Can the Rubik's Cube be solved with a specific color pattern?

[MarneMath]

My friend has been working on mastering the Rubik's cube lately. He can solve it now and make new designs. One day he asked me for a pattern so, I told him I wanted the red center piece enclosed by the green cubes, and the green center piece enclosed by the red cubes. I think this is impossible, but he believe it is. Does anyone know if this can be done?

The Cube looks like this:

Placing the cube flat on a cube, the color of the face facing me is red. Moving right the colors are blue, orange and green and then back to red. The face at the bottom is yellow and the face on top is white.

 

[justsomeguy]

Not possible, as red and green are adjacent. It can be done with red and orange, blue and green, yellow and white, etc.

 

[wuliheron]

The center squares in a standard 3x3x3 cube don't move relative to each other. You can move all the other pieces around them, but the centers themselves form a jack or spindle shape that never changes.

 

[Containment]

You can do it but it involves taking the cube apart and putting it back together wrong :)

 

[jim mcnamara]

If the cube has color stickers, just move them around and pretend like you did it.

 

[Evo]

If the cube has color stickers, just move them around and pretend like you did it.

Evo likes how Jim thinks.

 

[Jimmy Snyder]

Start with the cube in the solved state. Then film yourself wildly twisting and turning it into some horrible mess that looks unsolvable. Then show the film backwards.

 

[Evo]

Start with the cube in the solved state. Then film yourself wildly twisting and turning it into some horrible mess that looks unsolvable. Then show the film backwards.

Lol!

 

[wuliheron]

One of my favorite patterns to make on the cube is hybrid between 90 degree rotated crosses and ordinary checkerboards. Combined when people try to solve for the ordinary checkerboard pattern which is easy it instead flip-flops to another checkerboard pattern as if they had simply made a mistake. You can keep some people busy forever that one.

 

[Borek]

Borrow a lego Rubik cube machine:

 

[wuliheron]

That's a odd video. The cube is usually about 13 to 17 at most moves from being solved and to make that video they must have programmed a more complicated solution. World record holders sometimes just happen to recognize the pattern the cube is scrambled in and solve it in maybe 17 moves. The cube is the shape everybody loves though and nothing else feels as good in your hands or brain.

 

[Borek]

Yes, it solves in way too much moves, which most likely means it uses some relatively simple and naive algorithm. I remember one such algorithm was to solve upper layer first, then repeat some sequence of moves that rotates corners of the middle layer in some particular order, till they happen to be in the right places, then repeat some other combination of moves that rotates some of the pieces of the bottom layer in some particular order till they fit, then use exactly the same approach for the remaining pieces - and you are ready. Key to success is a selection of particular sequence of moves for each stage - it needs to shift pieces you want to move without moving all other pieces.

Kind of a brute force approach that you can teach to a monkey, and it typically requires much more moves then the optimal solution, but it works.

 

[wuliheron]

I prefer to think of the cube as a closed system myself and, it is my toy after all. Once you understand the power of context over content the cartoons become obvious and it explains why children under five can have a better chance of success solving the cube.

 

[BruceW]

lulwut?

 

[rcgldr]

Rubiks Cube, 3x3x3, and Professor's Cube / Rubik's x5, 5x5x5, have centers mounted to a 3 axis "star" and can only be rotated. One variation of Rubik's Cube had 4 of the centers marked with arrows to provide a bit of an extra task to solve the cube. Rubik's Revenge, which was 4x4x4 had movable centers, and adjacent sets of 4 center pieces could be swapped.

The way I initially solved Rubiks Cube was top layer, middle layer, and bottom layer. The top was straight forward, the middle layer used a pattern of moves that could be repeated to solve the bottom layer. The bottom layer had to be solved by getting corners into correct position, then rotated so colors match centers. Another sequence could be used to rotate any 3 bottom edge pieces, and a final set of moves to flip them so colors matched centers.

The standard method is to do all 8 corners, then top, bottom, middle edge pieces. First all corners are moved into place then rotated so colors match. Then top and bottom edge pieces are moved and rotated into place. The middle edge pieces were moved into place, then a pattern called "Rubiks maneuver" was used to flip two middle pieces to get colors to match centers if needed.

For Rubik's Revenge, 4x4x4, the top and bottom sets of 4 center pieces could be fixed first. Then the middle sets of 4 center pieces could be fixed, but solving the corners first elminated having to remember the color orientation. Then the cube could be solved similar to a 3x3x3 cube, except two new patterns not possible with 3x3x3 cubes could occur since edge pieces are in sets of two. It is possible to have two pairs of edge pieces swapped (on different sides) and 1 pair of edge pieces rotated (colors on wrong side). The 4 center pieces have skinny "legs" to connect them into slots in an inner sphere and are easily broken, so you have to be careful with the 4x4x4 cubes.

The 5x5x5 cubes don't introduce any new moves. The only "new" pieces are the center "edge" pieces, but these solved in the same manner as the center "corner" pieces (like the 4x4x4). The Professor's Cube uses the one time standard of white and blue on opposite sides. The Rubik's x5 has white and blue adjacent.

