Archived Thread

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The 4D analog of Megaminx is here! This has been a work in progress for
some time really (the basics were done in 2006), but it is now evolved
enough for you crazy solvers to have at it! I like playing with it and it
will be fun to figure out sequences, but I'm not imagining I'll try a
solution myself. I do hope some of you will take it on, and that getting it
out now is good timing with the summer break for students and all. I'm
really pleased with how it is turning out, and have screenshots at the site
to check out if you don't feel like installing...

www.gravitation3d.com/magic120cell/

You'll see that I made some easier (or should I say just slightly less
ludicrous) puzzles which are only different in that they have multiple cells
set to the same color. These accentuate some of the 120cell symmetries and
could be more enjoyable simply because there are not so many colors to deal
with. The biggest feat is of course the full puzzle with 120 different
colored cells, but all of them are hard and I will post the name of anyone
who solves any of the puzzles in a Hall of Insanity.

I admit straight away there are still features that would be nice to have at
some point, and I do have my own wish list. Though I make no guarantees
when or if I can add stuff, please feel free to make suggestions or requests
you would find valuable. Bug reports will also be appreciated.

I hope you all really enjoy it. Good Luck!

Roice

P.S. This is what I alluded to this morning. I haven't figured it out
exactly yet and may not be able to, but a quick estimate shows it to have
well over 10^15000 (that's right, 10^15K) permutations!

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The 4D analog of Megaminx is here! This has been a work in progress for some time really (the basics were done in 2006), but it is now evolved enough for you crazy solvers to have at it! I like playing with it and it will be fun to figure out sequences, but I'm not imagining I'll try a solution myself. I do hope some of you will take it on, and that getting it out now is good timing with the summer break for students and all. I'm really pleased with how it is turning out, and have screenshots at the site to check out if you don't feel like installing...

You'll see that I made some easier (or should I say just slightly less ludicrous) puzzles which are only different in that they have multiple cells set to the same color. These accentuate some of the 120cell symmetries and could be more enjoyable simply because there are not so many colors to deal with. The biggest feat is of course the full puzzle with 120 different colored cells, but all of them are hard and I will post the name of anyone who solves any of the puzzles in a Hall of Insanity.

I admit straight away there are still features that would be nice to have at some point, and I do have my own wish list. Though I make no guarantees when or if I can add stuff, please feel free to make suggestions or requests you would find valuable. Bug reports will also be appreciated.

P.S. This is what I alluded to this morning. I haven't figured it out exactly yet and may not be able to, but a quick estimate shows it to have well over 10^15000 (that's right, 10^15K) permutations!

Roice rocks!!!

Don and I have been cooking up plans to support more puzzles including
this one in MC4D for some time but Roice beat us to the best one. The
120 cell holds a special place in the pantheon of regular polyhedra in
all dimensions. It has no analog in any dimension greater than 4. If
puzzles were planets, the 120 cell would be Saturn. Being the first to
tame this beauty will earn you a special place in history, though the
task may be as hard as being the first to visit Saturn. We're lucky that
Roice didn't solve it before publishing it. Please tell us why, Roice?
Did you try to but underestimate the difficulty or did you just want to
give everyone a chance? The screen shots are amazingly beautiful and it
is such a gift to the world that you would have been completely
justified to have kept it secret until first solve.

Thank you so much!
-Melinda

Roice Nelson wrote:
>
> The 4D analog of Megaminx is here! This has been a work in progress
> for some time really (the basics were done in 2006), but it is now
> evolved enough for you crazy solvers to have at it! I like playing
> with it and it will be fun to figure out sequences, but I'm not
> imagining I'll try a solution myself. I do hope some of you will take
> it on, and that getting it out now is good timing with the summer
> break for students and all. I'm really pleased with how it is turning
> out, and have screenshots at the site to check out if you don't feel
> like installing...
>
> www.gravitation3d.com/magic120cell/
>
>
> You'll see that I made some easier (or should I say just slightly less
> ludicrous) puzzles which are only different in that they have multiple
> cells set to the same color. These accentuate some of the 120cell
> symmetries and could be more enjoyable simply because there are not so
> many colors to deal with. The biggest feat is of course the full
> puzzle with 120 different colored cells, but all of them are hard and
> I will post the name of anyone who solves any of the puzzles in a Hall
> of Insanity.
>
> I admit straight away there are still features that would be nice to
> have at some point, and I do have my own wish list. Though I make no
> guarantees when or if I can add stuff, please feel free to make
> suggestions or requests you would find valuable. Bug reports will
> also be appreciated.
>
> I hope you all really enjoy it. Good Luck!
>
> Roice
>
> P.S. This is what I alluded to this morning. I haven't figured it out
> exactly yet and may not be able to, but a quick estimate shows it to
> have well over 10^15000 (that's right, 10^15K) permutations!
>

