This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31
76 is a composite number. 76 = 1 x 76, 2 x 38, or 4 x 19. Factors of 76: 1, 2, 4, 19, 38, 76, Prime factorization: 76 = 2 x 2 x 19, which can also be written 2² x 19.
76 is never a clue in the FIND THE FACTORS puzzles.
This week’s puzzles for you to solve
70 is a composite number. 70 = 1 x 70, 2 x 35, 5 x 14, and 7 x 10. Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Prime factorization: 70 = 2 x 5 x 7.
Sometimes 70 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 70 = 7 x 10 in the puzzles.
Here are the factors for today’s puzzle:
Level 3 puzzles are designed to be solved starting from a row or column at the top of the puzzle with 2 clues. First, find the common factor of those two clues that will allow you to write only numbers from 1 to 10 in the first column of the puzzle. Then factor row by row to the bottom of the puzzle making sure each number from 1 to 10 is written only once in both the factor row and the factor column. You will notice a rhythm for the answers as you work.
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 is the solution to last week’s level 3 puzzle:
Since level 3 puzzles are designed to be solved from the top of the puzzle to the bottom, they are fairly easy to solve. I will not usually write step by step instructions for them, but I will today.
Last week’s puzzle was solved by beginning with the clues 28 and 36 because they are both in the same row. Their common factors are 1, 2, and 4. Choose 4 because it is the only common factor that will allow us to write only numbers 10 or less in the factor row. Below is a chart showing the steps IN ORDER that were used to solve this puzzle using logic. The chart states what to write in the factor column and the factor row for each clue.
Just as a Find the Factor puzzle can be solved step by step, so can a 4 x 4 Magic Square:
34 is the magic sum of a 4 x 4 magic square. There are 880 ways times 8 different orientations to make a 4 x 4 square. I made a gif that shows one way. Look and you will see that half the numbers are in the correct positions right from the start:
make science GIFs like this at MakeaGif
34 is a composite number. 34 = 1 x 34 or 2 x 17. Factors of 34: 1, 2, 17, 34. Prime factorization: 34 = 2 x 17.
34 is never a clue in the FIND THE FACTORS puzzles.
32 is a composite number. 32 = 1 x 32, 2 x 16, or 4 x 8. Factors of 32: 1, 2, 4, 8, 16, 32. Prime factorization: 32 = 2 x 2 x 2 x 2 x 2, which can also be written 32 = 2⁵.
When 32 is a clue in the FIND THE FACTORS puzzles, always use 4 and 8 as the factors.
Here is a hook-shaped FIND THE FACTORS 1-10 puzzle, and YOU can solve in by hook or by crook!
The puzzle has a few square numbers: 16, 4, 81, 36, 36, 9, 1, and 25. Several square numbers can only be factored one way in a FIND THE FACTORS puzzle, while others can be factored 2 or 3 ways.
Obviously, 6 x 6 = 36 will not let us off the hook both times that 36 appears. One of those times we must use 4 x 9 = 36. This puzzle reminds us that 36 is sometimes a square and sometimes a rectangle.
The 16 isn’t a square in this puzzle either. Sometimes 4 and 9 are rectangles, but not in this case.
81 and 25 are always square in the FIND THE FACTORS puzzles because even though 1 and 3 are allowed to be factors, 25, 27, and 81 are greater than 10 and not allowed as possible factors to solve the puzzle.
To solve this puzzle, write the numbers from 1 to 10 in the top row and again in the first column so that those numbers are the factors of the given clues. Don’t fall for those slightly tricky clues hook, line, and sinker!
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. I’m hooked on factoring. I hope you’ll get hooked, too!
Here are all the factors to last week’s level 2 FIND THE FACTORS 1-12 snowflake puzzle:
Here’s a nice pattern
“Every 5th power ends in the last digit of the base.”
