A Multiplication Based Logic Puzzle

Posts tagged ‘centered triangular number’

901 and Level 4

Print the puzzles or type the solution on this excel file: 10-factors-897-904

901 is the 25th centered triangular number because (23×24 + 24×25 + 25×26)/2 = 901. That is the same as saying that 901 is the sum of the 23rd, the 24th, and the 25th triangular numbers.

901 is the sum of two squares two different ways:

  • 30² + 1² = 901
  • 26² + 15² = 901

901 is the hypotenuse of FOUR Pythagorean triples:

  • 60-899-901, calculated from 2(30)(1), 30² – 1², 30² + 1²
  • 424-795-901, which is (8-15-17) times 53
  • 451-780-901, calculated from 26² – 15², 2(26)(15), 26² + 15²
  • 476-765-901, which is 17 times (28-45-53)

Two of those were primitives. That can only happen because ALL of 901’s prime factors are Pythagorean triple hypotenuses.

  • 901 is a composite number.
  • Prime factorization: 901 = 17 × 53
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 901 has exactly 4 factors.
  • Factors of 901: 1, 17, 53, 901
  • Factor pairs: 901 = 1 × 901 or 17 × 53
  • 901 has no square factors that allow its square root to be simplified. √901 ≈ 30.016662

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694 and Level 6

  • 694 is a composite number.
  • Prime factorization: 694 = 2 x 347
  • 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 694 has exactly 4 factors.
  • Factors of 694: 1, 2, 347, 694
  • Factor pairs: 694 = 1 x 694 or 2 x 347
  • 694 has no square factors that allow its square root to be simplified. √694 ≈ 26.34388.

Level 6 puzzles aren’t easy, but do give it a try! There are hints at the bottom of the page.

694 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here are some ways that 694 is a special number:

694 is the 22nd centered triangular number because it is the sum of these consecutive triangular numbers 210, 231, 253.

Notice that 21(22)/2 is 231, the number in the middle of those consecutive triangular numbers. Also notice that 3(231) + 1 = 694.

694 BASE 10 becomes palindrome 1C1 in BASE 21 because C Base 21 is equivalent to 12 Base 10, and 1(21²) + 12(21) + 1(1) = 694.

 

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694 Logic

631 and Level 3

631 is a prime number that happens to also be the sum of the eleven prime numbers from 37 to 79.

Sometimes mathematicians talk about triangular numbers, centered triangular numbers, hexagonal numbers, and centered hexagonal numbers. Is there any relationship between those terms?

  • The sum of any three consecutive triangular numbers is a centered triangular number.
  • ALL hexagonal numbers are triangular numbers.
  • Most centered hexagonal numbers are NOT centered triangular numbers.

In my last post, I pointed out that 630 is a triangular number and a hexagonal number.

Well, amazingly the next number, 631, is a centered triangular number AND a centered hexagonal number.

How often does one more than a triangular number equal a centered triangular number? Also how often does one more than a hexagonal number equal a centered hexagonal number?

Let’s compare a couple of lists of these types of numbers up to 631. Red means a number is in both lists. Blue means the number in the 1st list is one less than the number in the 2nd list.

  • Triangular numbers (with hexagonal numbers in bold type):
  • 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630.
  • Centered triangular numbers (with numbers that are also centered hexagonal numbers in bold type):
  • 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631.

1 is the only number in the lists that is a triangular, centered triangular, hexagonal, and centered hexagonal number all at the same time. (1 is also a square number, a pentagonal number, etc.)

(630, 631) is the first combination of numbers that has BOTH a triangular number immediately followed by a centered triangular number AND a hexagonal number immediately followed by a centered hexagonal number.

631 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 631 is a prime number.
  • Prime factorization: 631 is prime.
  • The exponent of prime number 631 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 631 has exactly 2 factors.
  • Factors of 631: 1, 631
  • Factor pairs: 631 = 1 x 631
  • 631 has no square factors that allow its square root to be simplified. √631 ≈ 25.11971.

How do we know that 631 is a prime number? If 631 were not a prime number, then it would be divisible by at least one prime number less than or equal to √631 ≈ 25.1. Since 631 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 631 is a prime number.

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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 10.)  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.

631 Factors

571 Family Time

For the last two weeks we’ve been spending time with our grandchildren in Salt Lake City, Utah; Portland, Oregon; Houston, Texas; and Hartford, Connecticut. We also spent time with my brother’s daughter and her two children. AND we celebrated my sister’s 70th birthday party and reconnected with all of her children in Portland. We have been busy, and I am tired, but happy.

We’ve had a wonderful time with all of them, and it was so painful to say goodbye. We and a few other people took pictures except we didn’t get any pictures of the grandkids that live within 10 miles of our home. What’s with that? I’m as guilty as anyone else who takes people and things for granted.

Here’s a picture of me holding my newest grandson. My daughter is in the background holding one of her nieces.

 

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Since 171 + 190 + 210 = 571, it is the sum of the 18th, 19th, and 20th triangular numbers. That makes 571 the 20th centered triangular number.

  • 571 is a prime number and a twin prime with 569.
  • Prime factorization: 571 is prime.
  • The exponent of prime number 571 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 571 has exactly 2 factors.
  • Factors of 571: 1, 571
  • Factor pairs: 571 = 1 x 571
  • 571 has no square factors that allow its square root to be simplified. √571 ≈ 23.8956

How do we know that 571 is a prime number? If 571 were not a prime number, then it would be divisible by at least one prime number less than or equal to √571 ≈ 23.9. Since 571 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 571 is a prime number.

 

 

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