Level 6 Puzzle

766 and Level 6

/ ivasallay / Leave a comment
  • 766 is a composite number.
  • Prime factorization: 766 = 2 x 383
  • 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 766 has exactly 4 factors.
  • Factors of 766: 1, 2, 383, 766
  • Factor pairs: 766 = 1 x 766 or 2 x 383
  • 766 has no square factors that allow its square root to be simplified. √766 ≈ 27.6767.

766-factor-pairs

Here’s a level 6 puzzle for you to try:

766 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here are a few more reasons to be interested in the number 766:

766 is the sum of the twelve prime numbers from 41 to 89.

766 is palindrome 23332 in BASE 4; note that 2(256) + 3(64) + 3(16) + 3(4) + 2(1) = 766.

766 is the sum of three squares six different ways:

  • 27² + 6² + 1² = 766
  • 26² + 9² + 3² = 766
  • 21² + 18² + 1² = 766
  • 21² + 17² + 6² = 766
  • 21² + 15² + 10² = 766
  • 19² + 18² + 9² = 766

766 is also the 18th centered pentagonal number because (5⋅17² + 5⋅17 + 2)/2 = 766.

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This table shows a way to solve puzzle #766 using logic:

766 Logic

759 and Level 6

/ ivasallay / Leave a comment
  • 759 is a composite number.
  • Prime factorization: 759 = 3 x 11 x 23
  • 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 759 has exactly 8 factors.
  • Factors of 759: 1, 3, 11, 23, 33, 69, 253, 759
  • Factor pairs: 759 = 1 x 759, 3 x 253, 11 x 69, or 23 x 33
  • 759 has no square factors that allow its square root to be simplified. √759 ≈ 27.5499546.

759-factor-pairs

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

759 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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What else can I say about the number 759?

759 can be written as the sum of consecutive numbers seven different ways:

  • 379 + 380 = 759; that’s 2 consecutive numbers.
  • 252 + 253 + 254 = 759; that’s 3 consecutive numbers.
  • 124 + 125 + 126 + 127 + 128 + 129 = 759; that’s 6 consecutive numbers.
  • 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 = 759; that’s 11 consecutive numbers.
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45  = 759; that’s 22 consecutive numbers.
  • 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44  = 759; that’s 23 consecutive numbers.
  • 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 + 39 = 759; that’s 33 consecutive numbers.

759 is also the sum of five consecutive prime numbers: 139 + 149 + 151 + 157 + 163 = 759.

759 is a palindrome in two bases:

  • 3C3 BASE 14 (C is 12 base 10)
  • NN BASE 32 (N is 23 base 10)

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759 Logic

758 and Level 5

/ ivasallay / Leave a comment
  • 758 is a composite number.
  • Prime factorization: 758 = 2 x 379
  • 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 758 has exactly 4 factors.
  • Factors of 758: 1, 2, 379, 758
  • Factor pairs: 758 = 1 x 758 or 2 x 379
  • 758 has no square factors that allow its square root to be simplified. √758 ≈ 27.5317998.

758-factor-pairs

I started to feel like I was getting behind blogging so I decided over the weekend that it is okay NOT to blog everyday. I didn’t have time to post this puzzle until today, and I have a level 6 puzzle and another whole set of puzzles to post as well. I decided I don’t need to catch up. I’ll post them all soon enough. A logical way to find the solution is in a table at the end of the post.

758 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s more about the number 758:

758 is the sum of four consecutive numbers: 188 + 189 + 190 + 191 = 758.

758 is the sum of three squares six different ways:

  • 27² + 5² + 2² = 758
  • 26² + 9² + 1² = 756
  • 23² + 15² + 2² = 758
  • 22² + 15² + 7² = 758
  • 21² + 14² + 11² = 758
  • 19² + 19² + 6² = 758

758 is a palindrome in two different bases:

  • 464 BASE 13; note that 4(13²) + 6(13) + 4(1) = 758.
  • 262 BASE 18; note that 2(18²) + 6(18) + 2(1) = 758.

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758 Logic

 

 

752 and Level 6

/ ivasallay / Leave a comment
  • 752 is a composite number.
  • Prime factorization: 752 = 2 x 2 x 2 x 2 x 47, which can be written 752 = (2^4) x 47
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 752 has exactly 10 factors.
  • Factors of 752: 1, 2, 4, 8, 16, 47, 94, 188, 376, 752
  • Factor pairs: 752 = 1 x 752, 2 x 376, 4 x 188, 8 x 94, or 16 x 47
  • Taking the factor pair with the largest square number factor, we get √752 = (√16)(√47) = 4√47 ≈ 27.422618.

752-factor-pairs

Here’s today’s puzzle. A logical way to solve it is given in the table at the end of the post.

752 Puzzle

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

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Here’s a few more thoughts about the number 752:

52 is divisible by 4 so 752 is also divisible by 4. However, 52 is not also divisible by 8, but since 7 is odd, 752 IS divisible by 8.

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 + 39 = 752; that’s 32 consecutive numbers.

752 is the sum of two consecutive primes: 373 + 379 = 752.

