How Is 1730 the Sum of Consecutive Squares?

Today’s Puzzle:

Write all the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.

Factors of 1730:

  • 1730 is a composite number.
  • Prime factorization: 1730 = 2 × 5 × 173.
  • 1730 has no exponents greater than 1 in its prime factorization, so √1730 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1730 has exactly 8 factors.
  • The factors of 1730 are outlined with their factor pair partners in the graphic below.

More about the number 1730:

1730 is the sum of two squares in two different ways:
41² + 7² = 1730, and
37² + 19² = 1730.

1730 is the hypotenuse of FOUR Pythagorean triples:
520 1650 1730 which is 10 times (52-165-173)
574 1632 1730 calculated from 2(41)( 7), 41² – 7², 41² + 7²,
1008 1406 1730 calculated from 37² – 19², 2(37)(19), 37² + 19², and
1038 1384 1730 which is 346 times (3-4-5).

Finally, OEIS.org informs us that 1730 is the sum of consecutive squares in two different ways. What are those two ways? I figured it out. Can you?

Here’s a hint: It is the sum of three consecutive squares as well as twelve consecutive squares. That means √(1730/3) rounded is included in one sum and √(1730/12) rounded is included in the other. The solution can be found in the comments. Have fun finding them yourself though!

One thought on “How Is 1730 the Sum of Consecutive Squares?

  1. 23² + 24² + 25² = 1730, and
    6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² + 15² + 16² + 17² = 1730. 

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