Thread: "MC4D 4.0 is really fantastic!"

For the record, I think the permutation count formulas have been some =
of the=A0richest contributions to the group :D=A0 They are definitely as he=
lpful and important as the many other ways people have chosen to help out.=
=A0 Like Melinda said, it definitely takes a village!

=A0

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.=
=A0 I bet a general formula for the number of pieces on any {n}x{m} duopris=
m would be fun to work through, and it would cover=A0a great many of=A0the =
puzzles in the list!=A0 There is also a known problem for large n or m on e=
ven length=A0duoprism puzzles, and so having such a formula would help us b=
e able to know what exactly that cutoff is that people shouldn't go pas=
t.

=A0

All the best,

Roice

=A0

On 11/2/09, =
David Smith <djs314djs314@=
yahoo.com> wrote:

px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">

=A0=20

BOTTOM: 0px; MARGIN: 0px; WIDTH: 470px; PADDING-TOP: 0px">

Hi everyone, Sorry I haven't chimed in until = now!=A0 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!=A0 As a nov= ice programmer (and I mean extremely novice), I can appreciate how much wor= k you all put into this.=A0 I would also like to congratulate Chris, Remi, = Anthony, Roice, and Melinda for their solving accomplishments.=A0 I'm n= ot the type of person to offer suggestions or improvements to others' w= ork, which is probably a bad thing.=A0 But I would be happy to help with th= e piece-counting efforts if that is still needed. Also, thanks to Roice for your wonderful essay!=A0 It has been a lot of= fun to be a member of this community and read about all of your experience= s and contributions.=A0 I know I have not been much of an active member, on= ly presenting my obscure mathematical results here and there, and for that = I apologize.=A0 Such results are not of much use to the community, and defi= nitely not as helpful or important as everyone else's contributions reg= arding MC4D 4.0. Anyway, sorry I can't provide any experiences with the new program,= as I have not tried to solve the puzzles.=A0 I respect everyone else very = much for their efforts in supporting the new program.=A0 Congratulations ag= ain to Melinda, Don, Jay, and Roice, and to everyone else who has successfu= lly solved the puzzles. Best wishes, David

te>

--0015175cd9fc56ed74047765ecfc--

---

From: David Smith <djs314djs314@yahoo.com>
Date: Mon, 2 Nov 2009 15:04:59 -0800 (PST)
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!

=A0




=20=20=20=20
For the record, I think the permutation count formulas ha=
ve been some of the=A0richest contributions to the group :D=A0 They are def=
initely as helpful and important as the many other ways people have chosen =
to help out.=A0 Like Melinda said, it definitely takes a village!

=A0
I haven't heard back from anyone on piece counting yet, so I think those pr=
oblems are still open if you'd like to play with any of them.=A0 I bet a ge=
neral formula for the number of pieces on any {n}x{m} duoprism would be fun=
to work through, and it would cover=A0a great many of=A0the puzzles in the=
list!=A0 There is also a known problem for large n or m on even length=A0d=
uoprism 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.

=A0
All the best,
Roice

=A0
On 11/2/09, David Smith wrote:

=A0=20







Hi everyone,

Sorry I haven't chimed in until now!=A0 I would like to thank and congratul=
ate Melinda, Don, Jay, and Roice for their hard work and efforts in getting=
the beta out to us!=A0 As a novice programmer (and I mean extremely novice=
), I can appreciate how much work you all put into this.=A0 I would also li=
ke to congratulate Chris, Remi, Anthony, Roice, and Melinda for their solvi=
ng accomplishments.=A0 I'm not the type of person to offer suggestions or i=
mprovements to others' work, which is probably a bad thing.=A0 But I would =
be happy to help with the piece-counting efforts if that is still needed.


Also, thanks to Roice for your wonderful essay!=A0 It has been a lot of fun=
to be a member of this community and read about all of your experiences an=
d contributions.=A0 I know I have not been much of an active member, only p=
resenting my obscure mathematical results here and there, and for that I ap=
ologize.=A0 Such results are not of much use to the community, and definite=
ly not as helpful or important as everyone else's contributions regarding M=
C4D 4.0.


Anyway, sorry I can't provide any experiences with the new program, as I ha=
ve not tried to solve the puzzles.=A0 I respect everyone else very much for=
their efforts in supporting the new program.=A0 Congratulations again to M=
elinda, Don, Jay, and Roice, and to everyone else who has successfully solv=
ed the puzzles.


