Helpmate, Selfmate, Reflexmate

Helpmate (h#n: helped mate in n moves). The blacks play and help the whites to mate the black king in n moves. n.1.1.1... means that n different first black moves each lead to a different solution. 0.p.1.1... means that the leading-off whites have p different first moves, all of which lead to a solution.

Selfmate (s#n: selfmate in n moves), the whites play and force the blacks to checkmate the white king in n moves.

Reflexmate (r#n: reflexmate in n moves), selfmate with, as an added condition, the obligation for each side to checkmate in one move if the situation arises.