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?

  If each side has an arrow that can point north, south, west or east; and there are six sides, then a maximum of 24 orientations can occur.

I was able to arrive at 20 orientations using five sides:

  Label one side A, and each adjacent side in order clockwise B, C, D, and E.  Orient the arrows as follows. A points to B. B to C, C to D, D to E, and E to A.

  If side A is up and its arrow points north, then rotate the die south to show side B. Side B’s arrow points east toward C. Rotate west to put C on top, then D on top, then E. E’s arrow must point to A which is south.  Rotate north to put A on top.  A should now point west.

  Repeat an additional 3 times. Total of 20 moves.  The die will have ended back at it’s starting location.  All movements are inside of a 6x6 square.

Posted by Leming on 2006-03-22 13:26:41