Best wishes,
David





=20

=20=20=20=20=20=20

=20=20=20=20
=20=20=20=20
=09
=09=20
=09
=09








=09


=09
=09


=20=20=20=20=20=20
--0-1742062798-1257203099=:4327
Content-Type: text/html; charset=iso-8859-1
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">Note: Melinda sent me a message similar to Ro= ice'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 th= ink I'll leave the message as-is.  Thanks! Thank you Melinda an= d 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 co= unt pieces of the puzzles as soon as possible; that's the first step in finding permutati= on counts anyway. Thanks again, and I look forward to continuing my work! All the best, David --- On Mon, 11/2/09, Roice Nelson <roice3@gmail.com&g= t; wrote: , 255); margin-left: 5px; padding-left: 5px;"> From: Roice Nelson <ro= ice3@gmail.com> Subject: Re: [MC4D] MC4D 4.0 is really fantastic! = To: 4D_Cubing@yahoogroups.com Date: Monday, November 2, 2009, 11:32 AMr>   For the record, I think the permutation count= formulas have been some of the richest contributions to the group :D&= nbsp; They are definitely as helpful and important as the many other ways p= eople have chosen to help out.  Like Melinda said, it definitely takes= a village!   I haven't heard back from anyone on piece counting yet, so I think tho= se 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 woul= d 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 o= n even length duoprism puzzles, and so having such a formula would hel= p us be able to know what exactly that cutoff is that people shouldn't go p= ast.   All the best, Roice   On 11/2/09, = David Smith <o.com" target=3D"_blank" href=3D"/mc/compose?to=3Ddjs314djs314@yahoo.com">d= js314djs314@ yahoo.com> wrote: 204, 204);">  =20 Hi everyone, 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 n= ovice programmer (and I mean extremely novice), I can appreciate how much w= ork you all put into this.  I would also like to congratulate Chris, R= emi, Anthony, Roice, and Melinda for their solving accomplishments.  I= 'm not the type of person to offer suggestions or improvements to others' w= ork, which is probably a bad thing.  But I would be happy to help with= the piece-counting efforts if that is still needed. 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 experie= nces and contributions.  I know I have not been much of an active memb= er, 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 contribution= s regarding MC4D 4.0. 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 m= uch for their efforts in supporting the new program.  Congratulations = again to Melinda, Don, Jay, and Roice, and to everyone else who has success= fully solved the puzzles. Best wishes, David > =20 =20=20 =09=20 =09

=20=20=20=20=20=20
--0-1742062798-1257203099=:4327--

---

From: Melinda Green <melinda@superliminal.com>
Date: Mon, 02 Nov 2009 16:03:49 -0800
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!

 





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!

 
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.

 
All the best,
Roice

 
On 11/2/09, David Smith <djs314djs314@ yahoo.com> wrote:

 







Hi everyone,

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.


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.


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.


Best wishes,
David

--------------050401060801080906070507--

---

From: Roice Nelson <a class="moz-txt-link-rfc2396E" href="mailto:roice3@gmail.com"><roice3@gmail.com></a>
Date: Monday, November 2, 2009, 11:32 AM
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!

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!

 
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.

 
All the best,
Roice

 
On 11/2/09, David Smith <djs314djs314@ yahoo.com> wrote:

 







Hi everyone,

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.


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.


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.


Best wishes,
David




































--------------050401060801080906070507--

---

From: David Smith <djs314djs314@yahoo.com>
Date: Mon, 2 Nov 2009 21:36:45 -0800 (PST)
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!

=A0




=20=20=20=20
=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20



David,



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!=A0
:-)



Thanks!

-Melinda



David Smith wrote:

Note: Melinda sent me a message similar to Roice's that I did not realize=
was off of the group until this moment.=A0 But I would like to thank both =
Melinda and Roice for their support, so I think I'll leave the message as-i=
s.=A0 Thanks!

Thank you Melinda and Roice for your assurance that my math results
have a place in this group!=A0 Melinda, I especially appreciate that you
want to see my work continue. :)=A0 After reading that, it hit me that I
can find formulas for all of the polytopes in MC4D 4.0!=A0 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.=A0 The n^d simplex and {m}x{n}
duoprisms of any size would be a good place to start.=A0 Thanks again to
both of you for helping me realize that my projects are important to
those other than myself. :)=A0 Roice, I'll get started on formulas to count=
pieces of
the puzzles as soon as possible; that's the first step in finding permutati=
on counts anyway.=20



Thanks again, and I look forward to continuing my work!



