Part I: The Swap Puzzle Game Play: Randomly arrange the tiles in the boxes then try to put them in proper order using only legal moves. A variety of legal moves are: Legal Moves (variation 1): Swap the contents of any two boxes. Legal Moves (variation 2): Swap the contents of any box with box 1. Legal Moves (variation 3): Swap the contents of any two consecutive boxes (here we ll consider 8 and 1 as consecutive). Legal Moves (variation 4): Pick any 3 boxes and shift the contents either left or right one box. (We call this move a 3-cycle.) Legal Moves (variation 5): Pick any 4 boxes and shift the contents either left or right one box. (We call this move a 4-cycle.) Legal Moves (variation 6): Pick any 4 consecutive boxes and swap the contents of the inner two boxes, and then the outer two boxes. Legal Moves (variation 7): Pick any 4 boxes and split the contents into pairs. Then swap pairs. Think of some other variations that may be fun/challenging to play. Explore the puzzle using these different variations on the legal moves. For a particular variation some questions to explore are: Is the puzzle solvable for all random arrangements of the pieces? If not, can you characterize those arrangements that are not solvable, and those that are? Can you determine a strategy for solving the puzzle in the fewest number of moves?
Part I: The Swap Puzzle The Parity Theorem: For an arrangement of objects (permutation) if you can find a way to put the objects back in order using an even number of swaps, then it always must take an even number of swaps to put the objects back in order no matter how you do it. Similarly, if it took an odd number of swaps to put the objects back in order then it always must take an odd number. Fact 1: Swapping two boxes at a time (variation 1) is enough to solve any arrangement of the tiles. Fact 2: 3-cycles are enough to solve any even arrangement of the tiles. Part II: The 15-Puzzle Game Play: Randomly arrange the tiles in the boxes then try to put them in proper order by sliding the tiles around using the empty space.
Part III & IV: Rubik s Cube Move Notation notation pictorial description of basic moves F F = quarter turn of the front face in the clockwise F -1 = quarter turn of the front face in the counter-clockwise B B = quarter turn of the back face in the clockwise B -1 = quarter turn of the back face in the counter-clockwise R R = quarter turn of the right face in the clockwise R -1 = quarter turn of the right face in the counter-clockwise L L = quarter turn of the left face in the clockwise L -1 = quarter turn of the left face in the counter-clockwise U U = quarter turn of the top face in the clockwise U -1 = quarter turn of the top face in the counter-clockwise D D = quarter turn of the down face in the clockwise D -1 = quarter turn of the down face in the counter-clockwise M M = quarter turn of the middle slice in the upward M -1 = quarter turn of the middle slice in the downward
A (Basic) Strategy for Solving Rubik s Cube: Step 1: Solve the edges is the first layer. This is known as solving the cross. Step 2: Solve the corners is the first layer. Step 3: Solve the edges is the middle layer. (At this point things get challenging, so some useful moves are on the next page.) Step 4: Solve the corners is the last layer. Do this in two stages: 4a Place corners in the proper positions 4b Then twist them into the proper orientation. Step 5: Solve the edges is the last layer. Do this in two stages: 5a Place edges in the proper positions 5b Then twist them into the proper orientation. Congratulations, you ve solved the cube!
A Catalog of Useful Moves left: ( D L D -1 L -1 ) ( D -1 F -1 D F ) right: ( D -1 R -1 D R ) ( D F D -1 F -1 ) (or you could just modify the edge 3-cycle below. This means you really only need to understand the following four move-sequences to solve the cube.) Corner 3-cycle ( L D -1 L -1 ) U ( L D L -1 ) U -1 Double corner twist ( L D 2 L -1 F -1 D -2 F ) U ( F -1 D 2 F L D -2 L -1 ) U -1 Edge 3-cycle ( M -1 D -1 M ) U ( M -1 D M ) U -1 Double edge flip ( M -1 D M D -1 M -1 D 2 M ) U ( M -1 D 2 M D M -1 D -1 M ) U -1