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# Rolling blocks in 3D

### #1

Posted 23 April 2013 - 08:51 AM

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.

### #2

Posted 23 April 2013 - 07:19 PM

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.

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

- Bertrand Russell

### #3

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.

### #4

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.

**1 **2 **3** 4 **3 **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.

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

- Bertrand Russell

### #5

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 movesget to the other.

1 23 42 13 45 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

canstart from one andafter a series of movesget to the other.

123432145 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.**

### #6

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

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