that 37 is a prime number? If 37 were not a prime number, then it would be divisible by at least one prime number less than or equal to √37 ≈ 6.1. Since 37 cannot be divided evenly by 2, 3, or 5, we know that 37 is a prime number.

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

36 and The Six Word Challenge

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36 is a composite number, and it is 6 squared. 36 = 1 x 36, 2 x 18, 3 x 12, 4 x 9, or 6 x 6. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Prime factorization: 36 = 2 x 2 x 3 x 3, which can also be written 36 = 2² x 3².

Since √36 = 6, a whole number, 36 is a perfect square.

When 36 is a clue in the FIND THE FACTORS  1 – 10 puzzles, use either 4 x 9 or 6 x 6. When 36 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use either 3 x 12, 4 x 9 or 6 x 6. Only one of those combinations will work for any particular puzzle.

I enjoy this multiplication rhyme: Six times six, magic tricks, abracadabra, thirty-six!

A fellow blogger issued a challenge for her readers to write a puzzling story or tell a story about a puzzle using exactly six words. I’m choosing to give puzzle directions today using these six words:

Without guessing, logically decide puzzle’s factors.

2014-03 Level 5

It was a fun challenge. Her blog has a 6-word story written by Ernest Hemingway as well as a few written by other bloggers.

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 for last week’s level 5 snowflake puzzle:

2014-02 Level 5 Answer

There was more than one way to logically decide the snowflake puzzle’s factors. Some sets of clues needed to be avoided in the beginning: (18, 6) because both can be factored by 2, 3, and 6; (60, 40) because each can be factored by 5 and 10; and (16, 8) because both can be factored by 2, 4, and 8.

The column with 5 clues was a good place to start the puzzle after we narrowed down the common factor that would work for all the clues. The table below shows a way to find all the factors from 1 to 12 using logic without guessing and checking:

2014-02 L5 steps

Related article:

http://drieskewrites.wordpress.com/2014/01/22/six-word-story-challenge-puzzle/