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Honestly, the reason I wasn't planning on working through a solution was
that I am a bit scared of the sheer number of pieces! I just finished up
the final parts that I felt were needed for it to be solvable today, and I
actually haven't even figured out a single sequence yet. So as of this
evening, I only have the thoughts about it we've discussed in the past,
which is that it will be easier in some ways than MC4D because of the larger
space to sequester pieces, but that it will be a big effort in time. Also,
I think I am ready for a bit of a rest and was too excited to share to let
it sit on a shelf. Sarah will be happy to get my attention back now too
since I've been spending a lot of time on it lately :)

Since I haven't been through it, I hope there are enough features for a
solution (there are no macros yet, and I could see that being the first
request as people start attacking it, but it will be a while before I look
at that). I think my fear at this point is that some stupid bug would
somehow corrupt someones solution after they worked on it for a long time!
Working through MC5D myself was able to quell that worry last time. But the
scrambling and autosolve has had a decent amount of exercise as I've worked
on it, so I think all should be ok.

Writing a general 4D puzzle engine certainly would be a huge achievement.
It was nice to be able to hardcode 120 cell characteristics throughout the
code, and I certainly did - I can't imagine having worked without that
luxury at this point. I also realized through this effort that I understood
general 4D rotations very badly, and still do though I've improved a
little. It was so much easier when everything was coordinate axis aligned.
I was able to glean just enough from the results of a paper (
http://www.geometrictools.com/Documentation/RotationsFromPowerSeries.pdf),
which combined with some experimentation to figure out how to calculate the
rotation matrix coefficients allowed me to get working code for "simple
rotations". Wiki has some more info on that term if anyone is interested (
http://en.wikipedia.org/wiki/SO%284%29).

Anyway, I won't go on too much for now, but I look forward to more
discussion about this puzzle. Thank you so much for your highly positive
email :)

Roice

On Sun, May 4, 2008 at 9:44 PM, Melinda Green
wrote:

> Roice rocks!!!
>
> Don and I have been cooking up plans to support more puzzles including
> this one in MC4D for some time but Roice beat us to the best one. The
> 120 cell holds a special place in the pantheon of regular polyhedra in
> all dimensions. It has no analog in any dimension greater than 4. If
> puzzles were planets, the 120 cell would be Saturn. Being the first to
> tame this beauty will earn you a special place in history, though the
> task may be as hard as being the first to visit Saturn. We're lucky that
> Roice didn't solve it before publishing it. Please tell us why, Roice?
> Did you try to but underestimate the difficulty or did you just want to
> give everyone a chance? The screen shots are amazingly beautiful and it
> is such a gift to the world that you would have been completely
> justified to have kept it secret until first solve.
>
> Thank you so much!
> -Melinda
> Visit Your Group
>
> Find helpful tips
>
> for Moderators
>
> on the Yahoo!
>
> Groups team blog.
> Yahoo! Groups
>
> Dog Zone
>
> Connect w/others
>
> who love dogs.
> All-Bran
>
> 10 Day Challenge
>
> Join the club and
>
> feel the benefits.
> .
>
>
>

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Wow. Its really amazing. I like the ring coloring best. I'm not sure whether
I would attempt to solve it. I'm sure it would be possible and the moves
would be easy but there are so many pieces. The ghost piece feature is a
neat effect. How many hidden cells are there on this thing? I would imagine
there are a lot but that is just my intuition.

