You have a twenty-sided die (a regular icosahedron) 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?

(In reply to re(4): the longest I've gotten by Tristan)

I had to finally dig up one of my twenty-siders.  I mapped out the faces against the figure of my last post, and then carefully rotated the die and noted the direction of each "arrow".  Repeating this procedure more than once for confirmation.

All six directions (rotating clockwise) of each arrow of the 17 faces appeared.  The 3 remaining faces leading into the first face appeared once each.  Thus, 3 + (17 * 6) = 105 was the longest I've gotten so far.  Am I looking for longer?  

Posted by Dej Mar on 2006-04-09 16:37:19