63 is a composite number. 63 = 1 x 63, 3 x 21, or 7 x 9. Factors of 63: 1, 3, 7, 9, 21, 63. Prime factorization: 63 = 3 x 3 x 7, which can also be written 63 = 3² x 7.
Sometimes 63 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 7 x 9 = 63.
All Previously Published Puzzles
62 is a composite number. 62 = 1 x 62 or 2 x 31. Factors of 62: 1, 2, 31, 62. Prime factorization: 62 = 2 x 31.
62 is never a clue in the FIND THE FACTORS puzzles.
All Previously Published Puzzles
Here is the logic to solve the puzzle & a chart showing one way to get the solution:
9 has to be the factor of both the 27 and the 63. Therefore 3 x 6 = 18, (not 2 x 9).
3 is one of the factors of both the 18 and the 27. Therefore 9 x 1 = 9, (not 3 x 3).
How do we know that 61 is a prime number? If 61 were not a prime number, then it would be divisible by at least one prime number less than or equal to √61 ≈ 7.8. Since 61 cannot be divided evenly by 2, 3, 5, or 7, we know that 61 is a prime number.
61 is never a clue in the FIND THE FACTORS puzzles.
Wedding Bells are Going to Chime
My daughter is getting married in early March, and family will be coming in from out of town. I also need to study for a test at work. There is so much that needs to get done that hasn’t been getting done. I’m taking a month-long break from daily blogging and a few other things so I can catch up on the things that just can’t wait.
Over the last several weeks there has been several new people who have found this blog and have done some of the puzzles. Some of them have clicked on the older excel files to work on puzzles. All that clicking was probably inconvenient, but several people have still clicked anyway. Thank you for your interest and persistence. I want to make it easier to catch up on all those old puzzles.
To make it easier to access the older puzzles, I’ve put all previously published FIND THE FACTORS 1-10 and 1-12 puzzles in one excel file, All Previously Published Puzzles. The formatting isn’t printer friendly, but if you enable editing, you can type the factors directly onto the excel file. The answers are also in the file for easy comparison with your work. (Hopefully, I didn’t mess up any of the clues while I hurriedly created this large file. Please, tell me if you find any errors.)
While I catch up on all the things I need to do, I hope you will catch up a few things as well.
When solving a level 6 puzzle, trying to place the numbers from 1 to 10 in the top row as well as the first column can be frustrating. Every single clue appears to have more than one possible answer. How can someone guess right with every single number placement? Is seeing the answer the only way to solve the puzzle?
Here is this week’s level 6 puzzle. Usually, to solve a puzzle, you start with a row or column with 2 clues. There are 6 rows or columns with more than one clue, and every one of them is no help!
You can also find this and a few other puzzles 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.
“Pay no attention to the that man behind the curtain!” is a famous line from The Wizard of Oz. Everyone in Oz thought the Wizard was powerful enough to solve any problem. When Dorthy and her friends looked behind the curtain they discovered that he was really an ordinary man who was able to find an advantage in what other people didn’t know. It turns out that other people actually had the abilities they thought only the Wizard had. Today we will take a peek behind the curtain of a level 6 puzzle and discover that it really isn’t as intimidating as we might imagine and that you have the ability to solve such a puzzle, too. This is what last week’s level 6 puzzle looks like from behind the curtain: 
Many people try to start with the row that contains the clue 4 x 8 = 32. (When I solve this puzzle, that row is actually the one I FINISH with!) Instead, look at the row with no clues. If you study all the clues in the puzzle, you will notice that 6 is listed as a factor only once. If there were a clue in the row without any clues, that clue would be a multiple of 6. That hint will keep you going for several clues. It also eliminates 8 as a possible factor of 72 and ensures that 1 x 8, not 2 x 4, must be used for the clue 8. This means that 2 x 5, not 1 x 10, must be used for the clue 10 because both ones will have been used. That means that 2 x 10, not 4 x 5, must be used for the clue 20 because the number 10 has to be used as a factor twice. Here is the completed puzzle: 
60 is a composite number. 60 = 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, or 6 x 10. Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Prime factorization: 60 = 2 x 2 x 3 x 5, which can also be written 60 = 2² x 3 x 5.
Sometimes 60 is a clue in the FIND THE FACTORS puzzles. Even though it has many other factors, we only use 6 x 10 for the FIND THE FACTORS 1-10 puzzles and 6 x 10 or 5 x 12 for the FIND THE FACTORS 1-12 puzzles.
60 has six factor pairs. One of those factor pairs adds up to 17, and another one subtracts to 17. If you can identify those factor pairs, then you can solve this puzzle:
How do we know that 59 is a prime number? If 59 were not a prime number, then it would be divisible by at least one prime number less than or equal to √59 ≈ 7.7. Since 59 cannot be divided evenly by 2, 3, 5, or 7, we know that 59 is a prime number.
59 is never a clue in the FIND THE FACTORS puzzles.
Valentine’s Day is almost here. Here is a Valentine you can share with your friends who enjoy solving puzzles.