On Sun, May 4, 2008 at 11:38 PM, Roice Nelson wrote:

> Honestly, the reason I wasn't planning on working through a solution was
> that I am a bit scared of the sheer number of pieces! I just finished up
> the final parts that I felt were needed for it to be solvable today, and I
> actually haven't even figured out a single sequence yet. So as of this
> evening, I only have the thoughts about it we've discussed in the past,
> which is that it will be easier in some ways than MC4D because of the larger
> space to sequester pieces, but that it will be a big effort in time. Also,
> I think I am ready for a bit of a rest and was too excited to share to let
> it sit on a shelf. Sarah will be happy to get my attention back now too
> since I've been spending a lot of time on it lately :)
>
> Since I haven't been through it, I hope there are enough features for a
> solution (there are no macros yet, and I could see that being the first
> request as people start attacking it, but it will be a while before I look
> at that). I think my fear at this point is that some stupid bug would
> somehow corrupt someones solution after they worked on it for a long time!
> Working through MC5D myself was able to quell that worry last time. But the
> scrambling and autosolve has had a decent amount of exercise as I've worked
> on it, so I think all should be ok.
>
> Writing a general 4D puzzle engine certainly would be a huge achievement.
> It was nice to be able to hardcode 120 cell characteristics throughout the
> code, and I certainly did - I can't imagine having worked without that
> luxury at this point. I also realized through this effort that I understood
> general 4D rotations very badly, and still do though I've improved a
> little. It was so much easier when everything was coordinate axis aligned.
> I was able to glean just enough from the results of a paper (
> http://www.geometrictools.com/Documentation/RotationsFromPowerSeries.pdf),
> which combined with some experimentation to figure out how to calculate the
> rotation matrix coefficients allowed me to get working code for "simple
> rotations". Wiki has some more info on that term if anyone is interested (
> http://en.wikipedia.org/wiki/SO%284%29).
>
> Anyway, I won't go on too much for now, but I look forward to more
> discussion about this puzzle. Thank you so much for your highly positive
> email :)
>
> Roice
>
>
> On Sun, May 4, 2008 at 9:44 PM, Melinda Green
> wrote:
>
> > Roice rocks!!!
> >
> > Don and I have been cooking up plans to support more puzzles including
> > this one in MC4D for some time but Roice beat us to the best one. The
> > 120 cell holds a special place in the pantheon of regular polyhedra in
> > all dimensions. It has no analog in any dimension greater than 4. If
> > puzzles were planets, the 120 cell would be Saturn. Being the first to
> > tame this beauty will earn you a special place in history, though the
> > task may be as hard as being the first to visit Saturn. We're lucky that
> >
> > Roice didn't solve it before publishing it. Please tell us why, Roice?
> > Did you try to but underestimate the difficulty or did you just want to
> > give everyone a chance? The screen shots are amazingly beautiful and it
> > is such a gift to the world that you would have been completely
> > justified to have kept it secret until first solve.
> >
> > Thank you so much!
> > -Melinda
> > Visit Your Group
> >
> > Find helpful tips
> >
> > for Moderators
> >
> > on the Yahoo!
> >
> > Groups team blog.
> > Yahoo! Groups
> >
> > Dog Zone
> >
> > Connect w/others
> >
> > who love dogs.
> > All-Bran
> >
> > 10 Day Challenge
> >
> > Join the club and
> >
> > feel the benefits.
> > .
> >
> >
>
>

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Wow. Its really amazing. I like the ring coloring best. I'm not sure wh=
ether I would attempt to solve it. I'm sure it would be possible and th=
e moves would be easy but there are so many pieces. The ghost piece feature=
is a neat effect. How many hidden cells are there on this thing? I would i=
magine there are a lot but that is just my intuition.