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752 Logic

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

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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

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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

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.

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744 Logic

737 and Level 6

/ ivasallay / 3 Comments
  • 737 is a composite number.
  • Prime factorization: 737 = 11 x 67
  • 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 737 has exactly 4 factors.
  • Factors of 737: 1, 11, 67, 737
  • Factor pairs: 737 = 1 x 737 or 11 x 67
  • 737 has no square factors that allow its square root to be simplified. √737 ≈ 27.14774392.

737-factor-pairs

Here is a challenging level 6 puzzle. Some possible steps to solve it are in a table at the end of this post.

737 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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The Boeing 737 is the best selling commercial airline jet. It has been produced continuously since 1967. If you’ve ever flown Southwest Airlines, Ryanair, United Airlines, or American Airlines, chances are the 737 took you to your destination.

The next time you fly in a 737, use the following number facts to impress your seatmates. They will be so glad they sat next to you: 😉

7 – 3 + 7 = 11 so 737 is divisible by 11.

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

  • 368 + 369 = 737; that’s 2 consecutive numbers.
  • 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 = 737; that’s 11 consecutive numbers.
  • 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 = 737; that’s 22 consecutive numbers.

737 is obviously a palindrome, but base 10 is not the only base that can make that claim:

  • 737 base 10; note that 7(100) + 3(10) + 7(1) = 737.
  • 515 BASE 12; note that 5(144) + 1(12) + 5(1) = 737.
  • 191 BASE 23; note that 1(529) + 9(23) + 1(1) = 737.

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737 Logic

731 and Level 6

/ ivasallay / Leave a comment
  • 731 is a composite number.
  • Prime factorization: 731 = 17 x 43
  • 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 731 has exactly 4 factors.
  • Factors of 731: 1, 17, 43, 731
  • Factor pairs: 731 = 1 x 731 or 17 x 43
  • 731 has no square factors that allow its square root to be simplified. √731 ≈ 27.03711669.

Here is a Level 6 puzzle for you try:

731 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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731 can be written as the sum of consecutive numbers 3 different ways:

  • 365 + 366 = 731; that’s 2 consecutive numbers.
  • 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 = 731; that’s 17 consecutive numbers.
  • 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 = 731; that’s 34 consecutive numbers.

Because 17 is one of its factors,  731 is the hypotenuse of Pythagorean triple 344-645-731 making 344² + 645² = 731².

239 + 241 + 251 = 731 so 731 is the sum of 3 consecutive prime numbers.

731 is palindrome 3A3 in BASE 14; note that 3(14²) + 10(14) + 3(1) = 731.

731 is also 123 in BASE 26; note that 1(26²) + 2(26) + 3(1) = 731.

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731 Logic

Thanks to Ricardo whose tweet I include here:

//platform.twitter.com/widgets.js

723 and Level 6

/ ivasallay / Leave a comment

7 + 2 + 3 = 12, a multiple of 3, so 723 is divisible by 3.

  • 723 is a composite number.
  • Prime factorization: 723 = 3 x 241
  • 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 723 has exactly 4 factors.
  • Factors of 723: 1, 3, 241, 723
  • Factor pairs: 723 = 1 x 723 or 3 x 241
  • 723 has no square factors that allow its square root to be simplified. √723 ≈ 26.88866.

Finding the factors to solve this Level 6 puzzle will be more challenging than that. . .

 

723 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here is a little more about the number 723:

723 is the sum of consecutive numbers three different ways:

  • 361 + 362 = 723; that’s 2 consecutive numbers.
  • 240 + 241 + 242 = 723; that’s 3 consecutive numbers.
  • 118 + 119 + 120 + 121 + 122 + 123 = 723; that’s 6 consecutive numbers.

Because 241 is one of its factors, 723 is the hypotenuse of the Pythagorean triple 360-627-723. What is the greatest common factor of those three numbers? (It is one of the four factors of 723.)

723 is repdigit and palindrome 333 in BASE 15; note that 3(15²) + 3(15) + 3(1) = 723.

OEIS.org informs us that  (1!)! + (2!)! + (3!)! = 723. That looks more exciting than it really is. It is only equivalent to saying 1 + 2 + 720 = 723.

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723 Logic

Thank you, Ricardo, for completing the puzzle and tweeting your work:

//platform.twitter.com/widgets.js

716 and Level 6

/ ivasallay / Leave a comment
  • 716 is a composite number.
  • Prime factorization: 716 = 2 x 2 x 179, which can be written 716 = (2^2) x 179
  • 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 716 has exactly 6 factors.
  • Factors of 716: 1, 2, 4, 179, 358, 716
  • Factor pairs: 716 = 1 x 716, 2 x 358, or 4 x 179
  • Taking the factor pair with the largest square number factor, we get √716 = (√4)(√179) = 2√179 ≈ 26.758176.

You may find this Candy Cane puzzle more challenging than sweet:

 

716 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are two more facts about the number 716:

86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 = 716; that’s 8 consecutive numbers.

716 is palindrome 282 in BASE 17; note that 2(17²) + 8(17) + 2(1) = 716

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716 Logic