Sum of "freaky" squares

Riffing off of "Freaky Square Number!" by Mark Mottian:

Find the sum of all five-digit positive integers that

  1. are square numbers
  2. have all digits square
  3. and are NOT ordered numbers.

Clarifications:

  • An ordered number is one whose digits are in an increasing sequence. For example, \(155\) is an ordered number but \(515\) is not; \(11235\) is an ordered number but \(51321\) is not.
  • A square is one that can be obtained by multiplying a number by itself (in particular, \(0=0\cdot 0\) is a square).

This problem was inspired by "Freaky Square Number!" by Mark Mottian, which was inspired by Problem #3 of the 2014 South African Programming Olympiad.

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