answer:Here is some stuff that might be relevant, but I left them out of the details to prevent cluttering up the question. One of the first properties that I noticed was that the length of each line must have the same parity as the Manhattan distance between the endpoints. Also, the total length of all of the lines must have the same parity as the area of the board minus the number of different lines. (L1 + L2 + L3 + ....) mod 2 = (Board Area – # of lines) mod 2 This also means that any solvable board can be made unsolvable by simply moving one endpoint over to an adjacent square. This affects the parity of that line, but does not affect the parity of any of the other lines or the area of the board.