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 Post subject: Solving Alex's Black Hole
PostPosted: Sat Apr 07, 2012 3:25 am 
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Location: Marin, CA
At g4g I got from Oskar an Alex's Black Hole. Solving has been an interesting experience.

This puzzle is in the same vein as the Crossteaser, Patapata Cube/Cubedron, Bram's Black Hole, and Bram's Rocket, in that there's a gap but the pieces also reorient. The difficulty is a little hard to assess, because it's fairly straightforward to figure out enough that if you scramble it up you have a one in three chance of being able to solve it on each try, and it isn't such a big deal to just rescramble it and try again if you fail, so in that sense it's just a bit harder than the Cubedron, but figuring out an elegant way of solving the very last step is much more difficult, making it much closer to the Crossteaser in difficulty.

The first thing to know in this puzzle is that there's a parity to the orientation of the center. If it's rotated 90 degrees, then two pieces become swapped. The easiest way to avoid problems with this is to simply memorize how one of the pieces should be positioned. For example, in my copy the grooves under the red, white, and blue piece swirl clockwise, so if I start with that piece and get it positioned correctly then the parity will always be correct in the end. Also on mine the grooves under the gap swirl clockwise in the solved state, and I'm going to assume that that's the case for yours as well (although at the time of this writing you probably don't have one, because I think only one or two have ever been made).

For notation I'm going to start by saying where the gap is, with one each of Front or Back, Right or Left, and Up or Down, and specify movement by the direction each piece moves to fill the gap in turn. In cases where it matters, I'm assuming that the gap starts in a spot of the same parity of where it belongs in the solved state - that is, one where the grooves swirl clockwise.

The most basic sequence is to simply rotate three pieces in place four times, thereby changing the orientations of all of them, like (BUFD)3. You can use this sequence to always solve the first face (although you sometimes have to move pieces out, do the sequence, and move them back in again) and then use it to solve two of the three pieces on the second face, although you might have to do some inversions, for example if the gap is in UFR and the piece in UBL is oriented correctly, you can turn the piece at URB clockwise by doing (UBDF)3 RBL (UFDB)3 RFL.

The orientation of the final corner is where things get interesting. The thematic approach for this type of puzzle is to figure out a sequence where each piece except for one winds up retracing all of its steps during the sequence. I'll call these 'tree-style' sequences. The question of, for any given graph, what paths one piece can be made to take while returning all others to where they started regardless of what kind of reorientation the puzzle involves is an interesting one which to my knowledge has never been studied. As it happens, the Crossteaser and Alex's Black Hole are fairly analogous here, so I tried using my sequence for the Crossteaser, which happened to not work on this puzzle, so I had to come up with a new sequence, which it turns out is better not only because it works on Alex's Black Hole but because it's fewer moves as well.

To rotate a piece next to the gap counterclockwise, position it at UBR with the gap at UFR, and do the sequence UFDRBLFRBLFUBDRFLBUFDBRFLUBDRFLB. Note that F means *towards* the front. If you watch carefully what each of the pieces does in this sequence, you'll notice that all of them except for one repeatedly retrace their steps, resulting in them having the same orientation as when they started (I remember this sequence by visualizing where the two pieces which start on the bottom go). To turn one piece clockwise, do the same sequence either mirror imaged or in reverse.

The above is a perfectly good solution, but it can be improved on, using what I'd like to call Polo-style sequences, in honor of Alex Polonsky, who came up with this puzzle. I can't help but tell an old slightly off-color joke to explain what a Polo-style sequence is. Please appreciate that this is meant in good humor, and that the point is literally applicable in this case:

Q: How many Polonskys does it take to change a light buld?
A: One hundred. One to hold the light bulb, and 99 to rotate the room.

Let's say you've positioned the pieces of the bottom layer and wish to rotate the piece at FDL clockwise. You can do this the way described previously, which is fairly awkward, or you can move the gap to FUR and do the sequence UFRDBLUFRB. This has the curious property that the two pieces which are 'rotated' aren't moved once in the entire sequence. That's because the entire rest of the pieces literally move around them! Once you've done the sequence, you have to reorient the puzzle as a whole to get the pieces back where they started. The trailing RB in the above sequence can be left off when you're just doing the first layer. This is a fun and fast way of doing the first layer, because it allows you to simply position everything and then orient, without having to awkwardly undo the positioning to do the reorienting. Likewise it can be used to orient the next to last piece.

Polo-style sequences can also be used to make for fewer moves to reorient the very last piece. For example, you can position the gap at UFL and rotate the piece at UFR clockwise by doing ULFRDLBRULF D BRULFRDLFRU LFRBLFR. I put in spaces to divide this sequence into its logical parts - first the bottom face snakes onto the front, then it snakes onto the top, then the remaining pieces get back into place. At the end you have to flip the cube as a whole back over again. This sequence is two fewer moves than the previously given on, and is a lot zanier.


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 Post subject: Re: Solving Alex's Black Hole
PostPosted: Sat Apr 07, 2012 3:57 am 
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Joined: Mon Feb 27, 2012 10:57 am
Location: In my study drooling over my puzzle hoard - Precioussssss!
Bram wrote:
and do the sequence UFDRBLFRBLFUBDRFLBUFDBRFLUBDRFLB.
....
The above is a perfectly good solution, but it can be improved on...
by doing ULFRDLBRULF D BRULFRDLFRU LFRBLFR. I put in spaces to divide this sequence into its logical parts

:shock: :shock: :shock:
I struggle to remember who I am some days! Not only am I often found half way upstairs having forgotten what I am going for, but I have even been known to have forgotten whether I am going up or down!!!! :oops: :oops:

How can anyone remember this? As for logical parts all I can say is :lol: :lol: :lol:

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Hopelessly addicted to puzzles!! :oops:
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 Post subject: Re: Solving Alex's Black Hole
PostPosted: Sat Apr 07, 2012 10:18 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Puzzlemad wrote:
Bram wrote:
and do the sequence UFDRBLFRBLFUBDRFLBUFDBRFLUBDRFLB.
....
The above is a perfectly good solution, but it can be improved on...
by doing ULFRDLBRULF D BRULFRDLFRU LFRBLFR. I put in spaces to divide this sequence into its logical parts

How can anyone remember this? As for logical parts all I can say is :lol: :lol: :lol:


There are obvious geometric patterns to both of the above if you do them on the actual puzzle. For the first one the pieces on the DL both stay in place, and the two pieces on the DR basically come out and then go back in just about as quickly as they can without having everything simply retrace their steps. In the second one, the four pieces on the bottom form a snake where the body always exactly follows the head, and it winds its way to the top - the first section moves it to the front, then the D makes way for its head, then the next section winds it to the up, then the last bit positions the other pieces.

I figured out both of the above sequences by hand, and typed them in off the top of my head, so it is possible to remember them, probably kind of hard to do it as a verbatim sequence without visualizing what the pieces are actually doing though. (I did double-check both of them with the actual puzzle).


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 Post subject: Re: Solving Alex's Black Hole
PostPosted: Tue Apr 10, 2012 2:04 pm 
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Joined: Fri May 06, 2005 10:13 am
Location: Norway
What about writing a solver?? One could even define the puzzle/problem in KSolve and see what it finds ... :wink:

Per

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 Post subject: Re: Solving Alex's Black Hole
PostPosted: Thu Apr 12, 2012 8:45 am 
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Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
At g4g I got from Oskar an Alex's Black Hole. Solving has been an interesting experience.
I'm far from a good solver but I really need to look in this section more. This does really appear to be an interesting puzzle. I love the term Polo-style sequences and it does certainly seem fitting in this case.

Carl

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