All the best,

David

--- On Mon, 11/2/09, Roice Nelson wrote:

From: Roice Nelson
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!
To: 4D_Cubing@yahoogrou ps.com
Date: Monday, November 2, 2009, 11:32 AM






=A0




=20=20=20=20
For the record, I think the permutation count formulas ha=
ve been some of the=A0richest contributions to the group :D=A0 They are def=
initely as helpful and important as the many other ways people have chosen =
to help out.=A0 Like Melinda said, it definitely takes a village!

=A0
I haven't heard back from anyone on piece counting yet, so I think those pr=
oblems are still open if you'd like to play with any of them.=A0 I bet a ge=
neral formula for the number of pieces on any {n}x{m} duoprism would be fun=
to work through, and it would cover=A0a great many of=A0the puzzles in the=
list!=A0 There is also a known problem for large n or m on even length=A0d=
uoprism 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.

=A0
All the best,
Roice

=A0
On 11/2/09, David Smith wrote:

=A0=20







Hi everyone,

Sorry I haven't chimed in until now!=A0 I would like to thank and congratul=
ate Melinda, Don, Jay, and Roice for their hard work and efforts in getting=
the beta out to us!=A0 As a novice programmer (and I mean extremely novice=
), I can appreciate how much work you all put into this.=A0 I would also li=
ke to congratulate Chris, Remi, Anthony, Roice, and Melinda for their solvi=
ng accomplishments.=A0 I'm not the type of person to offer suggestions or i=
mprovements to others' work, which is probably a bad thing.=A0 But I would =
be happy to help with the piece-counting efforts if that is still needed.


Also, thanks to Roice for your wonderful essay!=A0 It has been a lot of fun=
to be a member of this community and read about all of your experiences an=
d contributions.=A0 I know I have not been much of an active member, only p=
resenting my obscure mathematical results here and there, and for that I ap=
ologize.=A0 Such results are not of much use to the community, and definite=
ly not as helpful or important as everyone else's contributions regarding M=
C4D 4.0.


Anyway, sorry I can't provide any experiences with the new program, as I ha=
ve not tried to solve the puzzles.=A0 I respect everyone else very much for=
their efforts in supporting the new program.=A0 Congratulations again to M=
elinda, Don, Jay, and Roice, and to everyone else who has successfully solv=
ed the puzzles.


