Sum to Squares

Find three [unique positive] numbers, such that their sum is square, and the sum of any pair is a square.
  • List all solutions with total sum less than 1000.
  • Does a solution exist, such that the original three numbers are also square? For example, without the restriction that the numbers be positive, the set (0, 9, 16) would be a possible solution.

    Source: Diophantos, as quoted in The Penguin Book of Curious and Interesting Puzzles, David Wells, 1992, #28.

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