5x5 Rubik 39;s Cube Possible Combinations
Pei Tebow
If you want to find the number of combinations for a larger-sized cube, you can do so using the form below. I should warn you that if you type in a ridiculous size for the cube (say more than about 30), it will bog down your computer, not mine.
To simplify this down to the 22 puzzle, you have 8 pieces to make up the puzzle. Each puzzle piece can have 3 orientations. So the answer is the number of different places to put the puzzle pieces times the number of different combinations of orientations that the pieces can have (divided by 2 because of the rotation dependency).
The cube has six faces and so there are 6! ways of choosing the order in which to twist the faces. Each face has four positions, so I get 6! x 4^6 = 2,949,120 ways of scrambling the cube. I feel that this should be an upper limit to the number of legal combinations of the squares, since the only way to get a legal combination is by twisting the faces.
I agree. If the cube has pictures or figures you will see that the center pieces are rotated from their original position. Just the combination of the center pieces is 4096. Ignoring this factor significantly reduces the number of patterns for a cube with solid colors
Are you talking about the properly assembled Rubiks cube that can be solved? If you do, your computation of the number of combinations is wrong, because it includes combinations where it is not possible to exist. As an example, if you dismantle the cube and put the pieces back randomly, there is a greater chance that it cannot be solved. In this case, any combinations derived from this improperly assembled cube is not counted as a valid combination.
Hi, I have a question. Does the math accounts for orientation of the cube? What I mean is, if I put a solved cube on a table with white on top and red facing me. I rotate the cube 90 degree and now have blue facing me. In the math, is that considered a different combination? We can look at the same solved cube 24 different ways. Is that 24 combinations? If I do one move. Is that only one new combination or 24 new combinations?
So 8! would have to be multiplied by 3^8 since all 8 pieces can be orientated in 3 ways. Then, divide that number by 3 since that is the number of parities the corner piece can be in, 2 of which are impossible.
Think of a 2 by 2 cube. It has 8 corners which can be rearranged in 8! combinations. We then multiply by 3^7 since the 7 pieces can individually rotate. We then divide this by 24 because there are 24 possible parities.
(8! * 3^7) / 24
A 3 by 3 cube uses the same 8! * 3^7 because the corners will always hold this property of rearrangement and individually rotating. We multiply this by 12! and 2^10 because it has a fixed center and 12 edge pieces which can be rearranged in any order and 10 pieces can individually flip.
(8! * 3^7 * 12! * 2^10)
Considering a single side of a standard 3x3 Rubik's Cube, are all possible color combinations attainable? Or are there certain combinations that can only be made by breaking apart the device or moving stickers?
To put another way, if I randomly selected the color for each of the 9 squares on one side, could standard solved cube always be shuffled in such a way that one of its faces contained that configuration?
The single side in question would need to be determined by the color of the middle square of course. Then you can simply slot in any color to the edges and to the corners. This is always possible because:
The restrictions to what you can move where with a cube only apply toward the end where, for example, re-orientating a single piece cannot be done. In this instance, and orientation is possible because the remaining pieces are not restricted and can be orientated either way.
it's possible to achieve any even permutation of the cubelets, subject to the obvious constraints that (1) faces stay fixed, (2) edges remain edges, and (3) corners remain corners. This is obviously enough to achieve a state in which each cubelet on one face has somewhere on it the colour that we want for its square. (Consider some pair of cubelets not on that face; with them one way around or the other, the permutation taking us there is even.)
we want to get the orientations right. We have a mod-3 constraint on the total "twist" of all the corners, and a mod-2 constraint on the total "flip" of all the edges, and that's all. So pick a corner and an edge that aren't on the target face; by letting these get twisted/flipped in whatever way is necessary, we can achieve whatever twists and flips we want on the target face.
The Rubik's cube is the best selling toy in the U.S.
some may think it is impossible to solve, even Erno Rubik didn't think there was a combination to
there are 43,000,000,000,000,000,000 (that is 43 quintillion) possible combinations.
So how can you solve the cube without memorizing 43 quintillion different combinations?
there are things called algorithms that are based on mathematical equations that get pieces to places without messing up the whole cube.