To solve the puzzle, the numbers 1 through 10 need to be written on the top row and also on the first column so that those numbers and the clues inside the puzzle make a multiplication table. Since it’s a level 5 puzzle, it may be a little challenging.
Your friend will not have to wait a whole week for the solution because there is a gif that will solve it quickly and logically. The gif could be the valentine, too.
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 to last week’s puzzle.
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58 is a composite number. 58 = 1 x 58 or 2 x 29. Factors of 58: 1, 2, 29, 58. Prime Factorization: 58 = 2 x 29.
58 is never a clue in the FIND THE FACTORS puzzles.
Hugs and Kisses, OXOXOXOXOXOXOX, these were some of my stepmother’s favorite things. This week in anticipation of Valentine’s Day next week, I’ve included both an X puzzle and this O puzzle. Valentine’s Day was also my dad’s and my stepmother’s wedding anniversary, and she always made sure it was a special day for everyone who lived in our house. I’m sure that you also have many happy Valentine’s Day memories.
To solve this O puzzle, consider the clues given in the puzzle. Then write the numbers 1-12 in the top row and again in the first column so that the puzzle becomes 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 for last week’s level 4 Puzzle:
Here is one way those factors could be found:
57 is a composite number. 57 = 1 x 57 or 3 x 19. Factors of 57: 1, 3, 19, 57. Prime factorization: 57 = 3 x 19.
57 is never a clue in the FIND THE FACTORS puzzles.
How do you feel about factoring?
If you will let Cupid hit you with this arrow, you might just fall in love with factors.
Factoring is very important in many levels of mathematics, so enjoying it can be a very good thing.
I love factoring. I hope you will let Cupid’s arrow hit you so you will love factoring, too.
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.
The factors for last week’s level 3 Puzzle:
Here’s how last week’s puzzle was solved:
56 is a composite number. 56 = 1 x 56, 2 x 28, 4 x 14, or 7 x 8. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. Prime factorization: 56 = 2 x 2 x 2 x 7, which can also be written 56 = 2³ x 7.
Sometimes 56 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 7 x 8 = 56.
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.
The factors for last week’s level 2 Southeast Arrow Puzzle:
Here are three different factor trees for the number 56:
55 is a composite number. 55 = 1 x 55 or 5 x 11. Factors of 55: 1, 5, 11, 55. Prime factorization: 55 = 5 x 11.
Sometimes 55 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, the puzzle only uses 5 x 11 = 55.
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What is the square root of a number? If you had a square made with 9 squares, how many squares would be on each side?
Even a third grader can count 3 squares on each side. That is one way to explain why √9 = 3 to someone so young. Here’s another explanation:
These are the kinds of square roots that third grade students may need to know. When they get older they will learn about other square roots like √2 ≈ ±1.414, but third graders don’t need to worry about those yet.
On January 24 someone hurriedly typed “aquare roots for third grade” into a search engine, and a post from this blog came up. The person took ONE quick look and closed the post and the blog just as hurriedly. I’m not exactly sure what aquare or square roots for third grade means, but I felt bad that the blog disappointed this unknown person. I thought, “That person MAY have been searching for something that actually might be on the blog SOMEWHERE…..Perhaps a puzzle with perfect square clues would have been a welcome sight.” Hey, I can dream, can’t I?
To solve today’s puzzle (It actually has perfect square clues!), you will have to take the square roots of the clues, and place those square roots in the top row as well as the first column. After you write down all those square roots, figure out what numbers go in the empty cells so that the entire puzzle works as a multiplication table.
These square roots may or may not be the square roots for third grade that person was looking for. Nevertheless, I have personally seen several bright THIRD grade students solve similar perfect square puzzles that I’ve made. 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 for last week’s level 1 puzzle: 
How many squares are in this puzzle? Finding that answer is too tedious for me to pursue. There is 1 square that is bigger than all the rest, 169 of the smallest size squares, and some different number between 1 and 169 for each size square in between. Also, I think the clues seem to form three little squares, and one of the clues, 16, happens to be a perfect square. It’s a trick question, and most people don’t like trick questions!
Solving the actual puzzle will actually be less work and more fun. The actual puzzle challenges you to write the numbers 1 through 12 in the top row and again in the first column so that the answers you write and the clues inside the puzzle work together as a multiplication table. Use logic to find the unique solution to the puzzle. 
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.
54 is a composite number. 54 = 1 x 54, 2 x 27, 3 x 18, or 6 x 9. Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54. Prime factorization: 54 = 2 x 3 x 3 x 3, which can also be written 2 x 3³.
Sometimes 54 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact the puzzle uses is 6 x 9 = 54.
The number 6 has two factor pairs. One of those factor pairs adds up to 5, and the other one subtracts to 5. Can you put those factors in the correct places in the first puzzle?
The number 54 has four factor pairs. One of those factor pairs adds up to 15, and another one subtracts to 15. If you can identify those factor pairs, then you can solve the second puzzle.
The second puzzle is really just the first puzzle in disguise. Why would I say that?
Here is one way those factors can be found using logic.
