A Multiplication Based Logic Puzzle

Archive for December, 2013

21 Factors of the Year 2013 and 2014

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

When 21 is a clue in the FIND THE FACTORS puzzles, use 3 x 7.

Scroll down the page to find factoring information about 2013 and 2014.

2013 year

Near the end of each year movie critics make lists of the ten best movies and the ten worse movies of the year. News agencies list the ten most significant news stories. Time magazine lists the ten most influential people of the year. The music industry lists the top ten songs of the year. As 2013 draws to a close, it is most appropriate for me to review the factors of the year.

2013 had exactly 8 positive factors. These factors were 1, 3, 11, 33, 61, 183, 671, and 2013.

There is no room for argument. I am absolutely certain this list is complete. No one will make any comments disagreeing with me, calling me names, or asking how I could have left Two or Five or Seven off the list. Also no one will wonder why I would include forgettable 671 on the list. Do the Math. 671 was clearly a factor in 2013. Three of the factors of 2013 were also prime factors. They were 3, 11, and 61. This graphic clearly shows those prime factors.

2013 tree

2013 also had 8 negative factors. The first negative factor on the list is no surprise: Minus One. Year in and year out we can count on Minus One being a negative factor. Some other factors were just as negative in 2013, namely -3, -11, -33, – 61, -183, -671, and -2013. Of course, many of those factors were so obscure that most people never gave them a second thought all year long. Again I expect no arguments or negative comments on these selections. Anyone who knows anything about factors will have to agree with this list.

Even though 2014 hasn’t even started, I am going to predict the factors of 2014, and I am absolutely positive that my predictions will be 100% correct. You will not even have to wait until the end of 2014 to verify my accuracy.

The positive factors of 2014 will be (drum roll) 1, 2, 19, 38, 53, 106,1007, and 2014.

Most people expect the number One to be a positive factor every single year, and it will not let us down in 2014. The number Two has a reputation of being a factor only about half the time. Since she was not a factor at all in 2013, I am confident that she will get her act together again in 2014 and become a factor once more. All the other factors I’ve listed have not been factors for a very long time, and each one of them is due to make a difference over and over again in 2014 until they have nothing leftover. I predict that 2014 will have three prime factors, namely 2, 19, and 53, as illustrated in the following graphic.

2014 tree

How can I make such accurate assessments and spot on predictions? I will tell you: I work with factors almost every single day, and I’ve spent years observing them. Every time I have been given an assignment to become acquainted with them, I have approached that assignment with enthusiasm and determination.

Regardless of my astounding record, YOU can become just as much an expert as I am with just a little bit of knowledge and effort. You may discover, as I have, that factoring can be great fun. Here are a couple of logic puzzles that require factoring to solve: 

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All you have to do to solve one of the puzzles is 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. Each puzzle has only one solution.

At the top of this post is a page titled How to Find the Factors, and it gives hints to solve the puzzles.   Click 12 Factors 2013-12-30 to find a printable version of these and a few other puzzles as well as the solutions for last week’s puzzles. Excel or comparable spreadsheet program is needed to open the file.

Have a great 2014 and happy factoring!

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20 Underestimation

20 is a composite number. 20 = 1 x 20, 2 x 10, or 4 x 5. Factors of 20: 1, 2, 4, 5, 10, 20. Prime factorization: 20 = 2 x 2 x 5, which can also be written 20 = 2² x 5.

When 20 is a clue in the FIND THE FACTORS puzzles, use either 4 x 5 or 2 x 10. Only one set of those factors will work for any particular puzzle.

Estimating is a very important skill in mathematics, and I have always done well on any homework or test question involving that skill. However, I have never been very good at estimating real life – like how much time it will take me to do something.

Having never blogged before writing this one, I seriously underestimated how much time each puzzle would take me to create and test, and how much time each post would take me to write. Even though I have a large number of level 4, 5, and 6 puzzles already created, I still like to test them all again before I publish them, and that takes time.

Like most people, I work to make a living, and I have only a limited amount of free time each day. There are also other things that I ought to do or want to do that I haven’t given sufficient time in the last two months.

Because I underestimated my time and overestimated other things, I’ve decided to downsize and only publish posts on Mondays. Therefore there will not be any new puzzles today. Here is the easiest puzzle and the hardest puzzle from last week:

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To solve them, place 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. Click 12 Factors 2013-12-26 to see the solutions for all of the puzzles from last Thursday. Excel or similar spreadsheet program is needed to open the file. This Monday I will publish more 12 Factor puzzles followed by 10 factor puzzles the following week.

