Given an $n\times m$ grid and $x$ elements on the grid, what's the maximum number of moves required to arrange the elements, for all possible starting permutations of the elements, into a symmetric pattern, given that the number of moves for each permutation is the minimum possible number of moves?
There's a fair bit to breakdown here, so let's work with an example. To explain what I mean by a "move", let's say the grid is an $m\times n$ chessboard, and the elements within that grid are queen chess pieces. However, these queens are superpowered. Instead of their normal limits in chess, these queen pieces can move to any other free space*, even if it requires passing through other queens.
As for symmetry, any form of rotational or axial symmetry is valid.
So, given our $n\times m$ chessboard and $x$ super queens placed on the chessboard, assuming each set of moves is the minimum possible number for the given arrangement of super queens, what is the maximum number of moves required to arrange any initial permutation of super queens into a pattern which is symmetrical?
*This is an important distinction. Typically, "super queens" are still limited to 8 directions of movement. However, I'm specifically asking about "queens" that can move to any free space on the board.