Can you see why? What about other powers? Other than base-ten? pic.twitter.com/6Wk5Vr0HNr
— Tamás Görbe (@TamasGorbe) January 1, 2023
How do we know that 31 is a prime number? If 31 were not a prime number, then it would be divisible by at least one prime number less than or equal to √31 ≈ 5.6. Since 31 cannot be divided evenly by 2, 3, or 5, we know that 31 is a prime number.
31 is never a clue in the FIND THE FACTORS puzzles.
Every third grader should know how to fill in a multiplication table:
By the middle of third grade, every third grade student has been taught the basic multiplication facts. Of course, that doesn’t mean that every third grader has LEARNED those facts. Corestandards.org/Math/Content/3/ lists EVERYTHING a student should learn in third grade math. Multiplication is very key in learning all of these concepts.
Many students, even beyond third grade, rely on skip counting to figure out what a multiplication product should be. Some students have learned rhymes to help them recall a fact. Given the opportunity some students even in middle school will use a calculator to find the answer to a basic multiplication fact.
Practicing every day will help students recall the facts instantly instead of needing a calculator or other crutch to help them remember one of the basic multiplication facts. For variety, every third grader should also know how to fill out a multiplication table that looks like this:
This kind of table doesn’t allow a student to simply count by twos, fives, nines or any other number to fill out a particular row or column. This type of table encourages a student to fully memorize all the multiplication facts so the table can be completed as quickly as possible.
About the middle of third grade, students are introduced to division. As soon as students learn that any number can be divided by 1, they are ready to tackle the easiest of the easiest FIND THE FACTORS puzzles. Here is the level 1 puzzle for the 3rd full week of 2014:
It is a level 1 puzzle because students only have to know 10 (or 12) division facts to solve the puzzle. Even though it is a simple puzzle, third grade students will need some instruction to complete it. Tell them that this puzzle is going to turn into a multiplication table as soon as they figure out what numbers go into the top row and the first column. Tell them the only numbers that will be written in those two places are numbers from 1 to 10 and that all of those numbers have to be written in both places.
Help the students know what numbers to write in each space on the top row. Some students will likely be confused when they get to the column with 8 clues and to the column with no clues. Help them use logic to figure out what numbers should go in those spaces. When all the factors have been found in both the factor row and the factor column, have them complete the multiplication table. It will be great practice for all of them.
Last week’s puzzle was fairly easy. Here is the puzzle with all of the factors found:
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. Have fun!
26 is a composite number. 26 = 1 x 26 or 2 x 13. Factors of 26: 1, 2, 13, 26. Prime factorization: 26 = 2 x 13.
26 is never a clue in the FIND THE FACTORS 1-10 or 1-12 puzzles.
Write the numbers 1-12 in the top row and again in the first column so that those numbers are the factors of the given clues. Use logic, not guessing and checking, to find its unique solution. Here is how to solve this level 4 “snow shovel” puzzle:
Begin by looking for perfect square clues, 1, 25, 49, 64, 81, 100, 121, or 144 because there is only one way to factor any of them so that both factors are numbers from 1 to 12. If you find one of those perfect square clues, write down its factors in the corresponding factor row and factor column.
Next locate a row or a column with at least 2 clues. Find the common factor of that row or column that will allow you to write only numbers from 1 to 12 in both the factor row and the factor column. If there is more than one row or column with at least 2 clues, repeat the previous step until only rows and columns with one clue remain.
Now you have at least 3 factors written down. Next look at ALL the remaining clues in the puzzle and find one that can only be factored using one of the factors you’ve written down already but haven’t used twice. Repeat the last step until all (or almost all) of the factors are found.
If a row or a column contains no clues, use logic is determine which number from 1 to 12 should be written there.
While working on these puzzles, occasionally count from 1 to 12 to make sure each of those numbers is written exactly once in both the factor row and the factor column. You will notice a rhythm for the answers as you work. Good luck!
This week’s puzzles are also available in an excel file here, if you have a spreadsheet program on your computer. If you enable editing in excel, you can type your answers directly onto the puzzle, and you can also easily print the puzzles.
Here is the solution to the week 1 level 4 puzzle:
