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Rolling blocks in 3D


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5 replies to this topic

#1 TimeSpaceLightForce

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Posted 23 April 2013 - 08:51 AM

Based on bonanova's  rolling 8 blocks and with the same mechanics of rolling 1/4 on axis.
This puzzle requires 7 blocks (dice)  on 2 x 2 x 2. How it can be done? It would be more
interesting to play and solve if there is a mini program for it but i can't find one.

cube7.JPG
 
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#2 bonanova

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Posted 23 April 2013 - 07:19 PM

An interesting first step is to examine the solution space.
I.e., are all configurations (all orientations of all dice in all positions) reachable?
(In the 15 puzzle you can't interchange two tiles, so there are two separated configuration spaces.)
Here one might ask, (a) can two dice switch positions? (b) can a die rotate 90 degrees?

Then, we could ask, do the 6's have to be aligned? But that issue seems moot.
If there is any solution that obtains 6's, aligning the 6's and reversing the steps give you a solvable starting position.
I.e., if there is any solution, there is an initial orientation that leads to aligned 6's.

I think this puzzle could be programmed (not visually) to search exhaustively.

Nice puzzle! Magnetic cubes might work.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#3 TimeSpaceLightForce

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Posted 24 April 2013 - 05:20 PM

An interesting first step is to examine the solution space.
I.e., are all configurations (all orientations of all dice in all positions) reachable?
(In the 15 puzzle you can't interchange two tiles, so there are two separated configuration spaces.)
Here one might ask, (a) can two dice switch positions? No, the edge always touches the blue axis

(b) can a die rotate 90 degrees? Yes! that is 1/4 roll

©   ?     No it can not slide

Then, we could ask, do the 6's have to be aligned?May be colored faced is better?You can shuffle then reface back..  But that issue seems moot.
If there is any solution that obtains 6's, aligning the 6's and reversing the steps give you a solvable starting position..
I.e., if there is any solution, there is an initial orientation that leads to aligned 6's.

I think this puzzle could be programmed (not visually) to search exhaustively. I noticed that a cube has always 2 options, and all are tied in 1 vertex and a corner cube (though they are all are)
can be maneuver to 3 positions without affecting the others
Nice puzzle! Magnetic cubes might work.


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#4 bonanova

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Posted 25 April 2013 - 07:22 AM

By switching positions, I meant as in the 15-square puzzle.

These two positions are in separate solution spaces.

You can't start from one and after a series of moves get to the other.

 

 1  2  3  4     2  1  3  4

 5  6  7  8     5  6  7  8

 9 10 11 12     9 10 11 12

13 14 15 __    13 14 15 __

 

 

But two switches changes the parity back again.

So these two positions are in the same solution space:

You can start from one and after a series of moves get to the other.

 

 2  3  4     2  1  4

 5  6  7  8     5  6  7  8

 9 10 11 12     9 10 11 12

13 14 15 __    13 14 15 __

 

So I was wondering whether there are similar sets of disjoint configurations present here.

It's not simple to visualize what happens even after two or three moves.

It's very interesting.


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#5 TimeSpaceLightForce

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Posted 25 April 2013 - 09:46 AM

By switching positions, I meant as in the 15-square puzzle.

These two positions are in separate solution spaces.

You can't start from one and after a series of moves get to the other.

 

 1  2  3  4     2  1  3  4

 5  6  7  8     5  6  7  8

 9 10 11 12     9 10 11 12

13 14 15 __    13 14 15 __

 

 

But two switches changes the parity back again.

So these two positions are in the same solution space:

You can start from one and after a series of moves get to the other.

 

 2  3  4     2  1  4

 5  6  7  8     5  6  7  8

 9 10 11 12     9 10 11 12

13 14 15 __    13 14 15 __

 

So I was wondering whether there are similar sets of disjoint configurations present here.

It's not simple to visualize what happens even after two or three moves.

It's very interesting.

I did not know that..when i had tried dismantling the 15 tiles and reassembled them 15 -1 i was able to do it back.

Maybe this 7cube must be assembled first with all faces in same direction i.e. all blue top,orange down,red left,green right,yellow front,purple back.(or 1-6 if numbers)

But wonder if the position of blocks adjacent  to vacancy is different if it is slided up or down in initial position.

 

this 3D rolling is new idea for me owing from your 8 cube. It is really interesting but somehow doubt if we are the first to tackle.

 

Looking like a 2x2 Rubik with a missing cube...

If we physically construct it the center mechanism must have a fix hold of all the 7 vertices.

when rolling it shall pivot without twisting the binder.. that is rigid construction.

 

Magnetic design involve magnetic strips on edges or center faces but + & - poles orientation is uncertain.

 

But Rubber band might work.., knot 4 cuts tightly ,make a small holes on vertices of hollow cardboard cubes

insert each rubber band end into the cube hole. A round bead as stopper will allow rolling without twisting..


Edited by TimeSpaceLightForce, 25 April 2013 - 09:49 AM.

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#6 TimeSpaceLightForce

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Posted 28 April 2013 - 08:15 PM

To make the spatial solutions we should first assign a fix  axes reference.
Naming the 8 cell : using Up Down  Front Back   Right Left  orientation
Cells  :               
LUB   BUR   
LUF   FUR    

LDB   BDR  
LDF   FDR   
and since all dice showing only 3 of its faces on out sides..

Dice : (initial & final position)   
312   214      263   365  
315   [    ]      264   465

362   264      213   315   
365   564      214   [    ]       

Turning  :   
y-axis to yaw
x-axis to pitch
z-axis to roll

Solution1:  Straight forward moves
1. z FDR x FUR  z FDR              FUR:465 fixed                 
2. make a vacancy on upper deck next to 465..rotate lower deck until
the correct sixer is below it . Another  3 moves max to fix BUR or LUF
3. repeat until all sixers are fixed.
4.  with all oners on lower deck, if chiralty is correct, all the remaining
dice can be fixed..

 

Not much of a challenge.  So lets involve all the 6 sides and roll 25-dice in 3d

 

roldice.JPG

 


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