Complete arrangement of chessboards
Given a 2^k×2^k chessboard in which all white squares lie in the upper half and all black squares lie in the lower half, this paper presents an efficient general procedure for arranging the squares given to form the standard chessboard configuration of alternating black and white squares, using only translations of 2×2 pieces of the chessboard at time.
J. Akiyama, M. Ruiz. Complete arrangement of chessboards. The Bulletin of the Research Institute of Tokai University, 6, 99-111 (1998).