A Multiplication Based Logic Puzzle

Posts tagged ‘Square Pyramidal Number’

650 is the sum of all the clues in this Level 1 puzzle

1² + 2² + 3² + 4² + 5² + 6² +7² + 8² + 9² +10² + 11² + 12²  = 650

Thus 650 is the 12th square pyramidal number and can be calculated using 12(12 +1)(2⋅12 + 1)/6.

If you add up all the clues in today’s Find the Factors puzzle, you will get the number 650. However, if you print the puzzle from the excel file, one of the clues is missing because it isn’t needed to find the solution.

650 is the hypotenuse of seven Pythagorean triples!

  • 72-646-650
  • 160-630-650
  • 182-624-650
  • 250-600-650
  • 330-560-650
  • 390-520-650
  • 408-506-650

Can you find the greatest common factor of each triple? Each greatest common factor will be one of the factors of 650 listed below the puzzle.

650 is the hypotenuse of so many Pythagorean triples because it is divisible by 5, 13, 25, 65, and 325 which are also hypotenuses of triples. The smallest three numbers to be the hypotenuses of at least 7 triples are 325, 425, and 650.

Since 25 x 26 = 650, we know that (25-1)(26 + 1) = 650 – 2. Thus 24 x 27 = 648.

650 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 650 is a composite number.
  • Prime factorization: 650 = 2 x 5 x 5 x 13, which can be written 650 = 2 x (5^2) x 13
  • 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 650 has exactly 12 factors.
  • Factors of 650: 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650
  • Factor pairs: 650 = 1 x 650, 2 x 325, 5 x 130, 10 x 65, 13 x 50, or 25 x 26
  • Taking the factor pair with the largest square number factor, we get √650 = (√25)(√26) = 5√26 ≈ 25.495098.

650 Trees

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506 and Level 5

506 is divisible by 11 because 5 + 6 – 0 = 11, and 11 obviously is divisible by 11.

506 is the 11th square pyramidal number because it is the sum of the first eleven square numbers.

Thus 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 506.

That was predictable because 506 = (11 x 12 x 23)/6 and 12 = 11 + 1 and 23 = 2(11) + 1.

Since 506 = 22 x 23, it is the sum of the first 22 even numbers which also happens to be exactly two times the 22nd triangular number, 253.

Now here’s a Level 5 puzzle for you to try:

506 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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  • 506 is a composite number.
  • Prime factorization: 506 = 2 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 506 has exactly 8 factors.
  • Factors of 506: 1, 2, 11, 22, 23, 46, 253, 506
  • Factor pairs: 506 = 1 x 506, 2 x 253, 11 x 46, or 22 x 33
  • 506 has no square factors that allow its square root to be simplified. √506 ≈ 22.49444

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506 Logic

385 Is a Square Pyramidal Number

385 is the sum of the squares of the first ten counting numbers. Let me demonstrate what that means. All of the following are square pyramidal numbers:

  • 1^2 = 1
  • 1^2 + 2^2 = 5
  • 1^2 + 2^2 + 3^2 = 14
  • 1^2 + 2^2 + 3^2 + 4^2 = 30
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 = 285
  • 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385

We could have found the total, 385, much more quickly by putting the number ten in for “n” in this formula

Which means the sum of the first ten square counting numbers = (10 x 11 x 21)/6 = 5 x 11 x 7 = 385. (Notice its prime factorization!)

Here’s a video explanation:

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