Online since 2002. Over 3300 puzzles, 2600 worldwide members, and 270,000 messages.

TwistyPuzzles.com Forum
 It is currently Wed Apr 23, 2014 10:02 pm

 All times are UTC - 5 hours

 Page 1 of 1 [ 7 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Number of positions of a 17x17Posted: Mon Jan 13, 2014 4:29 pm

Joined: Mon Jun 07, 2010 11:24 am
Location: Long Beach, CA USA
I'd like to figure out how many reachable and discernible positions are possible for Oskar's Over the Top 17x17x17 cube.

I remember years ago seeing Richard Carr and Chris Hardwick's formulas for finding the positions of an NxNxN cube: http://www.speedcubing.com/chris/cubecombos.html
http://www.ws.binghamton.edu/fridrich/Richard/Cubes.pdf

Using the formula from Chris I've gotten this: http://bit.ly/1a2Tdhy
However I don't trust the answer because when I try it for a 3x3x3 I get this: http://bit.ly/L3BOtR

Ultimately I'd like to know the number of positions for both Oskar's Over the Top 17x17x17, and for a generic 17x17x17. (Both ignoring edges and corners, and including them.)

Any help would be appreciated!

_________________
-Kenneth

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 4:47 pm

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
In another thread I calculated the positions of a super 17x17x17, which I did with my own approach of handling each piece type one at a time. So I think I'll show it here as well.
For Oskar's 17x17x17, there are 56 groups of centers(ignoring the central center pieces), each of which contain 24 centers. 4 in each set have the same color, and there are 6 colors for them as well. This means there are:
((24!^56)/(4!^336))=4.371*10^868 possible combinations.
For a normal 17x17x17, we must account for the edges and corners.
Corner orientations: (3^7)
Corner permutations: (8!)
Middle edge permutations: (12!/2)
Middle edge orientations: (2^11)
There are 7 groups of wing edges, each containing 24 edges, which gives (24!^7) for their permutations, which are not restricted to even parity.
Multiplying these all together gives:
(8!)*(3^7)*(12!/2)*(2^11)*((24!^56)/(4!^336))*(24!^7)=6.691*10^1054 possible combinations.
(EDIT: fixed an error after Brandon pointed it out)
Are you making a "Why is the Rubik's cube so hard" episode 3? If so, maybe you could use Planck volumes to make a comparison.

_________________
My Shapeways Shop

Last edited by benpuzzles on Mon Jan 13, 2014 5:36 pm, edited 2 times in total.

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 5:19 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
I'm a bit busy right now but I'll make a generic NxNxN function later.

In the mean time, my calculation is:

8 corners: 8! and 3^3 twist, modulo parity and twist restriction
12 middle edges: 12! and 2^12 twist modulo parity with corners and twist restriction
7 edge wing types: (24!)^7
56 center piece types: (24!)^42 modulo 4 duplicate colors on each face for each type

? ((8! * 12!) / 2) * (3^8 / 3) * (2^12 / 2) * ((24!)^7) * ((24!) / ((4!) ^ 6))^56 =
669092608710520096261408314575991967111408122691540707290601365
294496257802119618956938205705136041636028689428016336273634131
487726647385709719884121474908504692670910698985371460377688900
699349198842497638186290806683678986850334593701338440753224464
740484033975924212665646410310537811828359510439026667039347182
757336297730724281196033862808102327432941067250179060157266025
054048093556007135154007603434085100547748064670636958246371249
119454463174658330555208369758612382449403973332343369712706870
923838041336318861143098538193323362829868347779481784646568888
023722509270749811402466088245770360947102010990952406412565132
175988024238740278224215845876500391255162029122054815404278641
999475767222218668661025073508769221156288818802031152122167665
036654264459567862643991333029626496008847360000000000000000000
000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000

That's 6.69 * 10^1054

I need to double-check the calculation and make it generic for any size.

Ben: your 12 denominator term in ((24!/12)^56) is definitely wrong.

_________________
Prior to using my real name I posted under the account named bmenrigh.

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 5:24 pm

Joined: Sun Nov 03, 2013 4:15 pm
In Chris Hardwick's formula, the brackets that look like ⌊ and ⌋ represent the floor function, meaning round down to the next integer. The other brackets ⌈ and ⌉ in Richard Carr's formula represent the ceiling function, meaning round up to the next integer. See http://en.wikipedia.org/wiki/Floor_and_ceiling_functions for more information.

So for a 3x3x3 you want put in n = 3, and get: http://bit.ly/1cXrhJs, which is the right answer.

And for a 17x17x17, put n = 17, and get: http://bit.ly/1lWHV0S. The other formula gives exactly the same thing: http://bit.ly/1eEb11I.

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 5:26 pm

Joined: Mon Jun 07, 2010 11:24 am
Location: Long Beach, CA USA
Thanks guys for the help!

Ben: You guessed it! I'm doing brainstorming on the topic but we'll see if I ever get to making a video on it.

_________________
-Kenneth

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 6:03 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
All right, now how about a 17-layer minx?

Top

 Post subject: Re: Number of positions of a 17x17Posted: Mon Jan 13, 2014 6:17 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Jared wrote:
All right, now how about a 17-layer minx?

viewtopic.php?f=1&t=24058&p=285637&hilit=oskarminx#p285637

_________________
Prior to using my real name I posted under the account named bmenrigh.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 7 posts ]

 All times are UTC - 5 hours

#### Who is online

Users browsing this forum: Google [Bot], grigr and 6 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Announcements General Puzzle Topics New Puzzles Puzzle Building and Modding Puzzle Collecting Solving Puzzles Marketplace Non-Twisty Puzzles Site Comments, Suggestions & Questions Content Moderators Off Topic