Best wishes,
David





=20

=20=20=20=20=20=20

=20=20=20=20
=20=20=20=20
=09
=09=20
=09
=09








=09


=09
=09


=20=20=20=20=20=20
=20=20




=20

=20=20=20=20=20=20

=20=20=20=20
=20=20=20=20
=09
=09=20
=09
=09








=09


=09
=09


=20=20=20=20=20=20
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top" style=3D"font: inherit;">Wow, thanks for entrusting me with this task!=   Now I actually feel like I will be contributing to the program itsel= f! :)  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. About the Goldilocks function, this doesn't s= eem to be too difficult and I believe I will be able to produce numbers tha= t are neither too large nor too small, and without too much computation.&nb= sp; My basic idea is to approximate the number of positions of each puzzle.=   Of course, finding the actual number of positions would involve usin= g my formulas, and we all know how complex those can get. ;)  To get t= he approximation, I will simply calculate the total number of configuration= s (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 e= stimate 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 f= ormula for any possible puzzle, including user-created ones.  It would= be helpful if you have a list of all of the possible schlafli symbol confi= gurations that the program accepts, as I have not been able to find a compl= ete list offhand, only a list of the regular polychora.  If not, I wil= l be able to figure it out. :) Thanks again!  I will start work= ing on this and the piece-counting formulas tomorrow. Take care, = David --- On Mon, 11/2/09, Melinda Green <melinda@superlimi= nal.com> wrote: solid rgb(16, 16, 255); margin-left: 5px; padding-left: 5px;"> From: Me= linda Green <melinda@superliminal.com> Subject: Re: [MC4D] MC4D 4.= 0 is really fantastic! To: 4D_Cubing@yahoogroups.com Date: Monday, No= vember 2, 2009, 7:03 PM   David, 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 =3D"http://en.wikipedia.org/wiki/Shuffle#Randomization">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! = ; :-) Thanks! -Melinda David Smith wrote: Note: Melinda sent me a message similar to Roice's that I did not re= alize was off of the group until this moment.  But I would like to tha= nk both Melinda and Roice for their support, so I think I'll leave the mess= age as-is.  Thanks! Thank you Melinda and Roice for your assura= nce that my math results have a place in this group!  Melinda, I es= pecially appreciate that you want to see my work continue. :)  Afte= r reading that, it hit me that I can find formulas for all of the polyto= pes 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 a= ll of them, which would take care of the Create Your Own option.  T= he n^d simplex and {m}x{n} duoprisms of any size would be a good place t= o start.  Thanks again to both of you for helping me realize that m= y projects are important to those other than myself. :)  Roice, I'll get started on formulas to count pieces of the puzzles as s= oon as possible; that's the first step in finding permutation counts anyway= . Thanks again, and I look forward to continuing my work!r> All the best, David --- On Mon, 11/2/09, Roice = Nelson o:roice3@gmail.com" target=3D"_blank" href=3D"/mc/compose?to=3Droice3@gmail= .com"><roice3@gmail. com> wrote: From: Roice Nelson =3D"nofollow" class=3D"moz-txt-link-rfc2396E" ymailto=3D"mailto:roice3@gmai= l.com" target=3D"_blank" href=3D"/mc/compose?to=3Droice3@gmail.com"><roi= ce3@gmail. com> Subject: Re: [MC4D] MC4D 4.0 is really fantastic!= To: ailto:4D_Cubing@yahoogroups.com" target=3D"_blank" href=3D"/mc/compose?to= =3D4D_Cubing@yahoogroups.com">4D_Cubing@yahoogrou ps.com Date: Monda= y, November 2, 2009, 11:32 AM   = For the record, I think the permutation count formulas have been so= me of the richest contributions to the group :D  They are definit= ely as helpful and important as the many other ways people have chosen to h= elp out.  Like Melinda said, it definitely takes a village! &nb= sp; I haven't heard back from anyone on piece counting yet, so I think t= hose 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 wo= uld be fun to work through, and it would cover a great many of th= e 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 h= elp us be able to know what exactly that cutoff is that people shouldn't go= past.   All the best, Roice   On 11/2/09, David Smith <djs314djs314@ yahoo.com> wrote:   = Hi everyone, Sorry I haven't chimed = in until now!  I would like to thank and congratulate Melinda, Don, Ja= y, 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 apprecia= te how much work you all put into this.  I would also like to congratu= late Chris, Remi, Anthony, Roice, and Melinda for their solving accomplishm= ents.  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. r>Also, thanks to Roice for your wonderful essay!  It has been a lot o= f fun to be a member of this community and read about all of your experienc= es and contributions.  I know I have not been much of an active member, only presenting my obscure mathematical results here and there, an= d for that I apologize.  Such results are not of much use to the commu= nity, and definitely not as helpful or important as everyone else's contrib= utions regarding MC4D 4.0. Anyway, sorry I can't provide any exp= eriences with the new program, as I have not tried to solve the puzzles.&nb= sp; I respect everyone else very much for their efforts in supporting the n= ew program.  Congratulations again to Melinda, Don, Jay, and Roice, an= d to everyone else who has successfully solved the puzzles. Best= wishes, David = > =20 =20=20 =09=20 =09

=20=20=20=20=20=20
--0-1430645171-1257226605=:44571--

---

From: "djs314djs314" <djs314djs314@yahoo.com>
Date: Wed, 04 Nov 2009 04:14:30 -0000
Subject: Re: [MC4D] MC4D 4.0 is really fantastic!

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 >=3D 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 >=3D 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=20
stickers. Most of you probably already noticed the octahedral=20
stickers if you looked at that puzzle. In fact, for the simple=20
reason that higher-dimensional simplices are not easily tiled by=20
lower-dimensional simplices, it may be impossible to define a Rubik's=20
simplex puzzle for dimensions greater than 4. The other really cool=20
thing concerns the {3,3,3} 2 puzzle. It contains a 5-colored piece!=20=20
What at first appears to be five separate 1-colored pieces are in=20
fact a single piece. It's not hard to see once you load up the=20
puzzle and take a look, but this surprised me.

