In the following position, each side has made exactly 4 moves (8 in total).
[fen "rnbqkbnr/pp3ppp/2p1p3/8/4P3/8/PPPP1PPP/RNBQK1NR w KQkq - 0 1"]
I need to find what these moves were.
Dag Oskar Madsen's (quite generous) hint:
The last move was king takes e-eight.
And here's the full solution:
Pawn to e-four, pawn to e-six, bishop to b-five, king to e-seven, bishop takes d-seven, pawn to c-six, bishop to e-eight, king takes e-eight.
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1 1. e4 e6 2. Bb5 Ke7 3. Bxd7 c6 4. Be8 Kxe8
This puzzle was by Tibor Orban in 1976, and is a great introductory puzzle for those who have never come across a proof game before. In its miniature form, it still shows two elementary technical features: Tempo Loss and Switchback. Tempo is already discussed: there are shorter (non-unique) solutions in 3.0 & 3.5 moves. Switchback is where the Black King leaves his starting square and returns to it. Apart from their artistic effect, these two features also serve to conceal the solution better.
Chess problems are a bit like high diving: you are trying to harmoniously combine a sequence of challenging effects before you hit the water. (See here. One difference is that in chess problems there is not the same rigorous point counting: appreciation is more subjective.)
Interesting puzzle, particularly the requirement that the number of moves be exactly four per side. But for that, a three-move-per-side solution would be easy:
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1 1. e4 c6 2. Bc4 e6 3. Bxe6 dxe6
The difficulty is that all three of black's moves are pawn moves, and wasting a turn moving anything else would force Black to waste another turn moving it back. It becomes apparent therefore that two of black's moves must be with something other than pawns, which in turn implies that White must capture d7 at its original location. That capture cannot occur prior to White's third move, and cannot occur after c6. Thus, Black will have only one move available after c6, and must move something off its original position before C6. That piece must return to its original spot on its last move, and that spot must contain the bishop. The only Black piece whose original spot White's bishop could reach are his king and his light-square bishop; of those, only the former can move before turn 3.