A Rubik's cube that is a small number of moves, say 3, from solved makes for a nice puzzle. The player is challenged to find the optimal solution rather than applying a generic solving method. Similar in spirit to chess mate in N problems.
Ideas on how to set up the puzzle: The obvious method, make 3 random moves while looking, fails because it is too easy to remember the scramble you did.
Scramble 3 moves behind your back, turning the cube a lot between turns.
Have a friend do it where you can't see.
Build a machine to do it, concealed behind a screen.
Computer algorithm to get to the puzzle state by a roundabout way: first find a short sequence to a random state (utilizing God's Number), then solve for a short sequence (via Kociemba algorithm or similar iterative deepening) from that random state to the target 3-move-scrambled state. Hopefully the short sequence does not go through 0- 1- or 2-move scrambled states.
Rubik's Touch (if it could be modified) could offer such puzzles. It has the advantage of being easily able to be reset to the puzzle start state.
Deep cut puzzles change a lot even in a single move, so would provide challenging short puzzles.
A small number of moves away from the cross, on some side, might be a useful puzzle to practice speedsolving.
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