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        riddles >> medium >> Dicube
        
(Message started by: Disoriented on Aug 10^th, 2004, 2:10pm)

Title: Dicube
Post by Disoriented on Aug 10^th, 2004, 2:10pm

This is an old Martin Gardner doozie. I know the answer, but not how to arrive at it! ???

A dicube is a 1x1x2 rectangular solid that can be generated by gluing two unit cubes together at their face. An ant starts at one of the eight corners of the dicube (for discussion's sake, let this point be called A).

Which point on the dicube's surface requires the ant to travel the farthest to reach?

Second (bonus) part: What two points on the dicube's surface are furthest apart from each other, as the ant walks?

Title: Re: Dicube
Post by JocK on Aug 10^th, 2004, 3:43pm

:: [hide] fold open the dicube and apply Pythagoras. The distance between corner point A and the opposite corner point is sqrt(8.). (Not sqrt(10)... beware to identify different points at the edge of the 2D shape as identical corner points on the dicube!)

The two most distant points are the centres of the 1x1 faces (distance = 3).[/hide]::

Title: Re: Dicube
Post by Disoriented on Aug 10^th, 2004, 5:30pm

Hint 1: The opposite corner of the dicube is *not* the furthest point from A.

Hint 2 (For the bonus part): The two most distant points on the dicube are > 3.01 apart.

Title: Re: Dicube
Post by JocK on Aug 14^th, 2004, 1:11am

You're right!

The point furthest away from a given corner point is - I think - a distance :: [hide]sqrt(221)/5[/hide] :: away.

Have to think a bit more about the bonus question...

J8)CK

Title: Re: Dicube
Post by Disoriented on Aug 14^th, 2004, 9:56am

I got a smaller number for the first part ;) Wondering where your point is.

Title: Re: Dicube
Post by JocK on Aug 14^th, 2004, 2:38pm

Am I again wrong..? :-[

Anyway, my point is at distance 1/5 from the opposite corner. See attached picture. (Note that the point under consideration is shown three times to ensure that the different 'straight' routes an ant can take all map onto a straight line.)

J8)CK

Title: Re: Dicube
Post by JocK on Aug 14^th, 2004, 3:05pm

Aaaahhhg... forgot the fourth image... :-[

(Must have been a bit disoriented... ) Corrected it now. The point at maximum distance from a given cornerpoint is at distance sqrt(130)/4. (Don't bother to hide anymore... past performance indicates this is most likely wrong.. he, he.. ::) )

J8)CK

Title: Re: Dicube
Post by JocK on Aug 14^th, 2004, 3:33pm

Based on the above, it seems that the maximum possible distance between two points on this dicube is 2.sqrt(4-sqrt(3)).

J8)CK

Title: Re: Dicube
Post by Disoriented on Aug 15^th, 2004, 9:49am

You got it (!), but I'm still not sure how you calculated the 2nd part! :)