A Multiplication Based Logic Puzzle

Archive for February, 2014

65 is the Magic Sum of a 5 x 5 Magic Square

65 is a composite number. 65 = 1 x 65 or 5 x 13. Factors of 65: 1, 5, 13, 65. Prime factorization: 65 = 5 x 13.

65 – 56 = 9, a square number and 65+56 = 121, another square number. Stetson.edu informs us that 65 and 56 are the smallest pair of mirror image numbers that produce a square number when either added or subtracted.

65 = (1 + 2 + 3 +. . . + 23 + 24 + 25)/5 so it is the magic sum of a 5 × 5 magic square. I solved this 5×5 magic square game using a time honored algorithm. Notice that 11, 12, 13, 14, and 15 are on a diagonal. Actually all of the numbers are sort of on a diagonal if you follow the instructions given here. Also notice that the first number, 1, is in the center of the top row and the last number, 25, is in the center of the bottom row.

You can try to solve it, too. You will know you have solved it when the sum of each row, column, and diagonal totals 65.

65 is also the sum of two squares two different ways:

  • 8² + 1² = 65
  • 7² + 4² = 65

That makes 65 the hypotenuse of FOUR Pythagorean triples:

  • 16-63-65 which was calculated from 2(8)(1), 8² – 1², 8² + 1²
  • 25-60-65 which is 5 times 5-12-13
  • 33-56-65 which was calculated from 7² – 4², 2(7)(4), 7² + 4²
  • 39-52-65 which is 13 times 3-4-5

 

 

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64 and Can You See the Logic Here?

64 is a composite number, and it is 8 squared. 64 = 1 x 64, 2 x 32, 4 x 16, or 8 x 8. Factors of 64: 1, 2, 4, 8, 16, 32, 64. Prime factorization: 64 = 2 x 2 x 2 x 2 x 2 x 2, which can also be written 64 = 2⁶.

Since √64 = 8, a whole number, 64 is a perfect square.

Sometimes 64 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, when it is a clue, the factors are always 8 x 8.

2014-08 Level 6

All Previously Published Puzzles

2014-08 Level 6 Answer

This solution is complicated, but here is a logical explanation:

There are only two 3s available, so at most one 9 can be 3 x 3.

Can both 9s be 9 x 1? If both were, then both 1s would be used, leaving 8 = 2 x 4 and 10 = 2 x 5 as the only possibilities for 8 and 10. But if both 9s were 9 x 1, then the 18 in the same row as the 9 would have to be 18 = 9 x 2, and that would use too many twos. Therefore, EXACTLY ONE of the nines can use 9 = 9 x 1.

Since one 9 = 3 x 3, using both 3s, 30 must equal 5 x 6. And since both 30 and 50 must use both 5s, 10 must equal 10 x 1.

Since 10 = 10 x 1 and a 9 = 9 x 1, 8 must equal 2 x 4. So now finally we can begin filling out the puzzle:

2014-08 Level 6 Logic

63 and Twelve Quick Clues

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.

2014-07 Level 6

All Previously Published Puzzles

2014-07 steps

62 and Love: Challenging But Worth It

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.

valentine 2014

All Previously Published Puzzles

square root mems

 

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).

Valentine 2014 steps

 

61 and Opportunity to Catch Up

  • 61 is a prime number.
  • Prime factorization: 61 is prime.
  • The exponent of prime number 61 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 61 has exactly 2 factors.
  • Factors of 61: 1, 61
  • Factor pairs: 61 = 1 x 61
  • 61 has no square factors that allow its square root to be simplified. √61 ≈ 7.8102

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.

2014-06 Level 4

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 puzzles in one excel file here. 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.) The file does not contain the factors for today’s puzzle.

2014-06 Level 4 Answer

The FIND THE FACTORS 1-12 puzzles were not included, but sometime in the next few weeks, I might compile them into one file and put them on the blog. (If enough people request it, I will definitely do it.)

While I catch up on all the things I need to do, I hope you will catch up a few things as well.

Image

60 and Pay No Attention to That Man Behind the Curtain

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.

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!

2014-05 Level 6

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: 2014-04 Level 6 Hint 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 used for the clue 8. Which means that 2 x 5, not 1 x 10, must be used for the clue 10 because both ones will have been used. Which 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: 2014-04 Level 6 Answer

59 and Valentine Gif

  • 59 is a prime number.
  • Prime factorization: 59 is prime.
  • The exponent of prime number 59 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 59 has exactly 2 factors.
  • Factors of 59: 1, 59
  • Factor pairs: 59 = 1 x 59
  • 59 has no square factors that allow its square root to be simplified. √59 ≈ 7.6811

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 Puzzle 2014

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.

2014-04 Level 5 Answer

2014-04 L-5 steps

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