Month: January 2016

747 Happy Birthday, Steve Morris!

/ ivasallay / 21 Comments
  • 747 is a composite number.
  • Prime factorization: 747 = 3 x 3 x 83, which can be written 747 = (3^2) x 83
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 747 has exactly 6 factors.
  • Factors of 747: 1, 3, 9, 83, 249, 747
  • Factor pairs: 747 = 1 x 747, 3 x 249, or 9 x 83
  • Taking the factor pair with the largest square number factor, we get √747 = (√9)(√83) = 3√83 ≈ 27.3313007.

747-factor-pairs

It’s Steve Morris’s birthday so I thought I’d make him a cake, but no regular size cake will do. He has been one of my earliest supporters, and I know that sometimes even a level 6 puzzle is just too easy for him. Once he sent out this tweet:

//platform.twitter.com/widgets.js

Admittedly that puzzle was easier than most level 6’s, but recently he tweeted me a puzzle that I have had to start over more than once and still haven’t conquered:

//platform.twitter.com/widgets.js

I think Steve Morris is due for an extra difficult Find the Factors puzzle for his birthday, one that all the numbers from 1 to 16 can be the factors. I’ll wait at least a week before I give any hints to complete it, too. As always, there is only one solution, but it can be found using logic.

747 Birthday Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

The possible factors for each clue is given below. Adding 14, 15, and 16 as possible factors really complicates the puzzle!

747 Puzzle Clues

—————————————

Now I’ll write a little about the number 747:

747 is a palindrome in base 10. Boeing’s most recognizable airplane also bares that number.

747 can be written as the sum of consecutive numbers five different ways:

  • 373 + 374 = 747; that’s 2 consecutive numbers
  • 248 + 249 + 250 = 747; that’s 3 consecutive numbers
  • 122 + 123 + 124 + 125 + 126 + 127 = 747; that’s 6 consecutive numbers
  • 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 = 747; that’s 9 consecutive numbers
  • 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 747; that’s 18 consecutive numbers

747 is the sum of three squares six different ways. Three of those ways repeat squares.

  • 27² + 3² + 3² = 747
  • 25² + 11² + 1² = 747
  • 23² + 13² + 7² = 747
  • 21² + 15² + 9² = 747
  • 19² + 19² + 5² = 747
  • 17² + 17² + 13² = 747

—————————————

Since today’s puzzle is the biggest puzzle I have ever published, it could be a little difficult just noticing that a factor had been duplicated in the top row or first column. Here is the tweet Steve Morris sent out once he finally solved the puzzle:

//platform.twitter.com/widgets.js

Now after waiting over a week, I now reveal one of the ways to solve this difficult puzzle logically:

747 birthday Logic

746 and Level 1

/ ivasallay / Leave a comment
  • 746 is a composite number.
  • Prime factorization: 746 = 2 x 373
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 746 has exactly 4 factors.
  • Factors of 746: 1, 2, 373, 746
  • Factor pairs: 746 = 1 x 746 or 2 x 373
  • 746 has no square factors that allow its square root to be simplified. √746 ≈ 27.313000567.

746-factor-pairs

Here’s today’s puzzle:

746 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

—————————————

Here’s a little more about the number 746:

746 is the sum of 4 consecutive numbers: 185 + 186 + 187 + 188 = 746.

746 is the hypotenuse of a Pythagorean triple triangle because 504² + 550² = 746².

746 is also the sum of three squares six different ways:

  •  27² + 4² + 1² = 746
  • 24² + 13² + 1² = 746
  • 24² + 11² + 7² = 746
  • 21² + 17² + 4² = 746
  • 21² + 16² + 7² = 746
  • 20² + 15² + 11² = 746

OEIS.org shared this fun number fact: 1^7 + 2^4 + 3^6 = 746.

—————————————

746 Factors

745 Pythagorean Triple Puzzle

/ ivasallay / 2 Comments
  • 745 is a composite number.
  • Prime factorization: 745 = 5 x 149
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 745 has exactly 4 factors.
  • Factors of 745: 1, 5, 149, 745
  • Factor pairs: 745 = 1 x 745 or 5 x 149
  • 745 has no square factors that allow its square root to be simplified. √745 ≈ 27.294688.

745-factor-pairs

PUZZLE DIRECTIONS: This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

Today’s Pythagorean triple puzzle has only 2 more triangles than last week’s puzzle, but it shouldn’t be any more difficult. Please, give it a try!

