All Four Can Come Back

White makes one move, Black makes one move, and White makes one more. In the final position, the black knight has gone from g8 to f6, and all of White’s pieces are back where they started. If we look only at this static board, White’s two moves have at least four complete possibilities:

g1 → h3 → g1
g1 → f3 → g1
b1 → c3 → b1
b1 → a3 → b1

None involves a capture, and none needs any exceptional rule: choose either white knight, jump to an empty square, then return along the same path; on the intervening move, the black knight arrives at f6. All four sequences are legal, and the final position is identical square by square. The game is so short that it contains only two White moves, yet it still leaves no trace distinguishing the route taken. From the board, one cannot tell which knight has just gone out and come back.

Another proof game in three and a half moves tightens the problem further. It gives the final position in advance: White’s e-pawn is on e3, White’s bishop is on g6; Black’s h-pawn has reached h4, and the f-pawn has reached f5. From the initial position, White moves four times and Black three times, alternating turns, and they must arrive at exactly this board.

The first two moves by each side are natural enough: the white pawn goes to e3, the black pawn advances from h7 to h5 in one move; the white bishop follows the newly opened diagonal to d3, and the black pawn then goes from h5 to h4. White has two moves left, and the bishop also seems to be only two moves from g6. e4 lies on the diagonal toward g6: the bishop can go from d3 to e4, then on the next move to g6, no more than one square away.

But those two moves are not adjacent. After the bishop stops on e4, it is Black’s turn; the f-pawn drops from f7 to f5. It occupies exactly the square that must be crossed between e4 and g6. The second move, complete just a moment ago, is not retrospectively judged wrong; it only becomes illegal during the waiting turn. If the bishop looks for room on the other side from d3, the first square, c2, has been occupied all along by its own white pawn.

This game therefore supplies a pressure opposite to that of the first: a shorter route does not make its passage easier to determine. The four white-knight round trips have already shown that few moves do not automatically reduce the number of lines; on the bishop’s side, the shortcut of exactly two moves cannot withstand the intervening black move. For the bishop to arrive, the target square cannot be treated as the place to stop as soon as possible; it has to pass through while f5 is still empty.

The white pawn first clears the diagonal, and Black pushes the h-pawn to h5. The bishop goes to d3, and the h-pawn then moves from h5 to h4. The bishop continues along the diagonal, crossing the still-empty f5, also passing beyond the destination g6, and goes to h7. Then the f-pawn lands on f5, and the bishop retreats from h7 to g6.