perplexus.info :: Shapes : Arrow dice 2

Arrow dice 2 (Posted on 2006-04-06)

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?

Submitted by Tristan

Rating: 4.5000 (4 votes)

Solution: (Hide)

Question 1: 105 turns (counting the starting position as a turn). Thanks to Dej Mar for finding it.
Question 2: 12 * edge of icosahedron (as far as I know)
Note that, given any face on the dice, the next face is actually opposite direction of the arrow.
Question 1: / \ / \ / \ / \ / \ Start with ABC / A \ / E \ / F \ / M \ / N \ Cycle D through T six times ----------------------------- \ B / \ D / \ G / \ L / \ O / \ \ / C \ / T \ / H \ / K \ / P \ ------------------------------ \ R / \ S / \ I / \ J / \ Q / \ / \ / \ / \ / \ / Question 2: / \ / \ / \ / \ / \ Start and end at A / T \ / S \ / R \ / Q \ / P \ ----------------------------- \ A / \ C / \ E / \ G / \ O / \ \ / B \ / D \ / F \ / H \ / N \ ------------------------------ \ L / \ K / \ J / \ I / \ M / \ / \ / \ / \ / \ /

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Answer	K Sengupta	2024-05-10 11:21:19
Puzzle Thoughts	K Sengupta	2023-04-16 23:55:26
re(4): the longest I've gotten	Tristan	2006-04-09 17:02:29
re(5): the longest I've gotten	Dej Mar	2006-04-09 16:37:19
re(4): the longest I've gotten	Tristan	2006-04-09 12:14:54
re(3): the longest I've gotten	Dej Mar	2006-04-09 06:01:53
re(2): the longest I've gotten	Tristan	2006-04-08 13:36:38
re: the longest I've gotten	Dej Mar	2006-04-07 14:52:29
the longest I've gotten	Charlie	2006-04-06 22:06:43
a start	Charlie	2006-04-06 15:49:08

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