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Topic: hollow 3x3x3 knights tour (Read 1313 times) |
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Noke Lieu
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hollow 3x3x3 knights tour
« on: Dec 5th, 2007, 6:31pm » |
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Yes, I know that your standard 3x3x3 is impossible- how can the knight get to the centre?
But what if that centre piece is not there?
Is it possible?
I only ask because the thought came to me in a meeting, a quick scribble on some paper got me to 23 moves before getting stuck, and don't quite have time to investigate properly.
Thought it might be an interesting little distraction for you.
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towr
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Re: hollow 3x3x3 knights tour
« Reply #1 on: Dec 6th, 2007, 12:50am » |
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There's only 20 cubes you need to visit, right? And the goal is to visit each at least and at most once. How could you get to 23 then?
Parity doesn't seem to pan out, anyway. 8 corner-cubes, 12 edge-cubes. And each step chances the kind you're on.
Or am I misinterpreting the puzzle?
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Grimbal
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Re: hollow 3x3x3 knights tour
« Reply #2 on: Dec 6th, 2007, 1:01am » |
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Actually, there are 27 cubes less the center, it makes 26 cubes.
But you are right that parity is a problem. There are 12 of one parity and 14 of the other.
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| « Last Edit: Dec 6th, 2007, 1:02am by Grimbal » |
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towr
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Re: hollow 3x3x3 knights tour
« Reply #3 on: Dec 6th, 2007, 1:28am » |
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Ah, I thought the centers of the faces had to be excluded as well (as on a 3x3 board).
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mikedagr8
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Re: hollow 3x3x3 knights tour
« Reply #4 on: Dec 6th, 2007, 1:53am » |
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I seem to be able to do this by using a clue from a previous riddle.  Well, that's how I solved it.
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Grimbal
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Re: hollow 3x3x3 knights tour
« Reply #5 on: Dec 6th, 2007, 2:37am » |
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Hm... you solved it? You mean you actually found a path that goes through all 26 cubes? I would like to see your solution.
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mikedagr8
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Re: hollow 3x3x3 knights tour
« Reply #6 on: Dec 6th, 2007, 2:44am » |
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Ah, maybe I misunderstood the puzzle. I drew a net diagram, of a 3*3*3. I thought that's how I should be able to do it. I'll try again. 
I have a strong feeling I misdrew my diagram anyhow, not a visual kind of person.
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| « Last Edit: Dec 7th, 2007, 2:27pm by mikedagr8 » |
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ThudnBlunder
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Re: hollow 3x3x3 knights tour
« Reply #7 on: Dec 7th, 2007, 2:26pm » |
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There are 12 Edges (with 2 faces showing).
There are 8 Corners (with 3 faces showing).
There are 6 Centres (with 1 face showing).
Corners and Centres can only be reached by jumping to and from an Edge, eg.
--Edge---<--->---Corner---<--->---Edge---<--->---Centre---<--->---Edge--
As there are 14 Corners and Centres we need 14 Edges.
But there are only 12.
Therefore it is impossible.
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| « Last Edit: Dec 7th, 2007, 2:29pm by ThudnBlunder » |
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Hippo
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Re: hollow 3x3x3 knights tour
« Reply #8 on: Dec 11th, 2007, 12:12am » |
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And what about when you remove whole main diagonal.
So there are
12 black ... edges
and
12 white ... 6 corners and 6 face centers?
... solvable for example
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14 23 xx | 05 10 13 | 16 03 18
07 02 21 | 12 xx 06 | 19 08 01
22 09 24 | 15 04 11 | xx 17 20
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| « Last Edit: Dec 11th, 2007, 5:11am by Hippo » |
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