You have a six-sided die with an arrow on each face. You play a little game with the die. You place the die flat on the table. You rotate the die in the direction of the arrow on the top face. This step is repeated each time by looking at the arrow on the top face of the die. The game is over when you see the same arrow pointing in the same direction twice.

If you can choose the directions of the arrows and the starting position of the die, what is the longest this game can last?

What is the farthest the die can go from the start to the end of the game?

There are six sides, each with 4 possible orientations, so there can be at most 24 moves before you repeat one.  Here's a solution that uses 20 moves:

      +-----+
      |     |
      | up  |     <-- TOP
      |     |
      +-----+
      |     |
      | up  | 
      |     |
      +-----+
      |     |
      | up  | 
      |     |
+-----+-----+-----+
|     |     |     |
| X X |left | up  |
|     |     |     |
+-----+-----+-----+

If you were to fold that up into a cube and start with the "TOP" face pointing up, you would go through 20 moves until you repeated a position.  Interestingly, it doesn't matter what you put on the side marked "XX" because you will never turn towards that side.

I don't know if this is the most, but the few times I tried to incorporate the sixth side it ended in less than 20 moves.

For the second part of the question, the cube ended in the same exact spot where it started.  However, the game could end farther away from the starting point, but it would happen in fewer moves.  My gut reaction is that the only way to get the cube to end in a different place than it started is to have two sides pointing at each other or have 4 in a row all pointing in the same direction, which means the game would end just four or five "squares" away.  That's just a guess though.

Posted by tomarken on 2006-03-22 12:40:32