Rotation and Reflection of Magic Square Type Sliding Games 

A magic square type sliding game is a puzzle that has pieces of numbers in a grid with exactly one empty space, in which each row, each column, and each of the two diagonals sum up to the same number. To play this game, we slide one piece at a time into the empty space and try to rearrange all numbers to increasing order from the left grid to the right grid in each row, and from the top row to the bottom row. In general, not all sliding games can be rearranged (solved) this way. In our study, we create a sequence from each of the magic square and use inversion number to analyze the sequence. We then can use the change of inversion number to explain how the rotation and the reflection impact the solvability of this type sliding games.