Rubik's Cube

davegage
Posers and Puzzles 27 May '04 03:38
1. 27 May '04 03:38
How many different (distinguishable) combinations are possible with a standard 3X3 Rubik's Cube?

If that is too easy for you, try finding the number of combinations of a 20X20 cube or generalize to nXn.

I think it is an interesting exercise to do at least once since it is truly astounding how many combinations are possible. if you ask any random person to wager a guess off the top of their head, their guess is likely to be many orders of magnitude off of the real answer.

by the way, if you have never tried cubing before, i urge you to start. there are many decent beginner solutions on the web that are good places to start. i started speed cubing last month and have gotten my average solve time down to just over 1 minute using a layer-by-layer method -- neither good nor bad, but i am always trying to improve.
2. TheMaster37
Kupikupopo!
28 May '04 20:41
58 seconds ðŸ™‚

I seem to remember a figure along the size of 80,000 lessee...

20 movable cubes. Wich gives at most 20! possible arrangements. Six colors, and it doesn't matter how you arrange cubes of the same color, wich cancels out 6 times 8! arrangements. Four rotations around 3 axes is 12 ways to mirror an arrangement. So i'd be on around the 838 053 216 000 arrangements left. It's late and i can't narrow it down any further with this state of mind.
3. 29 May '04 03:58
58 seconds...nice. do you use a layer method? i think the problem with the layer methods are that they involve too many turns...i am thinking about going to a more advanced method.

the way i like to think about the cube is that first of all you have six face centered tiles which are immobile, ie. they are always in the same position with respect to each other. so that leaves you with 12 edge pieces, each having 2 colored sides and 8 corners pieces, each having 3 colored sides. each of these pieces can be placed anywhere on their respective sub-lattice and in any orientation with respect to their colored sides. then you also have to account for the fact that, like you said, there are 12 mirrors for each arrangement.

i won't post the answer just yet, but i can tell you that your posted estimate is way too low (several orders of magnitude).
4. 29 May '04 08:38
Originally posted by TheMaster37
58 seconds ðŸ™‚

I seem to remember a figure along the size of 80,000 lessee...

20 movable cubes. Wich gives at most 20! possible arrangements. Six colors, and it doesn't matter how you arrange cubes of the same color, wich cancels out 6 times 8! arrangements. Four rotations around 3 axes is 12 ways to mirror an arrangement. So i'd be on around the ...[text shortened]... 0 arrangements left. It's late and i can't narrow it down any further with this state of mind.
Um, we have here a 20x20x20 cube, but only the sides have cubes. Furthermore, the corners are always corners etc. So you have 6! possibilities for the corners, (12*18)! possibilities for the edges and (6*18*18)! possibilities for the rest of the cubes. That's a total of 6!*12!*(18!)^3. Then allow for mirroring, same coloring(!) etc. You calculate, TM37! (I don't have a calculator nearby ðŸ˜‰)
5. TheMaster37
Kupikupopo!
29 May '04 11:52

I do use a layer method. first the sides of the top layer, then the corners. Then the sides of the middle layer. After that i arrange the corners of the last layer to that they are on the right place. Then i rotate the corners in their place until their are completely correct. After that i move and turn the sides of the last layers.
6. 29 May '04 21:18
Originally posted by piderman
Um, we have here a 20x20x20 cube, but only the sides have cubes. Furthermore, the corners are always corners etc. So you have 6! possibilities for the corners, (12*18)! possibilities for the edges and (6*18*18)! possibilities for the rest of the cubes. That's a total of 6!*12!*(18!)^3. Then allow for mirroring, same coloring(!) etc. You calculate, TM37! (I don't have a calculator nearby ðŸ˜‰)
The 20X20X20 cube is a lot trickier than the standard cube. one reason it is tougher to solve is that, unlike odd number cubes (for example 7X7X7 or 3X3X3), there are no immobile center tiles on each face. if you want to see some facts on the 20 cube, including the number of possible combinations, you can go to

http://www.speedcubing.com/chris/20cube.html

the actual number of combinations is about 1400 orders of magnitude more than your estimate! it is truly a mind boggling number. i have really no concept of just how big the number is -- i don't think many people do.
7. 01 Jun '04 17:19
For the 3x3x3 Cube:
There are 8 corner cubies which can be permuted 8! ways The 8 corners can be oriented 3^8 / 3 ways (we divide by 3 because the orientation of the last cube is fixed be those of the remaining 7)

There are 12 edge cubies which can be permuted 12! ways and oriented 2^12 / 2 ways

Furthermore the permutation of the whole cube must be &quot;even&quot; so we have one more division by two, giving:

8! x 3^7 x 12! x 2^11 x 1/2 = 43252003274489856000 ~ 4.3 x10^19

You could use similar logic to work out the 20^3 case, but it might take a while to identify all the different types of cubie (the sub-cubes).
8. 01 Jun '04 18:08
Don't forget to eliminate all those cases where exactly one edge piece is reversed. These orientations are impossible to achieve starting from a solved cube.

If you have a cube that, when solved, yields an edge that is reversed, then don't fret for someone has used a screwdriver to reverse that edge to drive you batty.

-Ray.
9. 01 Jun '04 20:19
Ray,

I didn't forget: that's why the respective factors for corner and edge orientations are 2^11 and 3^7 (and not 2^12 and 3^8)

Howard.
10. 01 Jun '04 20:34
yes, howard's answer is correct and can be proven formally through group theory methods. the answer of ~ 43 quintillion is a mind boggling number. to put it in perspective somewhat, if you handed a mixed up cube to a blind-folded person who then just started making random moves, say three random moves per second without taking any breaks, 365 days a year, then it would literally take him (on average) billions of lifetimes to randomly hit the one and only solution to the cube (assuming some first-order naive approximation that all moves are equally likely and uncoupled).

i guess that is why it reportedly took Rubik a full month of cubing to solve his cube when he first invented it in the early 80s.
11. 02 Jun '04 02:20