Rolling Cube Tour
A cube, shown unfolded below, has a number on each face. The cube is put on any cell of a checkerboard in any orientation.
To "move" the cube, read the number on the cube's top face, pick a direction (up, down, left, or right), and roll the cube (tipping it on its edge and making it fall squarely in the adjacent cell) in that single direction that number of times. Since the cube is confined to the board, it sometimes may not be able to move in a certain direction.
If only the cells the cube begins and ends on on a move count as being "visited," what is the minimum number of moves which the cube will have to make to visit each cell of the checkerboard at least once?
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1 solution
If the cube has a 4 on its top face at any point on its tour, it will be caught in an infinite loop and will only be able to visit the same nine cells repeatedly. If the number 2 is on the top face, the cube will have a 4 there on its next move. Therefore the goal is to avoid the numbers 2 and 4 being on the top face for as long as possible.
Start with any other number on the cube's top face. If the number is odd, we will be able to move in only two directions (either left/right or up/down) without landing on a 2 or a 4, restricting us to either the row or the column which the cube started in. The only way to safely exit that row or column would be for the number 6 to be on the top face, which will result in a 1 after moving the cube to a different row/column six spaces away. These two rows/columns, however, are the only two which the cube can safely travel to without a 2 or a 4 showing up on its top face, so the entire board cannot be completely toured by this particular cube.