I will do the obligatory retro and proof game plug:
A type of problem where you are to reach the diagram position within a given number of moves.
A good introductory PDF by Alain Brobecker, featuring many "classic" problems that are well-known among the proof game community.
Jeff Coakley has a lot of shorter (but still tricky) 4-move proof games on chesscafe (although the site put them behind a paywall since 2014, the resourceful reader can probably figure something out...)
Another simple example (original by me): Reach the position after black's 4th move.
rnbqkbnr/1ppppppp/8/p7/3P4/8/PPP1PPPP/RNB1KBNR w - - 0 1
Shorter proof games of about 4 to 7 moves are pretty good for regular chess players, and you literally only need to know how the pieces move. Longer ones can go up to 30 moves (for a "normal" long proof game) or even over 180 moves (the longest I know of).
Given only the diagram position and the fact that it was reached in a legal chess game, determine some feature of the position like side to move, castling rights, en passant rights, whether a piece is promoted...
An excellent collection of simpler problems that slowly rise in difficulty.
A magnificent introductory text by Nikolai Beluhov, an excellent composer of such problems. (Link is to a post containing PDF attachment.)
Another easy example by me again (although the idea is not new): What was the last move played? (I don't even need to give you the side to move for you to solve it.)
8/8/8/8/8/4PP2/1PPPRKPk/4nQ2 w - - 0 1