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 Post subject: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 4:28 pm

Joined: Mon Nov 30, 2009 1:03 pm
Hi Twisty Puzzles fans,

Irreversible Cube is based on an idea by Christoph Lohe:
"It is just a 2x2x2 Cube, however, with the following constrains:
- only clockwise quarter turns are allowed
- after a turn on one axis (out of three), the next move, if any, will need to be done on another axis
Obviously, it is not a normal twisty puzzle as you cannot simply reverse your move sequence. It also means: the puzzle is already scrambled after a first single move from the solved state."

So if we encode
"U" = upper face 90 degrees clockwise
"F" = front face 90 degrees clockwise
"L" = left face 90 degrees clockwise
then one could legally turn e.g. "UFLUFLULUFLUFLUL" as all turns are clockwise and there are no two-same-turns-in-a-row. However, "FF" would be illegal and the second "F" move should be blocked.

Irreversible Cube exactly achieves Chris Lohe's goal. There are only single clockwise quarter turns, and any second quarter turn is blocked, only to be unlocked when another face is turned 90-degrees clockwise.

Note that the puzzle has no easy undos. Once a quarter turn clockwise has been made, one cannot undo it by turning anti-clockwise. Anti-clockwise turns are blocked by the mechanism.

Analysing the mechanism, the follow steps happen when turning a face clockwise.
1) First the turning face locks itself against turning back to it original position. From this point, one can only move forward.
2) Next the turning face unlocks the previously locked adjacent face.
3) Then, the turning face locks itself against turning back to the adjacent-faces-unlocking state.
4) Finally, the turning face blocks. Now it can only be unlocked by turning an adjacent face.

Buy the puzzles at my Shapeways Shop.
Read a solving analysis in this thread at the Twisty Puzzles Forum.
Read more at the Shapeways Forum.
Check out the photos and sketches below.

Enjoy!

Oskar
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Last edited by Oskar on Sat Apr 20, 2013 6:19 am, edited 1 time in total.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 4:37 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Hey Oskar, can you design a 2x2x2 which must alternate clockwise and counterclockwise quarter turns?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 4:50 pm

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Absolutely incredible. You have once again obliterated a long-held assumption about the way twisty puzzles must behave.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 6:00 pm

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
Now expand the concept to a 3x3x3! or 4x4x4! Or more!

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 6:10 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
WOW!!! Very interesting mechanism and it appears to work very well. Congrats. To answer the question posed in the video I think I can undo the first move with 7 other moves.

Carl

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 6:24 pm

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
Here is another question:

If this puzzle were turned by a small amount say, 5 degrees, it can be reversed right? So, what is the largest angle you can turn and be reversible?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 6:32 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
At a recent meet-up with Matt Galla (Allagem) we were discussing this puzzle and a question about the initial bandaging state came up. Which axis is bandaged when the puzzle is solved? Of course, the related question is, if you scramble and solve it, can you ever solve it so that a different axis is bandaged. If you can, doesn't that mean this puzzle actually has 3x the states of a normal 2x2x2?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 6:35 pm

Joined: Mon Mar 30, 2009 5:13 pm
rubikcollector123 wrote:
Here is another question:

If this puzzle were turned by a small amount say, 5 degrees, it can be reversed right? So, what is the largest angle you can turn and be reversible?

... and what's the minimum corner-cutting angle - in case you can avoid irreversibility by cutting corners?

Amazing puzzle by the way!

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 7:31 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Also, can you design a mechanism where the same face can't do two 90 degree turns in a row, but turns can be either clockwise or counterclockwise?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 9:04 pm

Joined: Mon Mar 30, 2009 5:13 pm
Or how about a "forced reversal puzzle", where each turn *must* be the reverse of the previous turn, so that you can never actually scramble it?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 10, 2013 11:00 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
bmenrigh wrote:
At a recent meet-up with Matt Galla (Allagem) we were discussing this puzzle and a question about the initial bandaging state came up. Which axis is bandaged when the puzzle is solved? Of course, the related question is, if you scramble and solve it, can you ever solve it so that a different axis is bandaged. If you can, doesn't that mean this puzzle actually has 3x the states of a normal 2x2x2?

