Message #764
From: djs314djs314 <djs314djs314@yahoo.com>
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!
Date: Wed, 04 Nov 2009 04:14:30 -0000
Tonight I started working on the piece-counting formulas. To be
complete I have decided to find formulas for all acceptable
polytopes. Unfortunately, I started with the simplex rather than
the more important duoprisms, and it took longer than expected.
I have found formulas for the number of pieces and stickers of
a {3,3,3} n puzzle for n >= 2. When I finish I will upload a pdf
file with all the formulas, but for now here are the simplex ones:
{3,3,3} n, n >= 2
Piece Count: 5 + 5*ceil((n-2)/n) + floor(2/n) +
(20n - 50)ceil((n-2)/n) + 10((n-3)^2)ceil((n-3)/n) +
5(C(n,3) + C(n-2,3) + C(n-3,3))
Sticker Count: 5n(n^2 + 1)/2
where ceil is the ceiling function, and c(n,k) is the binomial
coefficient.
These formulas completely agree with Roice’s counts. As you can
see, the piece count is quite complex, which is why I did not get
further with the task tonight. Tomorrow I will start on the more
pertinent duoprism formulas, although I will not have much free
time, and will most likely finish Thursday.
There were some interesting things I found out about the simplex
puzzles. For one thing, there are 2 different types of pieces,
one with tetrahedral stickers and the other with octahedral
stickers. Most of you probably already noticed the octahedral
stickers if you looked at that puzzle. In fact, for the simple
reason that higher-dimensional simplices are not easily tiled by
lower-dimensional simplices, it may be impossible to define a Rubik’s
simplex puzzle for dimensions greater than 4. The other really cool
thing concerns the {3,3,3} 2 puzzle. It contains a 5-colored piece!
What at first appears to be five separate 1-colored pieces are in
fact a single piece. It’s not hard to see once you load up the
puzzle and take a look, but this surprised me.
I apologize for the way my last post appears, I’m not sure why that
happened. I sent this post from the website to prevent it from
happening again. I’ll write back when I get the duoprism formulas,
hopefully they won’t be too complex and will be helpful for fixing
any even-length duoprism piece counting issues. Thanks Melinda for
your assistance, and will work on the puzzle scrambling function
once I finish this (it will actually depend upon this work), but
hopefully the algorithm won’t be as complex as this simplex piece
count; I will try to keep it computation-friendly.
Best wishes,
David
— In 4D_Cubing@yahoogroups.com, David Smith <djs314djs314@…> wrote:
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> Wow, thanks for entrusting me with this task! Now I actually feel like I will be contributing to the program itself! :) By the way, I (believe I) was able to resolve one of the issues with the program (although a trivial one). I am very happy to be of as much help as I can.
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> About the Goldilocks function, this doesn’t seem to be too difficult and I believe I will be able to produce numbers that are neither too large nor too small, and without too much computation. My basic idea is to approximate the number of positions of each puzzle. Of course, finding the actual number of positions would involve using my formulas, and we all know how complex those can get. ;) To get the approximation, I will simply calculate the total number of configurations (i.e. all positions, regardless of whether they can be reached or not.) Then I would find the number of face turns necessary to reach that number of positions. I don’t believe we need to worry that the estimate of the number of positions is always larger than the actual value, as making face turns in succession produces numbers of possibilities which grow exponentially. The only difficulty will be to find the general formula for any possible puzzle, including user-created
> ones. It would be helpful if you have a list of all of the possible schlafli symbol configurations that the program accepts, as I have not been able to find a complete list offhand, only a list of the regular polychora. If not, I will be able to figure it out. :)
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> Thanks again! I will start working on this and the piece-counting formulas tomorrow.
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> Take care,
> David
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> — On Mon, 11/2/09, Melinda Green <melinda@…> wrote:
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> From: Melinda Green <melinda@…>
> Subject: Re: [MC4D] MC4D 4.0 is really fantastic!