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The full scramble is certainly terrifying to behold. OTOH, a scramble
with only 5 random twists didn't look at all hard. Easier in many ways
to the 3^4 since the odds of two random twists interacting with each
other are much smaller as the number of faces grows. I've even attached
my own solution file! I was actually surprised that any of the random
twists of my 5 scramble interacted at all. Did you try to ensure this
would happen? That would seem like a good idea. Was it supposed to beep
or something when I reached the solved state? I didn't notice anything.
Just like with MC4D I find the challenge of backing out a small number
of random twists to be an educational challenge. Fun and satisfying too.

Here are some suggestions after my short time with the puzzle.
* Allow users to supply the number of random twists. Scramble->Custom
perhaps?
* Detect when a user's click is the exact inverse of their previous one
and substitute an undo instead.
* Create the ability to move the front clipping plane in. I'm often
operating on a zoomed-in view of the central cell and often fight to get
clear views of it.
* I agree with Jenelle that the ghost piece highlighting is a cool
effect. Maybe you could use that effect for as long as ctrl-shift is
held down? That way there would be no need to cancel it.
* How about a method of doing the inverse? I.E. for any selected
sticker, highlight the one that belongs in its place.
* Add Help->About with your name, copyright, version number?

Lastly, where did you get your colors from for the full puzzle? It looks
fantastic!
-Melinda

Roice Nelson wrote:
> Honestly, the reason I wasn't planning on working through a solution
> was that I am a bit scared of the sheer number of pieces! I just
> finished up the final parts that I felt were needed for it to be
> solvable today, and I actually haven't even figured out a single
> sequence yet. So as of this evening, I only have the thoughts about
> it we've discussed in the past, which is that it will be easier in
> some ways than MC4D because of the larger space to sequester pieces,
> but that it will be a big effort in time. Also, I think I am ready
> for a bit of a rest and was too excited to share to let it sit on a
> shelf. Sarah will be happy to get my attention back now too since
> I've been spending a lot of time on it lately :)
>
> Since I haven't been through it, I hope there are enough features for
> a solution (there are no macros yet, and I could see that being the
> first request as people start attacking it, but it will be a while
> before I look at that). I think my fear at this point is that some
> stupid bug would somehow corrupt someones solution after they worked
> on it for a long time! Working through MC5D myself was able to quell
> that worry last time. But the scrambling and autosolve has had a
> decent amount of exercise as I've worked on it, so I think all should
> be ok.
>
> Writing a general 4D puzzle engine certainly would be a huge
> achievement. It was nice to be able to hardcode 120 cell
> characteristics throughout the code, and I certainly did - I can't
> imagine having worked without that luxury at this point. I also
> realized through this effort that I understood general 4D rotations
> very badly, and still do though I've improved a little. It was so
> much easier when everything was coordinate axis aligned. I was able
> to glean just enough from the results of a paper
> (http://www.geometrictools.com/Documentation/RotationsFromPowerSeries.pdf),
> which combined with some experimentation to figure out how to
> calculate the rotation matrix coefficients allowed me to get working
> code for "simple rotations". Wiki has some more info on that term if
> anyone is interested (http://en.wikipedia.org/wiki/SO%284%29).
>
> Anyway, I won't go on too much for now, but I look forward to more
> discussion about this puzzle. Thank you so much for your highly
> positive email :)
>
> Roice

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27576 3236 8420 25332 8682 3082 3080 3238 3236 3238
3238 3236 3238 3236 3238 3238 18702 18700 18702 18700
18702 18700 18700 18702 18700 18702 18700 18702 8624 8626
8624 8626 8680 8682 8680 8428 25334 27442 27550 27548
27550 27548 27482 27442

--------------080700050905000200030108--

Hi Roice,

This is incredible! What you have done with the MagicCube5D
and Magic120Cell programs is truly amazing, and they are a great
gift to this community!

I agree with Melinda that the 120-cell is a unique polyhedron,
being the only higher-dimensional analog of the dodecahedron,
and now we have the only higher-dimensional analog of the
Megaminx!

Also, you have given us a great new permutation problem,
which figuring out surely would be impossible without this
program. I think I'm going to postpone my work on the
n^4 cube and start working on this right away!

As for my 3-cycle algorithm for the n^4 cube, I would be
happy to describe it for the group, if anyone is interested.
It's just a commutator and I'm sure it could be improved upon,
as I have never used MagicCube4D before. I'll start writing it
up.