I was chatting with Daniel Kwan (dkwan) about this in the IRC channel and he pointed out that using Jaap's trivial "undo" trick of [UR]xN you can trivially change the bandaged axis.

From the solved state, if the A axis is bandaged then doing [BC]xN will transfer the bandaging to the C axis. In this way you can always choose which axis you want bandaged.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 12:13 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
bmenrigh wrote:
bmenrigh wrote:
I was chatting with Daniel Kwan (dkwan) about this in the IRC channel and he pointed out that using Jaap's trivial "undo" trick of [UR]xN you can trivially change the bandaged axis.

From the solved state, if the A axis is bandaged then doing [BC]xN will transfer the bandaging to the C axis. In this way you can always choose which axis you want bandaged.

What value of N are you suggestion? That trick also allows you to turn the same face twice.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 12:27 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Bram wrote:
Hey Oskar, can you design a 2x2x2 which must alternate clockwise and counterclockwise quarter turns?

As a suggested approach, you could have everything but the center be exactly the same as in this puzzle, then inside the center have a free-floating ball which has grooves in it and feet under three of the axles similar to the latest Polo Shift such that the ball rotates 90 degrees whenever a face rotates 90 degrees. Then you put additional grooves in the ball in exactly the pattern that they are in the core of Alex's Black Hole, and fixed posts sticking into half the corners (a half which would all be colored the same with checkerboard coloring).

That should do it, and it doesn't require any spring loaded anything.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 1:45 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Bram wrote:
What value of N are you suggestion? That trick also allows you to turn the same face twice.
15 will do. [RU]x15 (30 total moves) is an identity sequence.

Also, if U is locked then [RU]x14 R (29 moves) will perform the effect of a U'

[RUF]x10 is a similar sequence.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 1:56 am

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
All this makes me wonder: what is the sequence that solves the cube into a different internally bandaged situation?

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 5:06 am

Joined: Sun Nov 23, 2008 2:18 am
My question is could this concept be applied to a Skewb or UFO.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 5:16 am

Joined: Sun Oct 08, 2006 1:47 pm
Location: Houston/San Antonio, Texas
bmenrigh wrote:
bmenrigh wrote:
At a recent meet-up with Matt Galla (Allagem) we were discussing this puzzle and a question about the initial bandaging state came up. Which axis is bandaged when the puzzle is solved? Of course, the related question is, if you scramble and solve it, can you ever solve it so that a different axis is bandaged. If you can, doesn't that mean this puzzle actually has 3x the states of a normal 2x2x2?

I was chatting with Daniel Kwan (dkwan) about this in the IRC channel and he pointed out that using Jaap's trivial "undo" trick of [UR]xN you can trivially change the bandaged axis.

From the solved state, if the A axis is bandaged then doing [BC]xN will transfer the bandaging to the C axis. In this way you can always choose which axis you want bandaged.

So the answer would be yes, there are actually 3 solved states and here is a fairly simple algorithm to move from one to either of the other two (using n=15 as stated above).

Bram wrote:
bmenrigh wrote:
bmenrigh wrote:
I was chatting with Daniel Kwan (dkwan) about this in the IRC channel and he pointed out that using Jaap's trivial "undo" trick of [UR]xN you can trivially change the bandaged axis.

From the solved state, if the A axis is bandaged then doing [BC]xN will transfer the bandaging to the C axis. In this way you can always choose which axis you want bandaged.

That trick also allows you to turn the same face twice.

Excellent point! So U is possible from any position. U' is possible from any position. And U2 is possible from any position. Thus the Irreversible Cube has exactly the same number of positions as a standard 2x2x2 has. Three times that number if you count the bandaged axis as part of the state. Cool

Peace,
Matt Galla

Edit: Wait a second... All of this has been discovered already in the thread Oskar linked in the first post! Whoops.....
Jaap, you're awesome

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 5:38 am

Joined: Mon Jan 14, 2013 5:37 pm
Location: UK
bmenrigh wrote:
[RUF]x10 is a similar sequence.

[RUF] need only be done 5 times. Also, [RU']x9 is shorter, and more finger-trickable (than [RU])

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 12:14 pm

Joined: Sun Mar 15, 2009 12:00 am
Hi Oskar,

Congratulation, I am very happy that the brilliant mechanics worked out! It's a real masterpiece Will I see you in Guetersloh on the 20th?