745 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

Sorted Triples

————————————-

Here are some fun facts about the number 745:

745 can be written as the sum of consecutive numbers three different ways:

  • 372 + 373 = 745; that’s 2 consecutive numbers
  • 147 + 148 + 149 + 150 + 151 = 745; that’s 5 consecutive numbers
  • 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 = 745; that’s 10 consecutive numbers

745 can also be written as the sum of the thirteen prime numbers from 31 to 83. See if you can name all those primes while you add them up.

745 is the sum of two squares two different ways:

  • 27² + 4² = 745
  • 24² + 13² = 745

Both of 745’s prime factors are hypotenuses of Pythagorean triples, so 745 is the hypotenuse of FOUR Pythagorean triples:

  • 216-713-745; calculated from 2(27)(4), 27² – 4², 27² + 4².
  • 255-700-745
  • 407-624-745; calculated from 2(24)(13), 24² – 13², 24² + 13².
  • 447-596-745

5 is the greatest common factor of one of the non-primitive triples while 149 is the greatest common factor of the other. Which is which?

If you check any of those triples, you will see that 745² is 555025, which is a cool looking number, too.

745 is also the sum of three squares three different ways:

  • 18² + 15² + 14² = 745
  • 22² + 15² + 6² = 745
  • 24² + 12² + 5² = 745

745 is a palindrome in three different bases, two of which are consecutive:

  • 454 BASE 13; note that 4(13²) + 5(13) + 4(1) = 745.
  • 3B3 BASE 14 (B= 11 base 10); note that 3(14²) + 11(14) + 3(1) = 745.
  • 171 BASE 24; note that 1(24²) + 7(24) + 1(1) = 745.

————————————-

 

744 Taiwan elects its first female president!

/ ivasallay / 3 Comments
  • 744 is a composite number.
  • Prime factorization: 744 = 2 x 2 x 2 x 3 x 31, which can be written 744 = (2^3) x 3 x 31
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 744 has exactly 16 factors.
  • Factors of 744: 1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744
  • Factor pairs: 744 = 1 x 744, 2 x 372, 3 x 248, 4 x 186, 6 x 124, 8 x 93, 12 x 62, or 24 x 31
  • Taking the factor pair with the largest square number factor, we get √744 = (√4)(√186) = 2√186 ≈ 27.27636339397171.

744-factor-pairs

Today the majority in Taiwan defied mainland China and affirmed their belief in democracy as they overwhelmingly elected their first female president, Tsai Ing-wen. CNN reported that “her supporters filled streets, waving party banners and cheering to victory announcements made from a stage.”

Democratic Progressive Party presidential candidate Tsai Ing-wen casts her ballot.

Taiwan’s first female president, Tsai Ing-wen; Source CNN

Apparently another way they celebrated was sharing my 689 post on Facebook over 4000 times and viewing that post 8346 times today. Since it was written, that post has been shared on Facebook over 7000 times, and today I recorded a record high 9017 views on my blog with 6289 views from Taiwan, 1122 from Hong Kong, 844 from the United States, and 33 views from China.

The Hong Kong Free Press gives a complete and up-to-date explanation in English of why 689 is the magic number and why so many people are interested in it. The 689 Coincidence explains it in Chinese.

Update: As mentioned in the comments, Tsai Ing-wen was elected with 6894744 votes. That’s an amazing coincidence! The election results not only repeated the number 689 but its last three digits are the same number as this post, 744. I now give some fun facts about that number:

744 has 16 factors, and the first 16 digits in its square root are 27.27636339397171. 

Square root 744

Here is today’s puzzle. A logical way to find its solution is given in the table at the end of the post.

 

744 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

Here’s a little more about the number 744:

744 can be written as the sum of consecutive numbers three different ways:

  • 247 + 248 + 249 = 744; that’s 3 consecutive numbers.
  • 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 = 744; that’s 16 consecutive numbers.
  • 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 744; that’s 31 consecutive numbers.

744 is also the sum of 4 consecutive prime numbers: 179 + 181 + 191 + 193 = 744.

744 can also be written as the sum of three square numbers three different ways:

  • 26² + 8² + 2² = 744
  • 22² + 16² + 2² = 744
  • 22² + 14² + 8² = 744

Notice that the squares of 8, 2, and 22 were each used two different times in those sums of squares. 26, 16, and 14 were each used once.