There are much worse things to underestimate:

19 Last-Minute Gift

19 is a prime number. 19 = 1 x 19. Its only factors are 1 and 19. Prime factorization: none.

How do we know that 19 is a prime number? The square root of 19 is an irrational number approximately equal to 4.36. If 19 were not a prime number, then it would be divisible by at least one prime number less than or equal to 4.36. Since 19 is not divisible by 2 or 3, it is a prime number.

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

It’s Christmas Eve or even Christmas day, and maybe all of your shopping didn’t get done. Maybe you didn’t want to drive anyplace because of bad weather, or your favorite stores were closed early for the holiday. Well, if someone on your list likes number placing puzzles (like Sudoku or Kakuro), then I have a last-minute gift idea for you, and it’s free. I design a number placing logic puzzle based on the multiplication table called FIND THE FACTORS. If you have a computer, the internet, and a printer, you can print a little holiday booklet filled with these puzzles and give it as a gift. If the person on your gift list is many miles away, you can even send the booklet electronically. This last minute gift is good for the brain and can be good for the memory. The level 1 and level 2 puzzles can be solved by children 3rd grade and up, but most of the higher level puzzles will be challenging for everyone regardless of age. 

Here is puzzle created to look a little like an angel just for the holidays:

2nd angel

To solve the puzzle above simply write the numbers 1 – 12 in the top row and also in the first column so that those numbers are the factors of the given clues. Okay, maybe it isn’t quite that simple. You have to know basic multiplication facts and use logic to figure out where the numbers go, and yes, I may try to trick you. But you and the people on your gift list have enough skills and persistence to find the one and only correct solution.

Now glancing at the puzzle above you may think you know all the answers, but…

This is what the solved puzzle looks like. Some of those factors may surprise you. That is why using logic is so important when solving the puzzles. (Once the factors are found, filling out the rest of the table is optional.)

angel factors found

Click 2013 Factor Holiday to download a copy of the puzzle booklet. Some of the puzzles in the booklet are a little easier than the one above because they are a lower level or they only use factors up to 10. Have a very Merry Christmas and a Happy New Year!

Related articles with other ideas for last-minute gifts:

18 Up on the Housetop

18 is a composite number. 18 = 1 x 18, 2 x 9, or 3 x 6. Factors of 18: 1, 2, 3, 6, 9, 18. Prime factorization: 18 = 2 x 3 x 3, which can also be written 18 = 2 x 3².

When 18 is a clue in the FIND THE FACTORS puzzles, use either 2 x 9 or 3 x 6. Only one set of factors will work for any particular puzzle.

Up on the housetop reindeer pause
Out jumps good old Santa Claus
Down through the chimney with lots of toys
All for the little ones, Christmas joys.

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Up on the housetop – to solve these puzzles will require you to have knowledge of the multiplication facts and just a little imagination to see the housetops (and maybe the ability to tilt your head for most of them.) Place the numbers 1 – 10 in both the top row and the first column so that those numbers are the factors of the given clues. Click 10 Factors 2013-12-23 to print these puzzles or see last week’s solutions. In order to view the file, you need excel or other comparable spreadsheet program.

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17 Christmas Angels

17 is a prime number. 17 = 1 x 17. Its only factors are 1 and 17. Prime factorization: none.

How do we know that 17 is a prime number? The square root of 17 is an irrational number approximately equal to 4.12. If 17 were not a prime number, then it would be divisible by at least one prime number less than or equal to 4.12. Since 17 is not divisible by 2 or 3, it is a prime number.

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

Many Christmas trees in the United States have been up and decorated for weeks. Some of them have a beautiful angel on the top to remind us of the angel that visited the shepherds. In Hungary, the angel is remembered in a different way. There the Christmas tree is put up on Christmas Eve. Tradition says that angels are the ones who decorate the tree with the delicious candies called szaloncukor. The candies are wrapped in specially prepared white tissue and fastened to the tree with white yarn. See the related articles at the end of the post for more information about this fascinating tradition.

The angel puzzles that I’ve made for this post have a few extra clues so they will be easier to solve. The first level 5 puzzle even has many of the same clues as the level 4 puzzle. Nevertheless, be careful because each level 5 angel has a few tricks up her sleeve. Still if you can write the numbers 1 to 12 in both the top row and the first column so that those numbers are the factors of the given clues, then you’ve solved the puzzle. There is only one solution to each puzzle. Click 12 Factors 2013-12-19 for a printable version of these and a few other puzzles.