Thank you Roice, for this wonderful and unique program!

Best Regards,

David

--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> The 4D analog of Megaminx is here! This has been a work in=20
progress for
> some time really (the basics were done in 2006), but it is now=20
evolved
> enough for you crazy solvers to have at it! I like playing with=20
it and it
> will be fun to figure out sequences, but I'm not imagining I'll=20
try a
> solution myself. I do hope some of you will take it on, and that=20
getting it
> out now is good timing with the summer break for students and=20
all. I'm
> really pleased with how it is turning out, and have screenshots at=20
the site
> to check out if you don't feel like installing...
>=20
> www.gravitation3d.com/magic120cell/
>=20
> You'll see that I made some easier (or should I say just slightly=20
less
> ludicrous) puzzles which are only different in that they have=20
multiple cells
> set to the same color. These accentuate some of the 120cell=20
symmetries and
> could be more enjoyable simply because there are not so many=20
colors to deal
> with. The biggest feat is of course the full puzzle with 120=20
different
> colored cells, but all of them are hard and I will post the name=20
of anyone
> who solves any of the puzzles in a Hall of Insanity.
>=20
> I admit straight away there are still features that would be nice=20
to have at
> some point, and I do have my own wish list. Though I make no=20
guarantees
> when or if I can add stuff, please feel free to make suggestions=20
or requests
> you would find valuable. Bug reports will also be appreciated.
>=20
> I hope you all really enjoy it. Good Luck!
>=20
> Roice
>=20
> P.S. This is what I alluded to this morning. I haven't figured it=20
out
> exactly yet and may not be able to, but a quick estimate shows it=20
to have
> well over 10^15000 (that's right, 10^15K) permutations!
>

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Hi Jenelle,

The number of hidden cells depend on the display settings chosen, so the
answer is different depending on which of the example settings are chosen or
any alterations you make. By default, the only cells that are hidden are
the inverted (or mirrored during projection) ones. With the default 4D
perspective point distance, this has the effect of hiding 13 cells (the one
antipodal to the puzzle "center" and the first layer surrounding that
antipodal cell). If you increase the 4D perspective point distance, more
cells get mirrored so you would see some of them disappear. To show every
cell, you can start the program with the installed defaults, go to the
visibility tab, then select the checkbox "Draw Inverted Cells". But like
MC4D (where the analogous face would be the hidden 8th face), the mirrored
cells tend to hide other portions of the puzzle...

Take Care,

Roice

On Mon, May 5, 2008 at 12:35 AM, Jenelle Levenstein <
jenelle.levenstein@gmail.com> wrote:

> Wow. Its really amazing. I like the ring coloring best. I'm not sure
> whether I would attempt to solve it. I'm sure it would be possible and the
> moves would be easy but there are so many pieces. The ghost piece feature is
> a neat effect. How many hidden cells are there on this thing? I would
> imagine there are a lot but that is just my intuition.
>
>
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Thanks for the great feedback! I like these ideas and they all should be
quick changes, so I should be able to address each one. To answer the
questions you posed:

- I randomly generate twists (a random cell, random sticker, random
direction) when scrambling, so I'm a bit surprised about a 5 twist scramble
having interacting pieces too!
- The puzzle does do something special when you reach the solved state, but
only if a full scramble has been performed. I like the beep idea and now
remember how MC4D does that, so I'll do that to let a user know in smaller
scramble cases too.
- The colors for the full puzzle are generated randomly as well (later it
might be cool to allow these all to be set by editing a file). I
did generate them with the more artistic HLS color scheme instead of
randomly generating RGB values, which I think helps make them look nicer. I
restricted luminance and saturation values to ranges that seemed to produce
better colors.

Interesting about the clipping plane troubles. I feel like I really have to
zoom way in to clip that cell, but I will definitely try to improve that.

Thanks again!