I apologize for the way my last post appears, I'm not sure why that=20
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=20
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 wrote:
>
> Wow, thanks for entrusting me with this task!=A0 Now I actually feel like=
I will be contributing to the program itself! :)=A0 By the way, I (believe=
I) was able to resolve one of the issues with the program (although a triv=
ial one).=A0 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.=A0 My basic idea is to approxi=
mate the number of positions of each puzzle.=A0 Of course, finding the actu=
al number of positions would involve using my formulas, and we all know how=
complex those can get. ;)=A0 To get the approximation, I will simply calcu=
late the total number of configurations (i.e. all positions, regardless of =
whether they can be reached or not.)=A0 Then I would find the number of fac=
e turns necessary to reach that number of positions.=A0 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.=A0 The only difficulty will be =
to find the general formula for any possible puzzle, including user-created
> ones.=A0 It would be helpful if you have a list of all of the possible s=
chlafli 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.=
=A0 If not, I will be able to figure it out. :)
>=20
> Thanks again!=A0 I will start working on this and the piece-counting form=
ulas tomorrow.
>=20
> Take care,
> David
>=20
> --- On Mon, 11/2/09, Melinda Green wrote:
>=20
> From: Melinda Green
> 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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> =A0
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>=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20=20
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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!=A0
> :-)
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> Thanks!
>=20
> -Melinda
>=20
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> David Smith wrote:
>=20
> Note: Melinda sent me a message similar to Roice's that I did not reali=
ze was off of the group until this moment.=A0 But I would like to thank bot=
h Melinda and Roice for their support, so I think I'll leave the message as=
-is.=A0 Thanks!
>=20
> Thank you Melinda and Roice for your assurance that my math results
> have a place in this group!=A0 Melinda, I especially appreciate that you
> want to see my work continue. :)=A0 After reading that, it hit me that I
> can find formulas for all of the polytopes in MC4D 4.0!=A0 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.=A0 The n^d simplex and {m}x{n}
> duoprisms of any size would be a good place to start.=A0 Thanks again to
> both of you for helping me realize that my projects are important to
> those other than myself. :)=A0 Roice, I'll get started on formulas to cou=
nt pieces of
> the puzzles as soon as possible; that's the first step in finding permuta=
tion counts anyway.=20
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> Thanks again, and I look forward to continuing my work!
>=20
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> All the best,
>=20
> David
>=20
> --- On Mon, 11/2/09, Roice Nelson wrote:
>=20
> From: Roice Nelson
> 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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> =A0
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> For the record, I think the permutation count formulas =
have been some of the=A0richest contributions to the group :D=A0 They are d=
efinitely as helpful and important as the many other ways people have chose=
n to help out.=A0 Like Melinda said, it definitely takes a village!
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> =A0
> 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.=A0 I bet a =
general formula for the number of pieces on any {n}x{m} duoprism would be f=
un to work through, and it would cover=A0a great many of=A0the puzzles in t=
he list!=A0 There is also a known problem for large n or m on even length=
=A0duoprism 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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> =A0
> All the best,
> Roice
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> =A0
> On 11/2/09, David Smith wrote:
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> =A0=20
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> Hi everyone,
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> Sorry I haven't chimed in until now!=A0 I would like to thank and congrat=
ulate Melinda, Don, Jay, and Roice for their hard work and efforts in getti=
ng the beta out to us!=A0 As a novice programmer (and I mean extremely novi=
ce), I can appreciate how much work you all put into this.=A0 I would also =
like to congratulate Chris, Remi, Anthony, Roice, and Melinda for their sol=
ving accomplishments.=A0 I'm not the type of person to offer suggestions or=
improvements to others' work, which is probably a bad thing.=A0 But I woul=
d be happy to help with the piece-counting efforts if that is still needed.
>=20
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> Also, thanks to Roice for your wonderful essay!=A0 It has been a lot of f=
un to be a member of this community and read about all of your experiences =
and contributions.=A0 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.=A0 Such results are not of much use to the community, and defini=
tely not as helpful or important as everyone else's contributions regarding=
MC4D 4.0.
>=20
>=20
> Anyway, sorry I can't provide any experiences with the new program, as I =
have not tried to solve the puzzles.=A0 I respect everyone else very much f=
or their efforts in supporting the new program.=A0 Congratulations again to=
Melinda, Don, Jay, and Roice, and to everyone else who has successfully so=
lved the puzzles.
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> Best wishes,
> David
>

---