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Hungary:

United States:

16 Silver Bells

16 is a composite number, and it is 4 squared. 16 = 1 x 16, 2 x 8, or 4 x 4. Factors of 16: 1, 2, 4, 8, 16. Prime factorization: 16 = 2 x 2 x 2 x 2, which can also be written 16 = 2⁴.

Since √16 = 4, a whole number, 16 is a perfect square.

When 16 is a clue in the FIND THE FACTORS puzzles, use either 2 x 8 or 4 x 4. Only one of those sets of factors will work for any particular puzzle.

Silver bells, silver bells.
It’s Christmas time in the city.

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Find the Factors is a type of logic puzzle. To solve one of the above puzzles, place the numbers 1 – 10 in both the top row and in the first column so that those numbers are factors of the given clues. For each puzzle there is only one solution. Click on 10 Factors 2013-12-16 to find these and a few more puzzles, as well as last Monday’s solutions.

Some of these Related articles have the lyrics or soundtrack to Silver Bells:

15 is the Magic Sum of a 3 x 3 Magic Square

15 is a composite number. 15 = 1 x 15 or 3 x 5. Factors of 15: 1, 3, 5, 15. Prime factorization: 15 = 3 x 5.

When 15 is a clue in the FIND THE FACTORS 1 – 10 or 1 – 12 puzzles, use 3 and 5 as the factors.

If you added the first nine counting numbers together, what sum would you get? What is 1 + 2 +3 + 4+ 5 + 6 + 7 + 8 + 9?

Would you get the same answer by adding (1 + 9) + (2 + 8) + (3 +7) + (4 + 6) + 5?

These are two of the many fun questions you can explore when you try to make a magic square. What is a magic square? If you can place the numbers from 1 to 9 in the box below so that the sum of any row, column, or diagonal will equal the sum of any other row, column, or diagonal, then you will have made a 3 x 3 magic square. The sum of a row, column, or diagonal in a magic square is called the magic sum.

1-9

Clearly it is not a magic square yet. In fact, only one of the numbers is positioned where it needs to be. Which number do you think is already in the correct position?

When it becomes a magic square, what will the magic sum be? One student noticed that in its current state the sums of the rows are 6, 15, and 24. The sums of the columns are 12, 15, 18. The sums of the diagonals are 15 and 15. Since 15 occurs most often, could the magic sum be 15? One way to determine what the magic sum should be is to add the sums of all three rows and then divide by the number of rows. Since 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 and 45 ÷ 3 = 15, then 15 is indeed the magic sum.

Here are a few easy-to-remember steps to construct a 3 x 3 magic square quickly.

Step 1: Draw a tic-tac-toe board and put 5 in the middle.

step 1 magic

Step 2: Put one of the even numbers in one of the corners.  You have four different choices, 2, 4, 6, or 8. The illustration is for the number 2, but any of the even numbers will work.

step 2 magic

Step 3: Subtract your even number from 10 to find its partner. 4 + 6 are partners and so are 2 + 8. Put the partner of the number you chose for step 1 in the corner that is diagonal to it.

step 1 magic

Step 4: Put the other two even numbers in the remaining corners. Yes, you have two choices where to put the numbers. Either choice will work.

step 4 magic

Step 5: Since 6 + 8 = 14 and 15 – 14 = 1, put 1 in the cell between the 6 and the 8. Do similar addition and subtraction problems on each side of the square to determine where to place the 3, 7, and 9. You can work clockwise or counter clockwise, or skip around the square doing the addition and subtraction problems; it doesn’t matter.

This finished magic square looks like this:

step 5 magic

Check it out! Every row, column, and diagonal adds up to 15!

As we created the square, we made choices. First we chose between 4 even numbers, and later we had 2 more choices. Notice that 4 x 2 = 8. There are 8 different ways to make a 3 x 3 magic square! (However, they are all really the same square turned upside down, rolled on its side, viewed from the back. etc.)

There are 880 different ways to make a 4 x 4 magic square. Look over the related articles at the end of this post to learn more about magic squares that are bigger than 3 x 3.

Speaking of magic squares, when I look at the square logic puzzle below, something magical happens. This puzzle has nine clues in it, and all of them are perfect squares. I can use those nine clues to construct a complete multiplication table. If you finish the same puzzle, your multiplication table will look exactly like mine because this puzzle has only one solution.

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The level 3 puzzle below is only a little bit more difficult. To solve it place the numbers 1 – 10 in the top row and again in the first column so that those placed numbers are the factors of the given clues. Again there is only one solution, and you will need to use logic to find it. Click 10 Factors 2014-01-06 for more puzzles and last week’s answers.

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May we all find a little bit more magic in our lives!

 

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