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Topic: Dicube (Read 834 times) |
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Disoriented
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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?
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JocK
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Re: Dicube
« Reply #1 on: Aug 10th, 2004, 3:43pm » |
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:: 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).::
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| « Last Edit: Aug 10th, 2004, 3:44pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Disoriented
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Re: Dicube
« Reply #2 on: Aug 10th, 2004, 5:30pm » |
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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.
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JocK
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Re: Dicube
« Reply #3 on: Aug 14th, 2004, 1:11am » |
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You're right!
The point furthest away from a given corner point is - I think - a distance :: sqrt(221)/5 :: away.
Have to think a bit more about the bonus question...
J CK
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IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Disoriented
Newbie
Posts: 6
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Re: Dicube
« Reply #4 on: Aug 14th, 2004, 9:56am » |
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I got a smaller number for the first part  Wondering where your point is.
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| « Last Edit: Aug 14th, 2004, 9:57am by Disoriented » |
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JocK
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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.)
JCK
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IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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JocK
Uberpuzzler
Gender:
Posts: 877
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Re: Dicube
« Reply #6 on: Aug 14th, 2004, 3:05pm » |
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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..  )
JCK
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IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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JocK
Uberpuzzler
Gender:
Posts: 877
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Re: Dicube
« Reply #7 on: Aug 14th, 2004, 3:33pm » |
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Based on the above, it seems that the maximum possible distance between two points on this dicube is 2.sqrt(4-sqrt(3)).
JCK
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IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Disoriented
Newbie
Posts: 6
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Re: Dicube
« Reply #8 on: Aug 15th, 2004, 9:49am » |
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You got it (!), but I'm still not sure how you calculated the 2nd part! 
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| « Last Edit: Aug 16th, 2004, 3:51pm by Disoriented » |
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