Kind Regards
Christoph

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 2:14 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Duskstar wrote:
[RUF] need only be done 5 times.
Can you elaborate? [RUF]x5 is 15 moves and an odd number of moves can never be an identity sequence due to parity.

Duskstar wrote:
Also, [RU']x9 is shorter, and more finger-trickable (than [RU])
Good find, this is better.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 2:29 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
bmenrigh wrote:
...
Duskstar wrote:
Also, [RU']x9 is shorter, and more finger-trickable (than [RU])
Good find, this is better.
Oskar wrote:
- only clockwise quarter turns are allowed

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 2:35 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
bmenrigh wrote:
Good find, this is better.
Oskar wrote:
- only clockwise quarter turns are allowed
It helps to pay attention I was only thinking about the irreversibility property, not the clockwise restriction too. My fault.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 4:14 pm

Joined: Mon Jan 14, 2013 5:37 pm
Location: UK
Whoops - I must have been sleepy

Edit - and x5 works just in regards to using it to eg. change the bandaged axis.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 11, 2013 7:16 pm

Joined: Sun May 29, 2011 2:56 pm
Location: New York
This is very nice! I especially like it because it defies the "standard" rules of Twisty Puzzles.

-Doug

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 Post subject: Re: Irreversible Cube by OSKARPosted: Fri Apr 12, 2013 2:51 am

Joined: Mon Nov 30, 2009 1:03 pm
rubikcollector123 wrote:
If this puzzle were turned by a small amount say, 5 degrees, it can be reversed right? So, what is the largest angle you can turn and be reversible?
You are correct. The sketch below should answer your second question. It also suggests a way to test the state of a face without making a move. If you can turn it backward a bit, the the face is locked in the forward direction. If you cannot turn it back, then it is unlocked.
KelvinS wrote:
... and what's the minimum corner-cutting angle - in case you can avoid irreversibility by cutting corners?
If you are asking me whether you can cheat by corner-cutting, the answer is no. As you can see from the sketch below, the four steps make sure that all locking and unlocking goes in lockstep.

Oskar
Attachment:

Irreversible Cube - view 4.jpg [ 40.08 KiB | Viewed 5538 times ]

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 Post subject: Re: Irreversible Cube by OSKARPosted: Wed Apr 17, 2013 7:30 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Oskar wrote:
rubikcollector123 wrote:
If this puzzle were turned by a small amount say, 5 degrees, it can be reversed right? So, what is the largest angle you can turn and be reversible?
You are correct. The sketch below should answer your second question. It also suggests a way to test the state of a face without making a move. If you can turn it backward a bit, the the face is locked in the forward direction. If you cannot turn it back, then it is unlocked.
KelvinS wrote:
... and what's the minimum corner-cutting angle - in case you can avoid irreversibility by cutting corners?
If you are asking me whether you can cheat by corner-cutting, the answer is no. As you can see from the sketch below, the four steps make sure that all locking and unlocking goes in lockstep.

Oskar
Attachment:
Irreversible Cube - view 4.jpg

Wow, that's pretty neat. I'm assuming that when the axis is forced downward by the slope, something happens which shoves the other two axes up, if they aren't in that position already. Is there some mechanism preventing the axis from shifting up when it's not supposed to? For example, if I held the puzzle in my hand and slapped it against a table, would the desired axis "pop up", thereby granting a free rotation which would otherwise not be permitted?

Also, I believe your "lock step" link refers to the wrong article, unless it involves prison/military marching formations.

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 Post subject: Re: Irreversible Cube by OSKARPosted: Thu Apr 18, 2013 2:39 am

Joined: Mon Nov 30, 2009 1:03 pm
stardust4ever wrote:
For example, if I held the puzzle in my hand and slapped it against a table, would the desired axis "pop up", thereby granting a free rotation which would otherwise not be permitted?
Yes, the puzzle has some cheats, which have been designed on purpose. As you can see in the video, you can reset blockers with a small metal pin. A bit of slapping and tapping works as well. You can also take it apart with a screwdriver.

Oskar

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