Series-movers ask for one side to play a long series of consecutive moves. In series-mates, White plays the series and try to mate Black at the end of the series. In series-helpmates, Black plays the series and try to reach a final position where White may mate in one. In series-selfmates, White plays the series and try to reach a final position where Black must mate him in one.
During the series, it is not allowed to walk into check, or to check the opponent before the last move of the series, when he can answer the check.
Consequent series-movers are series-movers where, at every moment during the series, the intermediary position must be legal as if we were in an orthodox game. The previous rules about walking into check still apply, but additionally it is not possible to enter positions with no retro-analytically "legal past". For retro-analytical purposes, all intermediary positions are evaluated independently.
Ded. G. Yacoubian
5+10. shc#15 (Consequent Series-Helpmate in 15 moves)
The try 1. Ra3 2. Rxb3?? 3. Rb5 4. Bb4 5-7. Kc4 8. Rc5 9. b5 b3# fails because Wh. has no last move after 2. Rxb3.
The solution is to first capture the wh. Pg2 so that the White King has -1. Kg2-h1 as a possible last move. 1. Rf8 2. Rf2 3. Rxg2!! 4. Rf2 5. Rf8 6. Ra8 and we go on as in the try.
Rex Multiplex 15-16, 1985
5+3. shc#8 (Consequent Series-Helpmate in 8 moves)
After 1. Kf8 2. Ke8 we reach a position where, if it was the starting point of a problem, Bl. could castle according to the usual conventions for castling rights.
In consequent series-movers, the intermediary positions are evaluated independently, so that we may go on as if we were in the starting position and play 3. O-O!!. The fact that we have just moved the bl. King from g8 is explicitly disregarded in consequent series-movers.
Now 4. Rxa8 5. Kf8 6. Ke8 7. O-O-O!! is legal according to the consequent series-mover logic. We conclude with 8. Rd7 Ra8#.