744 is also a palindrome in a couple other bases:

  • 2112 BASE 7; note that 2(7^3) + 1(7^2) + 1(7^1) + 2(7^0) = 744.
  • OO BASE 30 (O = 24); note that 24(30) + 24(1) = 744.

————————————-

744 Logic

743 and Level 5

/ ivasallay / Leave a comment
  • 743 is a prime number.
  • Prime factorization: 743 is prime.
  • The exponent of prime number 743 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 743 has exactly 2 factors.
  • Factors of 743: 1, 743
  • Factor pairs: 743 = 1 x 743
  • 743 has no square factors that allow its square root to be simplified. √743 ≈ 27.258026.

743-factor-pairs

How do we know that 743 is a prime number? If 743 were not a prime number, then it would be divisible by at least one prime number less than or equal to √743 ≈ 27.3. Since 743 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 743 is a prime number.

Here is today’s puzzle. It can be solved using logic as explained in the table at the end of the post.

743 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

————————————-

Here is just a little more about the number 743:

743 is prime so it can be written as the sum of consecutive numbers only one way: 371 + 372 = 743.

743 is a palindrome in two bases:

  • 616 BASE 11; note that 6(11²) + 1(11) + 6(1) = 743.
  • 212 BASE 19; note that 2(19²) + 1(19) + 2(1) = 743.

————————————-

743 Logic

742 and Level 4

/ ivasallay / Leave a comment
  • 742 is a composite number.
  • Prime factorization: 742 = 2 x 7 x 53
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 742 has exactly 8 factors.
  • Factors of 742: 1, 2, 7, 14, 53, 106, 371, 742
  • Factor pairs: 742 = 1 x 742, 2 x 371, 7 x 106, or 14 x 53
  • 742 has no square factors that allow its square root to be simplified. √742 ≈ 27.2396769.

742-factor-pairs

Here is a level 4 puzzle for you to try:

742 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

————————————-

Here’s more about composite number 742:

742 can be expressed as the sum of consecutive numbers three ways:

  • 184 + 185 + 186 + 187 = 742; that’s 4 consecutive numbers.
  • 103 + 104 + 105 + 106 + 107 + 108 + 109 = 742; that’s 7 consecutive numbers.
  • 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 742; that’s 28 consecutive numbers.

Because 53 is one of its prime factors, 742 is the hypotenuse of Pythagorean triple 392-630-742. The greatest common factor of three numbers in the triple is in the factor pair with 53.

742 is the sum of three squares two different ways:

  • 27² + 3² + 2² = 742
  • 25² + 9² + 6² = 742

————————————-

742 Logic

741 and Level 3

/ ivasallay / Leave a comment

Look at 741’s three digits: 4 = (1/2)(7 + 1) so 741 is divisible by 3.

  • 741 is a composite number.
  • Prime factorization: 741 = 3 x 13 x 19
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 741 has exactly 8 factors.
  • Factors of 741: 1, 3, 13, 19, 39, 57, 247, 741
  • Factor pairs: 741 = 1 x 741, 3 x 247, 13 x 57, or 19 x 39
  • 741 has no square factors that allow its square root to be simplified. √741 ≈ 27.221315.

741-factor-pairs

Here is today’s puzzle:

741 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

————————————-

Here are more 741 number facts:

19 x 39 = 741. Since 19 and 39 are both exactly 10 away from their average, 29, we know that 741 is exactly 100 away from 29² = 841.

19 x 39 = (38/2 )(38 + 1) so 741 is the 38th triangular number.

741 can be expressed as the sum of consecutive numbers seven ways:

  • 370 + 371 = 741; that’s 2 consecutive numbers.
  • 246 + 247 + 248 = 741; that’s 3 consecutive numbers.
  • 121 + 122 + 123 + 124 + 125 + 126 = 741; that’s 6 consecutive numbers.
  • 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 = 741; that’s 13 consecutive numbers.
  • 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 = 741; that’s 19 consecutive numbers.
  • 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 741; that’s 26 consecutive numbers.
  • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 741; that’s 38 consecutive numbers confirming that 741 is the 38th triangular number.

Since 13 is one of its factors, 741 is the hypotenuse of the Pythagorean triple 285-684-741. The greatest common factor of those 3 numbers can be found in the same factor pair as 13.

————————————-

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  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.

