Yet another puzzle: It's only 4 4 moves

Both positions seem impossible to achieve in 4 moves. It takes two moves by both sides just to get rid of all of the castling rights.

Ah, the castling rights missing is just to not give away any unnecessary hints to the solvers. Notice in the first one (Tibor Orbán 1976 by the way) if the rights were given it kind of reveals the solution.

The second position is in a way a tribute to Orbán. His PG was the first I'd encountered and took me a whole weekend to solve. Obviously I've improved since then. ;)

OK, now I also have the second one: 1.d4 g6 2.Qd2 g5 3.Qxg5 e6 4. Qg6 fxg6. This one was really tricky, especially after doing the first one. At the beginning, I tried to make a similar pattern that solved the first problem work, but the queen takes too long from d1! Then I thought about how to lose a tempo, in order to make the 3-move-solution a 4-move solution and finally, I found it :)

Magnificent puzzles. Thanks for posting them (Sargon, Illion) and for the solutions (Kabel).

Thx. Right @Illion, I removed the additional information (castling rights) from the FEN. It should be just the diagram, forget about the rest. Like in a book.

Good solutions, well done Kabeljaukrieger.
Proof game problems can be pretty deep - these bite-sized ones in 4.0 moves are good for a general audience.

Another one? 4 moves (7 plies as a hint). ;)

lichess.org/editor/rnb1kbnr/pppp1ppp/4p3/8/3P4/8/PPP1PPPP/RNB1KBNR b - -

Why does it feel like something's off with the FEN link?... Anyone able to solve it?

Yes, I found the answer within a minute. It's a simpler version of a problem by Martin Wolfgang Hoffmann, Schweizerische Schachzeitung 1989. Two parts to it, both after black's 4th move:

(a) >> lichess.org/editor/rnbqkbnr/pppp2pp/5p2/4p3/8/3P4/PPP1PPPP/RNB1KBNR_w_-_-
(b) >> lichess.org/editor/rnb1kbnr/pppp2pp/5p2/4p3/8/3P4/PPP1PPPP/RNB1KBNR_w_-_-

(It's only polite to state composer and publication date when citing problems - would you present a Carlsen-Nakamura game without giving the date and the names of the players?)

Yeah, the second is practically the same with ...f7-f6 added. I had the idea for this diagonal manoeuver somewhere in my brain and composed quickly a model game.