There are several solutions to the puzzle. Here's one:
The chessboard is eight squares by eight squares. Number the columns, starting left to right, one to eight. Then, letter each row from top to bottom A through H. If done correctly, the bottom left chessboard square would be at the intersection of H and 1. Perform the following moves:
H4 to G6
G6 to F4
F4 to H5
H5 to G3
G3 to H1
H1 to F2
F2 to D3
D3 to E1
E1 to C2
C2 to E3
E3 to D1
D1 to B2
B2 to A4
A4 to B6
B6 to A8
A8 to C7
C7 to E8
E8 to F6
F6 to G8
G8 to H6
H6 to F7
F7 to D8
D8 to B7
B7 to C5
C5 to B3
B3 to A5
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