You are about to play a game of chess with the ghost of Bobby Fisher (at age 24). Bobby has agreed to a concession: In a pre-game round, you may select any one of your pawns and activate it by advancing F₂ squares (here F₂ = second Fibonacci number = 1). Next, you activate the pawn located F₃=2 columns to the right, moving it F₄ squares forward, and so on. (Each pawn always stay within their native column.)

If at some point you return to a pawn which was previously moved, you must then move it in the opposite sense. That is, any given pawn will alternatively move forward ±F_2M rows (where M is the appropriate turn counter and a negative move corresponds to a move backward).

Whichever of Bobby's pieces (except the king) you land on with one of your pawns are removed from the board. If at some point you were to land on the square occupied by the Bobby's king, or by one of your own pieces, the pawn simply shares that square without a capture/removal.

You may continue this procedure until such time as you are no longer able to capture any further of Bobby's pieces. After these preliminaries, you return your pawns to their normal positions and Bobby will play white against you, using whatever pieces still remain to him. Bobby congratulates himself on having granting you a generous advantage.

How generous was Bobby and which pawn should you activate first?

Clarification and example: Wraparound is used to deal with “off the board” locations. For instance, if the formula requires you to select a pawn one column beyond the right edge of the board, you should revert back to the leftmost column.

An example may help clarify the procedure. After ten times moving pawns in this way, you would next activate the pawn located F₂₁ columns to the right of the most recently moved pawn. Since F₂₁=10946, we determine (using wraparound) that the pawn to activate lies two columns to the right (alternatively, six columns to the left) of the most recently activated pawn. Suppose for the sake of illustration that this pawn is currently positioned in the fifth row, having been moved exactly once previously. We now have to move it back by F₂₂=17711 rows which, using wraparound, would reposition it into the sixth row.

You're right, the revised cycle looks like this:

1             1             0
2
3             3             2
5
8             0             7
13
21            5             4
34
55            7             6
89
144           0             7
233
377           1             0
610
987           3             2
1597
2584          0             7
4181
6765          5             4
10946
17711         7             6
28657
46368         0             7
75025
121393        1             0
196418
317811        3             2
514229
832040        0             7
1346269
2178309       5             4
3524578
5702887       7             6
9227465
14930352      0             7
24157817
39088169      1             0
63245986
102334155     3             2

which means that it is the pawn four to the right or left of the original pawn moved that advances forward 5 and back 5 to knock out the opposing pawn, rather than the one 3 to the right or 5 to the left.