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http://en.wikipedia.org/wiki/helpmate. Linked, the last sentence of history spells out Helpmate as next in prevalence after Direct Mates.
/// http://www.chessbase.com/newsdetail.asp?newsid=6768. Talk about off-topic is Nunn's Sagittarian Trifid Nebula.///
Now composers go to great lengths to author/create odd board positions in
constructed Helpmates out of the so many millions possibilities. Actually,
the starting point for the 100-year-old field should have been the
500-year-old array itself, RNBQKBNR, the thirty themselves.  Consider that the field has not really begun until this moment.  Helpmate operational definition: Black does utmost to assist White.  Long-known from initial position is that thus Black Queen mates in two. Now we formalize: White Queen mates in three from initial array.  White Bishop-f1 is Helpmate in 3, as is Black Bishop-f8. The latter could go: 1 f3 e6; 2 g4 B-e2; 3 (N-c3) B e2-h4 Checkmate. Bishop on h4, or g3 if h2 moves, is checkmate after f3 and g4 in either order without White King move, and there are many actual move sequences solving the condition.  White's particular '3 N-c3' is one such precluding the King step. Okay, Black Queen takes 2 to mate, White Queen 3 to mate, Bishop-f1 3 to mate, and Bishop-f8 3 to mate. That makes four solved cases of the 30 from array.  Next, in how few can Rook-h1 mate?
There are the two styles or cases, assisted and unassisted, in this further development of Orthodox Helpmates.  These are all going to be extreme unassisted, 
http://www.chessvariants.org/index/displaycomment.php?commentid=26767.