Sum-Difference

150 and Level 3

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Today’s Puzzle:

2014-24 Level 3

Factors of 150:

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-16

150 is a composite number.
Factor pairs:150 = 1 x 150, 2 x 75, or 3 x 50, 5 x 30, 6 x 25, or 10 x 15
Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
Prime factorization: 150 = 2 x 3 x 5 x 5, which can also be written 150 = 2 x 3 x (5^2)

Sum-Difference Puzzle:

6 has two factor pairs. One of those factor pairs adds up to 5, and the other one subtracts to 5. Can you place those factors in the proper boxes to complete the first puzzle?

150 has six factor pairs. One of those factor pairs adds up to 25, and another one subtracts to 25. If you can identify those factor pairs, then you can complete the second puzzle!

The second puzzle is really just the first puzzle in disguise.  Why would I say that?

More About the Number 150:

I verified this for myself:

A Logical Way to Solve Today’s Puzzle:

A Logical Approach to FIND THE FACTORS: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

2014-24 Level 3 Factors

 

120 and Level 5

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Today’s Puzzle:

Write the numbers from 1 to 10 in both the top row and the first column so that this puzzle functions like a multiplication table.

2014-19 Level 5

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

Thinking process using divisibility rules to find the factor pairs of 120:

√120 is irrational and approximately equal to 10.95. Every factor pair of 120 will have one factor less than 10.95 and one factor greater than 10.95, and we will find both factors in each pair at the same time. The following numbers are less than 10.95. Are they factors of 120?

  1. Yes, all whole numbers are divisible by 1, so 1 x 120 = 120.
  2. Yes, 120 is an even number. 120 ÷ 2 = 60, so 2 x 60 = 120. (Since 60 is even, 4 will also be a factor of 120.)
  3. Yes, 1 + 2 + 0 = 3 which is divisible by 3 (but not by 9), so 120 is divisible by 3. 120 ÷ 3 = 40, so 3 x 40 = 120. Note 120 will not be divisible by 9.
  4. Yes, the number formed from its last two digits, 20, is divisible by 4, so 120 is divisible by 4, and 4 x 30 = 120. (Note since 30 is even, 8 will also be a factor of 120.)
  5. Yes, the last digit is 0 or 5, so 120 is divisible by 5, and 5 x 24 = 120.
  6. Yes, 120 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 20 = 120.
  7. No. The divisibility trick for 7 requires us to split 120 into 12 and 0. We double 0 and subtract the double from 12. 12 – (2 x 0) = 12 – 0 = 12. Since 12 is not divisible by 7, 120 also is not divisible by 7.
  8. Yes, see 4 above. 120 = 8 x 15. (This will mean that ANY number whose last 3 digits are 120 will also be divisible by 8.)
  9. No, see 3 above. 120 is not divisible by 9.
  10. Yes, 120 ends with a zero, so 10 is a factor of 120, and 10 x 12 = 120.

From this thinking process we conclude that the factor pairs of 120 are 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, and 10 x 12.

Factors of 120:

120  is a composite number. 120 = 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, or 10 x 12. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Prime factorization: 120 = 2 x 2 x 2 x 3 x 5, which can also be written 120 = 2³ x 3 x 5.

When 120 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 12 as the factors.

Sum-Difference Puzzle:

30 has four factor pairs. One of those factor pairs adds up to 13, and another one subtracts to 13. Can you write those factors in their proper places in the first puzzle below?

120 has eight factor pairs. One of those factor pairs adds up to 26, and another one subtracts to 26. 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?

More about the Number 120:

120 = 5! because 1·2·3·4·5 = 120

120 is also the smallest positive multiple of 6 that is neither preceded nor followed by a prime number! (119 = 7 ×1 7, and 121 = 11 × 11, so neither one is prime.)

What kind of shape is 120 in?

  • 120 is the 15th triangular number because 15(16)/2 = 120,
    it’s the 8th tetrahedral number because (8)(9)(10)/6 = 120 (That means
  • 120 is the sum of the first eight triangular numbers), and
  • it is the 8th hexagonal number because (8)(2·8-1) = 120.

120 is the hypotenuse of a Pythagorean triple:
72-96-120, which is 24 times (3-4-5).

A Logical Way to Find the Solution to Today’s Puzzle:

2014-19 Level 5 Logic

96 and Level 1

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the top row and the first column so that this puzzle will function like a multiplication table:

2014-16 Level 1

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

Factors of 96:

96 is a composite number. 96 = 1 x 96, 2 x 48, 3 x 32, 4 x 24, 6 x 16, or 8 x 12. Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96. Prime factorization: 96 = 2 x 2 x 2 x 2 x 2 x 3, which can also be written as 2⁵ x 3.

Sometime 96 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has many other factors, we only use 8 x 12 in the puzzles.

Sum-Difference Puzzle:

6 has two factor pairs. One of those factor pairs adds up to 5 and the other one subtracts to 5. Can you place its factors in the correct boxes in the first puzzle?

96 has six factor pairs. One of those factor pairs adds up to 20, and the another subtracts to 20. If you can identify those two factor pairs, then you can solve the second puzzle!

The second puzzle really is just the first puzzle in disguise. Why would I say that?

Solution to This Week’s Puzzle:

2014-16 Level 1 Factors

84 and Level 2

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that this puzzle functions like a multiplication table.

2014-14 Level 2

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-07

Factors of 84:

84 is a composite number. 84 = 1 x 84, 2 x 42, 3 x 28, 4 x 21, 6 x 14, or 7 x 12. Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Prime factorization: 84 = 2 x 2 x 3 x 7 which can also be written 2² x 3 x 7.

Sometimes 84 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 84 = 7 x 12 in the puzzles.

Sum-Difference Puzzle:

84 has six factor pairs. One of those factor pairs adds up to 25, and another one subtracts to 25. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 84:

84 is prominent in this cool pattern:

Answer to This Week’s Puzzle:

2014-14 Level 2 Factors

60 and Pay No Attention to That Man Behind the Curtain

/ ivasallay / 4 Comments

Today’s Puzzle:

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.

A Peek at the Possible Factors for Each of Last Week’s Clues:

“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 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: 2014-04 Level 6 Answer

Factors of 60:

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.

Sum-Difference Puzzle:

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:

 

54 and How Many Squares Are in This Puzzle?

/ ivasallay / 2 Comments

How Many Squares?

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. 2014-04 Level 6 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.

Factors of 54:

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.

Sum-Difference Puzzle:

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?

Solution to Last Week’s Level 6 Puzzle:

2014-03 Level 6 Answer

Here is one way those factors can be found using logic.

2014-03 L-6 steps

30 and Pieces of Pi

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Pieces of Pi:

Haha. There’s no easy way to cut a pie into seven equal slices. . .

But you can find beautiful representations of π at Paula Krieg’s blog. It is a bookbinding blog with just the right artistic touch for pi expressed as π, a fraction (2 ways), or a decimal!

Factors of 30:

30 is a composite number. 30 = 1 x 30, 2 x 15, 3 x 10, or 5 x 6. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Prime factorization: 30 = 2 x 3 x 5.

When 30 is a clue in the FIND THE FACTORS puzzle, use 3 x 10 or 5 x 6. Only one of those pairs will work for that clue in that particular puzzle.

Sum-Difference Puzzle:

30 has four factor pairs. One of those pairs adds up to 13, and another one subtracts to give us 13. If you can find those factor pairs, then you can solve this puzzle!