Consider this Martin Gardner/Raymond Smullyan puzzle just asked by Favst over on Chess.SE (Source: The Colossal Book of Puzzles and Games, problem A.12 "Doubled Knights"):
... Assume that a game is played with the new rule that a knight can on a single turn move twice. [I.e., both players' both knights are replaced by fairy pieces that move like knights but after their first knight-move may optionally make a second knight-move; either or both of those knight-moves may be capturing.] The first player, White, [is confident that he'll be able to win].
"Actually, I think I have a slight advantage," says Black, "so just to even the playing field, I'll play without my king pawn."
Is White still guaranteed to win?
"In fact," says Black, "I'll even play without my king and queen pawns!"
Is White still guaranteed to win?
Finally, an open problem: Is there any set of pawns/pieces that Black can give up, to escape the easy mate by White? (Define "escape" as "Black gets to make at least 4 moves.")