741 Factors

740 and Level 2

/ ivasallay / Leave a comment
  • 740 is a composite number.
  • Prime factorization: 740 = 2 x 2 x 5 x 37, which can be written 740 = (2^2) x 5 x 37
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 740 has exactly 12 factors.
  • Factors of 740: 1, 2, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740
  • Factor pairs: 740 = 1 x 740, 2 x 370, 4 x 185, 5 x 148, 10 x 74, or 20 x 37
  • Taking the factor pair with the largest square number factor, we get √740 = (√4)(√185) = 2√185 ≈ 27.202941.

740-factor-pairs

This level 2 puzzle isn’t too difficult to solve:

740 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

————————————-

Here’s more about 740:

740 is the sum of consecutive numbers several ways:

  • 146 + 147 + 148 + 149 + 150 = 740; that’s 5 consecutive numbers.
  • 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 = 740; that’s 8 consecutive numbers.
  • 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 740; that’s 37 consecutive numbers.

Because 5 and 37 are two of its factors, 740 is the hypotenuse of four Pythagorean triples:

  • 240-700-740
  • 228-704-740
  • 416-612-740
  • 444-592-740

740 is also a palindrome in two bases, one of which is double the other:

  • 252 BASE 18; note that 2(18²) + 5(18) + 2(1) = 740.
  • KK BASE 36 (K = 20 base 10); note that 20(36) + 20(1)

————————————-

740 Factors

739 and Level 1

/ ivasallay / Leave a comment

Today’s Puzzle:

739 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

Factors of 739:

  • 739 is a prime number.
  • Prime factorization: 739 is prime.
  • The exponent of prime number 739 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 739 has exactly 2 factors.
  • Factors of 739: 1, 739
  • Factor pairs: 739 = 1 x 739
  • 739 has no square factors that allow its square root to be simplified. √739 ≈ 27.1845544.

739-factor-pairs

How do we know that 739 is a prime number? If 739 were not a prime number, then it would be divisible by at least one prime number less than or equal to √739 ≈ 27.2. Since 739 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 739 is a prime number.

More about the Number 739:

739 is prime so there is only one way it can be written as the sum of consecutive numbers:

  • 369 + 370 = 739.

However, 739 can be written as the sum of 3 squares 3 different ways:

  • 27² + 3² + 1² = 739
  • 21² + 17² + 3² = 739
  • 17² + 15² + 15² = 739

739 is included in this list of prime numbers.

Factors for Today’s Puzzle:

739 Factors

738 Warning: These 3 right triangles do NOT form one large right triangle

/ ivasallay / 2 Comments
  • 738 is a composite number.
  • Prime factorization: 738 = 2 x 3 x 3 x 41, which can be written 738 = 2 x 3² x 41
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 738 has exactly 12 factors.
  • Factors of 738: 1, 2, 3, 6, 9, 18, 41, 82, 123, 246, 369, 738
  • Factor pairs: 738 = 1 x 738, 2 x 369, 3 x 246, 6 x 123, 9 x 82, or 18 x 41
  • Taking the factor pair with the largest square number factor, we get √738 = (√9)(√82) = 2√82 ≈ 27.166155.

738-factor-pairs

The puzzle below is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

Go ahead, give it a try!

738 Triple Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

Sorted Triples

Some time ago I published a rather ambitious Pythagorean triple logic puzzle. I didn’t post the answers but invited anyone who desired to post some or all of the answers and to do so in the comments. As of today, no one has posted any answers. Perhaps that puzzle was too difficult. I decided to post a SIMPLER one today. Just follow the instructions above the puzzle.

———————————————————

Now for some facts about the number 738:

Because 41 is one of its factors, 738 is the hypotenuse of Pythagorean triple 162-720-738. What is the greatest common factor of the three numbers in the triple? It’s the other number in the same factor pair as 41.

738 can be expressed as the sum of consecutive numbers four different ways:

  • 245 + 246 + 247 = 738; that’s 3 consecutive numbers.
  • 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 = 738; that’s 9 consecutive numbers.
  • 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 = 738; that’s 12 consecutive numbers.
  • 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38; that’s 36 consecutive numbers.

738 is a palindrome in base 15 and base 16, two consecutive bases:

  • 343 BASE 15; note that 3(225) + 4(15) + 3(1) = 738.
  • 2E2 BASE 16 (E = 14 base 10); note that 2(256) + 14(16) + 2(1) = 738.

From OEIS.org I learned that 6 + 66 + 666 = 738. Cool! Six 6’s were used!

———————————————————