An overview of the fifteen puzzle using the ancient Chinese concept of pairing

The ancient Chinese mathematician Yang Hui was interested in magic squares and apparently constructed them by pair‐wise transpositions based on opposites by position—top‐bottom, left‐right, etc. The transpositions considered for solutions to the modern‐day fifteen puzzle, however, are much more restrictive; only adjacent horizontal and vertical moves are permitted. Nevertheless, we can conclude that only configurations which represent an even number of transpositions produce possible configurations in the fifteen puzzle whereas any configurations which represent an odd number of transpositions of the integers from their natural order are not possible in the fifteen puzzle.