player is assumed to have castling rights unless it can be proved otherwise. In some retrograde analysis problems, the solver (who usually plays White) is required to prove that the opponent has previously moved their king or rook and therefore cannot castle. This is sometimes accomplished by castling or by capturing en passant, thereby disproving other possible game histories. The diagram shows a mate in two. 1.Rad1 0-0 does not work. The is 1.0-0-0 This demonstrates that the white king has not moved yet and that the rook on d4 must therefore be a promoted piece. Therefore, either the black king or black rook has previously moved to let the white rook off the back rank. Therefore Black cannot castle. After any move by Black, 2.Rd8 is mate. Novelty problems Some joke chess problems involve castling with a promoted rook of the opponent's color. In orthodox chess, this would be illegal since the rook would be giving check to the king, but under fairy chess conditions, this might not actually be check. The diagrammed problem involves castling with an opposing rook under the Koko fairy condition (each piece must end up adjacent to another piece when moving). The solution (Black's move being given first per helpmate convention) is: 1. Bg7 h8=R 2. Bf6 Kg6 3. 0-0 Kh7# where, after 3.0-0, White's rook is not checking the king; a hypothetical capture of the king would result in the rook not being adjacent to any other piece, which is illegal under the Koko condition. The allowance of castling with a "phantom rook" in handicap games has also been used in joke problems. Many other joke variations on castling are possible. Vertical castling In 1907, C. Staugaard composed a two-mover in which White promotes a pawn to a rook and then castles vertically with the newly promoted rook (placing the king on e3 and the rook on e2), since the rook has not moved. In the position on the right, White plays 1.e8=R, and after the forced move 1...Kxc2 castles vertically with the promoted rook, checkmating Black. Vertical castling, also known as "Staugaard castling" or "Pam–Krabbé castling", has been used in a few novelty chess problems. Tim Krabbé's 1985 book Chess Curiosities includes a problem featuring vertical castling, along with an incorrect claim that the problem's 1973 publication prompted FIDE to amend the castling laws in 1974 to add the requirement that the