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The aim of this puzzle is to move the white pieces to occupy the squares where the black pieces are and vice versa. The bishops can only move diagonally and cannot move onto squares where they can be taken by the opposing colour. This also means that the bishops on the white squares cannot affect the bishops on the black squares and vice versa.

Solution

First of all, work only with the pieces on one colour of squares and then use the mirror image of the moves, used to get them across, to get the pieces on the other colour squares across. Alternate the below moves, with Black and White:

BLACK E2→D3 A2→B3 D3→B1 B3→C4 A4→D1 D1→E2 B1→C2 C2→A4 E4→C2 C4→A2 E2→C4 C2→D1 A2→B1 B1→E4 C4→D3 D1→B3 B3→A2 D3→E2
WHITE E3→D2 A3→B2 D2→B4 B2→C1 A1→D4 D4→E3 B4→C3 C3→A1 E1→C3 C1→A3 E3→C1 C3→D4 A3→B4 B4→E1 C1→D2 D4→B2 B2→A3 D2→E3