Somewhere in my archives, I have a windows 3.1 version (it will run on any version of Windows) of a virtual Rubik's cube that let's you set the number of pieces.

 

[Borek]

The way I initially solved Rubiks Cube was top layer, middle layer, and bottom layer. The top was straight forward, the middle layer used a pattern of moves that could be repeated to solve the bottom layer. The bottom layer had to be solved by getting corners into correct position, then rotated so colors match centers. Another sequence could be used to rotate any 3 bottom edge pieces, and a final set of moves to flip them so colors matched centers.

That's what I wrote about. It is guaranteed to work, but it is far from optimal.

 

[rcgldr]

The way I initially solved Rubiks Cube was top layer, middle layer, and bottom layer.

That's what I wrote about. It is guaranteed to work, but it is far from optimal.

It was slower, but it required fewer patterns and you didn't have to know about "Rubik's maneuver" to rotate (to get colors to match) a pair of edge pieces. I figured out how to disassemble and reassemble the cube, then wrote down what patterns of moves did, then use that to solve the puzzle. I'm also slower in that I keep the top on the top, rather than flip the puzzle over once the top is done, to solve what was the bottom.

 

[2sin54]

That's a odd video. The cube is usually about 13 to 17 at most moves from being solved and to make that video they must have programmed a more complicated solution. World record holders sometimes just happen to recognize the pattern the cube is scrambled in and solve it in maybe 17 moves. The cube is the shape everybody loves though and nothing else feels as good in your hands or brain.

The cube is 20 moves at most from being solved. The world record for the fewest moves made to solve a scrambled cube was 20 moves.

 

[wuliheron]

The cube is 20 moves at most from being solved. The world record for the fewest moves made to solve a scrambled cube was 20 moves.

According to world record holder Tony Snyder, the cubes used in championships are scrambled using 13 moves and simply reversing them would solve the cube. His record in a competition is 19, but 13 to 17 is what a computer can do and his goal is to replicate the feat.

 

[Borek]

Wikipedia entry on optimal solutions for Rubik's Cube lists a "computer-assisted proof" that at most 20 moves are required.

 

[wuliheron]

Wikipedia entry on optimal solutions for Rubik's Cube lists a "computer-assisted proof" that at most 20 moves are required.

I believe that is correct, but 17 is more like the average and 13 the minimum.

 

[BruceW]

well, 13 is only minimum because in the championships, they only scramble them with 13 moves, right?

 

[wuliheron]

well, 13 is only minimum because in the championships, they only scramble them with 13 moves, right?

They use a formula that produces the most scrambled cube possible, so 13 is the minimum for a completely scrambled cube.

 

[collinsmark]

They use a formula that produces the most scrambled cube possible, so 13 is the minimum for a completely scrambled cube.

That doesn't seem right according to these:

http://www.engadget.com/2010/08/09/rubiks-cube-solved-in-twenty-moves-35-years-of-cpu-time/

http://www.cube20.org/

 

[BruceW]

nice links. 35 years of computer time. phew! So when the cube is in an initial configuration which takes the longest to solve, it takes exactly 20 moves to solve it? (if I understood right)

edit: and to put it in other words, even in the worst-case scenario, it is possible to solve in 20 moves?

 

[wuliheron]

I'm not sure what is confusing you. Computer programs exist which can solve the cube from the most scrambled state in 13 moves every time. The computer merely looks at all the pieces and figures out what orientation it was scrambled in. However, the program these links refers to actually solves the cube from any orientation and it is the computer that averages 17 moves to do so rather than the world champs who are still struggling to get 20 average. The champ who did it in 19 moves happened to just recognize the orientation it was scrambled in giving him a slight edge over the competition.

P.S.- Personally I find competitions boring and prefer to study the patterns you can make on a cube. A standard Rubik's has 40 patterns and a 4x4x4 Rubik's Revenge has 400. Many of these patterns are named after subatomic particles such as the "Large Meson" pattern.

 

[BruceW]

but where do you get this number 13? collinsmark has a couple of links which state 20 is the fastest solution in the worst-case scenario.

edit: do you mean 13 moves is the average minimum moves to solution, given that the initial configuration was chosen purely randomly? If this is what you mean, then it is definitely not solved in 13 moves every time. sometimes it will be 20, sometimes 13, and sometimes zero moves (and every number up to 20, I would guess).

 

[wuliheron]

but where do you get this number 13? collinsmark has a couple of links which state 20 is the fastest solution in the worst-case scenario.

Again, they use 13 moves to scramble the cube and if you do them exactly backwards that solves it. Therefore, 13 is the minimum number of moves that will work, 17 is the average the new computer program can manage, and 20 is the maximum the computer requires. The computer program is complex and mere mortals use simplified variations to beat their human competition.

 

[BruceW]

right. But you see what I mean. The number 13 has no significance whatsoever. It is not 'completely scrambled' either. Because for example, there are configurations that take 20 moves to solve. Therefore those configurations are 20 moves away from the solved configuration. Therefore, the cube is not completely scrambled, because some configurations (the hardest ones) are excluded. In other words, in the tournament, they specifically exclude the most difficult initial configurations. ps sorry if I sound hostile. I get carried away sometimes.

edit: I'm assuming here that they scramble the cube from the solved configuration, since someone would have solved it in the previous round of the tournament. So 13 'scrambling moves' from the solved configuration will not get the cube to the most difficult configurations. (and by 'most difficult' I just mean those that take from 14 to 20 moves to solve).