Some of you may be wondering why i wanted to learn to solve the Rubik's cube, well I am very big into math and I am Very good at memorizing things in fact I have an IQ of 120 but you don't need a high iq to learn how to solve the Rubik's cube. I learn this method last Christmas when I was bored on Christmas break and wanted something to do. It only took me a week to learn to solve it, but it wasn't like I was doing it for a week straight 24 hours a day, in fact i probably spent an hour a day max that's 7 hours max it took me to solve the Rubik's cube. So in this tutorial i will show you a beginners method that I used on how to solve the Rubik's cube and maby later i might teach you an advance method on how to solve it, but you will need to know how to solve it with the beginners method first before you move onto the advanced method.
What You Need
- A Rubik's cube
- an Attentive mindset
To get how to solve the Rubik's cube i will explain to you how this method solves it.
Most people think you solve the cube face by face but that is wrong you are supposed to solve one face then layer by layer.
the blue picture below shows the notation of moves that make up an algorithm
first things first you need to get the cross it doesn't matter what color it is as long as you have a cross where the colors on the sides match
Note: what I mean by the colors on the side matching is if you are solving the red faces cross the color of the edge piece should match up with it's side
Congratulations you have solved the first side but that isn't very impressive so lets go on.
Now you need to flip the solved side so its on the bottom so you can get the edge pieces aligned also known as the second layer
you need to find the edge pieces that belong in the middle row
you need to find an edge piece and line it up with its color
if it has to the right the algorithm you need to use is
U, R, Ui, Ri, Ui, Fi, U, F
if it has to go to the left use this algorithm
Ui, Li, U, L, U, F, Ui, Fi,
now do that to all the edge pieces
you are 2/3rd's of the way done
Now you have the first 2 layers finished
you now need the top cross
keep doing this algorithm until the top cross is solved
F, U, R, Ui, Ri, Fi
now you should have the top cross
to solve the top face of the Rubik's cube you will need to do this algorithm
until the top of you Rubik's cube looks like the picture then turn the whole cube to the right and do the again. if the top of you cube is solved your good but if it isn't get turn it so it looks like the picture then turn it to the right then do the algorithm again
R, U, Ri, U, R, U, U, Ri
now all you need to do is get the 3rd layer done
Congratulations You have now solved the Modern day mystery; The Rubik's Cube
but if all this went right over your head try checking out this link it does a very good job explaining how to solve it -center/3x3_guide/
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Both topics can involve advanced mathematics in combinatorics and group theory, but I will keep it somewhat light so to be accessible to a large audience. I hope this comes off as a fun read, with a sprinkling of mathematics and history.
The cube has 6 middle pieces each with one face, 8 corners each with three faces, and 12 edges each with two faces. You can physically pull out the corners and edges and you are left with only the 6 middle pieces that are fixed:
Now we can do the same thing with corner pieces. We grab the first corner and notice eight possible positions to put it in. Depending upon which face is up (or forward/down/backward), there are three different ways to insert it. That means 8 x 3 possible ways for the first corner. Likewise, there are 73 ways for the second corner. You can do the rest, the total possible combinations for the corners are:
So the total number ways of putting it back together is the product of these values: 12! 212 8! 38. Almost there, but as I said before this is not the real total number of mixups! The reason why is because when we turn the cube the ways that it allows us to turn it, it cannot reach all of these mixups. It turns out that out of all those combinations, the last corner that you insert cannot go all three possible ways: only one of them agrees with a valid mixup. Because of this, we need to divide the above value by 3. Similarly, there is a dependency on the edges that requires us to divide by 4. In other words, we have to divide the quantity above by 3 4 = 12, and hence the total number of combinations is:
Unfortunately, these online solvers work for computers, and are too complex for humans to memorise. Instead, the speed solvers of today use methods like CFOP or Roux, and average in the mid 40s to low 50s number of moves. The current official WCA (World Cubing Association) world record for average of 5 solves is 5.09 seconds by Tymon Kolasiński from Poland, however there are a handful of cubers that could beat the record on a really good day so we could be seeing sub-5 second averages in the near future. Tymon uses the CFOP method.
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