> To: 4D_Cubing@yahoogroups.com
> Date: Monday, November 2, 2009, 7:03 PM
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> David,
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> Yes, you’re quite valuable to this community. In fact I have another
> task that I’d love to put into your queue just after Roice’s request,
> and this one will be quite interesting to our power-solvers. What I
> need is a formula for the minimum number of random twists needed to
> fully scramble a given puzzle. It’s similar to the well-known question
> of how many shuffles it takes to fully mix a
> deck of cards. (The answer is six or seven.) I’ve implemented a
> function that seems to work reasonably well but I’m not at all
> convinced that it’s good enough. It’s important to not overestimate too
> much because that fills the log files rather quickly for large puzzles.
> It’s also important to not underestimate because that could bring into
> question the validity of some people’s solutions. I need a really good
> Goldilocks function, and you seem like just the person to produce one!
> :-)
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> Thanks!
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> -Melinda
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> David Smith wrote:
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> Note: Melinda sent me a message similar to Roice’s that I did not realize was off of the group until this moment. But I would like to thank both Melinda and Roice for their support, so I think I’ll leave the message as-is. Thanks!
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> Thank you Melinda and Roice for your assurance that my math results
> have a place in this group! Melinda, I especially appreciate that you
> want to see my work continue. :) After reading that, it hit me that I
> can find formulas for all of the polytopes in MC4D 4.0! Perhaps I can
> even categorize all of the possible polychora that can exist as a
> Rubik-like puzzle and find formulas for all of them, which would take
> care of the Create Your Own option. The n^d simplex and {m}x{n}
> duoprisms of any size would be a good place to start. Thanks again to
> both of you for helping me realize that my projects are important to
> those other than myself. :) Roice, I’ll get started on formulas to count pieces of
> the puzzles as soon as possible; that’s the first step in finding permutation counts anyway.
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> Thanks again, and I look forward to continuing my work!
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> All the best,
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> David
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> — On Mon, 11/2/09, Roice Nelson <roice3@gmail. com> wrote:
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> From: Roice Nelson <roice3@gmail. com>
> Subject: Re: [MC4D] MC4D 4.0 is really fantastic!
> To: 4D_Cubing@yahoogrou ps.com
> Date: Monday, November 2, 2009, 11:32 AM
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> For the record, I think the permutation count formulas have been some of the richest contributions to the group :D They are definitely as helpful and important as the many other ways people have chosen to help out. Like Melinda said, it definitely takes a village!
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> I haven’t heard back from anyone on piece counting yet, so I think those problems are still open if you’d like to play with any of them. I bet a general formula for the number of pieces on any {n}x{m} duoprism would be fun to work through, and it would cover a great many of the puzzles in the list! There is also a known problem for large n or m on even length duoprism puzzles, and so having such a formula would help us be able to know what exactly that cutoff is that people shouldn’t go past.
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> All the best,
> Roice
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> On 11/2/09, David Smith <djs314djs314@ yahoo.com> wrote:
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> Hi everyone,
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> Sorry I haven’t chimed in until now! I would like to thank and congratulate Melinda, Don, Jay, and Roice for their hard work and efforts in getting the beta out to us! As a novice programmer (and I mean extremely novice), I can appreciate how much work you all put into this. I would also like to congratulate Chris, Remi, Anthony, Roice, and Melinda for their solving accomplishments. I’m not the type of person to offer suggestions or improvements to others’ work, which is probably a bad thing. But I would be happy to help with the piece-counting efforts if that is still needed.
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> Also, thanks to Roice for your wonderful essay! It has been a lot of fun to be a member of this community and read about all of your experiences and contributions. I know I have not been much of an active member, only presenting my obscure mathematical results here and there, and for that I apologize. Such results are not of much use to the community, and definitely not as helpful or important as everyone else’s contributions regarding MC4D 4.0.
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> Anyway, sorry I can’t provide any experiences with the new program, as I have not tried to solve the puzzles. I respect everyone else very much for their efforts in supporting the new program. Congratulations again to Melinda, Don, Jay, and Roice, and to everyone else who has successfully solved the puzzles.
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> Best wishes,
> David
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