 

[wuliheron]

right. But you see what I mean. The number 13 has no significance whatsoever. It is not 'completely scrambled' either. Because for example, there are configurations that take 20 moves to solve. Therefore those configurations are 20 moves away from the solved configuration. Therefore, the cube is not completely scrambled, because some configurations (the hardest ones) are excluded. In other words, in the tournament, they specifically exclude the most difficult initial configurations. ps sorry if I sound hostile. I get carried away sometimes.

edit: I'm assuming here that they scramble the cube from the solved configuration, since someone would have solved it in the previous round of the tournament. So 13 'scrambling moves' from the solved configuration will not get the cube to the most difficult configurations. (and by 'most difficult' I just mean those that take from 14 to 20 moves to solve).

Yes, they scramble the cube from the solved position, however, the thing to remember is the cube is NEVER, EVER, EVER more than 13 steps from being solved. If it takes longer that means you don't have the ideal solution and doesn't necessarily reflect how scrambled the cube is.

 

[collinsmark]

Yes, they scramble the cube from the solved position, however, the thing to remember is the cube is NEVER, EVER, EVER more than 13 steps from being solved. If it takes longer that means you don't have the ideal solution and doesn't necessarily reflect how scrambled the cube is.

If during the official scrambling, only 13 moves are used, then Yes, I agree that the cube is never, initially more than 13 moves from being solved.

What the links are saying however, is if the initial scrambling has more than 20 moves, then there does potentially exist states that take up to 20 (minimum) moves to solve. 20 is the maximum, minimum number of moves that it takes to solve a Rubik's cube.

---

On a different note, I find interesting is that the entropy of a Rubik's cube (according to the link) is maximum at 18 ideal moves away from the solution. But there does exist some states that have a lower entropy, yet a larger number (19 or 20) of ideally solvable moves away from the solution (if we are to believe the link). In other words, if the cube starts in one of these states, the solver must first increase the cube's entropy to get it to 18 moves away from the solution before decreasing its entropy.

 

[BruceW]

@wuliheron: well collinsmark's links say otherwise. They say it can take 20 steps to be solved, even if you have the ideal solution.

 

[BruceW]

On a different note, I find interesting is that the entropy of a Rubik's cube (according to the link) is maximum at 18 ideal moves away from the solution. But there does exist some states that have a lower entropy, yet a larger number (19 or 20) of ideally solvable moves away from the solution (if we are to believe the link). In other words, if the cube starts in one of these states, the solver must first increase the cube's entropy to get it to 18 moves away from the solution before decreasing its entropy.

yep, that is interesting. So, I guess a completely 'shuffled' cube will be most likely 18 moves away from the solution? And I suppose entropy doesn't always agree exactly with the idea of how 'not useful' the state of the system is. I think entropy is one of those things where the mathematical definition is good, but the interpretation is not always clear.

 

[collinsmark]

yep, that is interesting. So, I guess a completely 'shuffled' cube will be most likely 18 moves away from the solution?

Yes, if we are to believe the link, it means that if you shuffled a Rubik's cube for a long time -- not just 13 moves, but shuffle it randomly until the cows come home -- it's most likely to fall into a state that takes 18 moves to solve, minimum.

It also means that if you were the shuffler, and you wanted to put the cube in a state that takes 19 or 20 moves to solve, minimum, you would probably have to put some thought into the process, particularly the 20 move configuration. (It's very unlikely that the cube would ever end up in the 20 move minimum configuration by pure random shuffle. Possible yes, but very unlikely.)

Putting it yet another way, it means that you were attempting to solve the cube, and you were able to recognize that it were in one of these configurations (19 or 20 ideal moves away from the solution), you would have a very high probability of getting closer to the solution by first making one or two totally random moves.

And I suppose entropy doesn't always agree exactly with the idea of how 'not useful' the state of the system is. I think entropy is one of those things where the mathematical definition is good, but the interpretation is not always clear.

I can think of a simple analogy to this situation. Suppose you had six coins lined up in a row, all heads. Randomly roll a six sided die and flip the corresponding coin over. Repeat indefinitely. You will find that after several rolls the coins converge to having around 3 heads and 3 tails (these are the states of the highest entropy). Although improbable, it is possible on occasion that all 6 coins will show tails. This is another lowest-entropy state, even though it's the furthest number of moves from the first lowest entropy state with all heads.

 

[BruceW]

I don't really understand the analogy. But I do understand the bit you were saying before then. The 'difficult' configurations (which take 19 or 20 moves to solve) are quite rare compared to the other configurations. And this is what seems weird in my intuition of entropy, because I would normally associate the most likely set of configurations with being the most difficult. For example, an ideal gas, the high entropy set of states are where the molecules take up all the space, flying in all directions, all mixed up. (i.e. difficult). But really, I guess is to be expected that what I think of as 'difficult' does not always match up with high entropy.