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By applying the 24 rotation symmetries to the corner facelets of the cube one may generate the Cross Pretty Pattern Group. These patterns may be arranged into five conjugate classes: the identity cube, six order two 6-cross patterns, eight order 3 6-cross patterns, six order 4 4-cross patterns and three order 2 4-cross patterns.

By applying the 24 Th symmetries to the edge facelets of the cube one may generate the Check (or Checkerboard ) Pretty Pattern Group. These patterns may be arranged into six conjugate classes: the identity cube, pons asinorum, eight order three 6-check patterns, eight order six 6-check patterns, three order two 4-check patterns and three order two 2-check patterns.

Of course, it had been previously proved that some positions of the Fifteen Puzzle require 80 moves to solve, but in that work it was assumed that a move only affects one tile at a time. Since people commonly slide up to 3 tiles in the same row or column at once, it seems natural to count such an action as a single move. With this way of counting, which we call the "multi-tile metric," the maximum number of required moves is only 43, and of the 16!/2 = 10,461,394,944,000 valid configurations of the puzzle, there are only 16 antipodes, i.e., positions that actually require 43 moves.

The 16 antipodes include two positions that are mirror-symmetric to themselves. These two positions are those that are obtained by transposing the rows and columns with respect to either diagonal. The other antipodes consist of 7 pairs of positions that are mirror-symmetric with the other. These 14 positions also include 4 pairs of neighboring positions. So only 8 of the antipodes are "strict" antipodes having the property that any move gets you one move closer to the solved state.