Response:
There is no widely known or popular game that combines chess and twisty puzzles like Rubik's Cube in the way you describe. However, the concept you've mentioned is quite interesting, and there is nothing stopping you or anyone else from creating a custom game with such rules.
Here's a simple outline for a game that combines chess and twisty puzzles:
1. Board: Use an 8x8 chessboard or a board of your preferred size.
2. Pieces: Instead of traditional chess pieces, use twisty puzzles of various sizes and types (e.g., 2x2, 3x3, 4x4 Rubik's Cubes, Pyraminx, etc.). Assign each puzzle piece a role similar to chess pieces (king, queen, rook, knight, bishop, pawn).
3. Movement: Define how each puzzle piece can move on the board, similar to chess rules.
4. Puzzle manipulation: In addition to moving the puzzle pieces on the board, players can perform twisty puzzle moves on their own pieces, the opponent's pieces, or both. You could limit the number of moves allowed per turn, or create special conditions under which puzzle manipulation is permitted.
5. Victory condition: Establish a winning condition, such as capturing the opponent's "king" puzzle piece or solving a certain number of the opponent's puzzle pieces.
Remember that this is just a basic outline, and you can modify it or come up with entirely different rules to create a unique game experience. Once you have developed the rules, you can playtest with friends to refine the gameplay and make any necessary adjustments.