Roice

2008/5/5 Melinda Green :

> The full scramble is certainly terrifying to behold. OTOH, a scramble
> with only 5 random twists didn't look at all hard. Easier in many ways
> to the 3^4 since the odds of two random twists interacting with each
> other are much smaller as the number of faces grows. I've even attached
> my own solution file! I was actually surprised that any of the random
> twists of my 5 scramble interacted at all. Did you try to ensure this
> would happen? That would seem like a good idea. Was it supposed to beep
> or something when I reached the solved state? I didn't notice anything.
> Just like with MC4D I find the challenge of backing out a small number
> of random twists to be an educational challenge. Fun and satisfying too.
>
> Here are some suggestions after my short time with the puzzle.
> * Allow users to supply the number of random twists. Scramble->Custom
> perhaps?
> * Detect when a user's click is the exact inverse of their previous one
> and substitute an undo instead.
> * Create the ability to move the front clipping plane in. I'm often
> operating on a zoomed-in view of the central cell and often fight to get
> clear views of it.
> * I agree with Jenelle that the ghost piece highlighting is a cool
> effect. Maybe you could use that effect for as long as ctrl-shift is
> held down? That way there would be no need to cancel it.
> * How about a method of doing the inverse? I.E. for any selected
> sticker, highlight the one that belongs in its place.
> * Add Help->About with your name, copyright, version number?
>
> Lastly, where did you get your colors from for the full puzzle? It looks
> fantastic!
> -Melinda
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Awesome, I didn't want to ask directly, but I'm excited to hear of your
interest in looking into the problem of the number of permutations! I
wonder if this particular calculation has ever been done before. Also, I've
seen and used big number calculators on the web before, so I also had to
wonder how easy it would be to find one that would be able to show all the
digits in the final answer to this problem. I hope one is out there.

cya,
Roice

On Mon, May 5, 2008 at 5:38 AM, David Smith wrote:

> Hi Roice,
>
> This is incredible! What you have done with the MagicCube5D
> and Magic120Cell programs is truly amazing, and they are a great
> gift to this community!
>
> I agree with Melinda that the 120-cell is a unique polyhedron,
> being the only higher-dimensional analog of the dodecahedron,
> and now we have the only higher-dimensional analog of the
> Megaminx!
>
> Also, you have given us a great new permutation problem,
> which figuring out surely would be impossible without this
> program. I think I'm going to postpone my work on the
> n^4 cube and start working on this right away!
>
> As for my 3-cycle algorithm for the n^4 cube, I would be
> happy to describe it for the group, if anyone is interested.
> It's just a commutator and I'm sure it could be improved upon,
> as I have never used MagicCube4D before. I'll start writing it
> up.
>
> Thank you Roice, for this wonderful and unique program!
>
> Best Regards,
>
> David
>
> Visit Your Group
>
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>
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>
> It's free and it's
>
> good for the planet.
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>
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>
> to discuss Curves.
> .
>
>
>

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Hi everyone,

I have just finished calculating an upper bound for the number
of permutations of the 120-Cell. To establish it is the exact answer
(which I am virtually certain of), we will have to find a number
of algorithms which I will describe in my explanation.

The calculation was only difficult in one aspect, namely the
possible orientations of the corners. Here is the upper bound:

(600!/2)*(1200!/2)*(720!/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)

To answer Roice's question, I did find an arbitrary-precision=20
calculator (Googol+, trial version) that displays the entire
number in its full glory:

23435018363697222779126210606140343600982219866708667227704291465940007
37743198001537086016413748065359228217622633869330769129523601891497799
90823414733250819032377663096727895392891107724676361939174468537213471
84699260131924584724938945790242680862147295113762851571432130901040238
96149551266842769465158629370618815995041314328829732432057176063611161
23422302770133676753359134856348612503635674252607065815753807941966366
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43170816415770010483963217284920963354265992129473309221874222731561178
16542353296798264919536869373254412130565886195935986667368898349834998
21329543130389608025077440970771216857898162084209764122888617411552472
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32568991232059877343852662991568615775896662637181748468409947337080580
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05260822568001712636170958894341958120566294218192269192589063843887982
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42102203322863117771608181202015567452798664995683183676667131261081040
30610697346947842941118989099929505001072885907143020380705671971970771
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71854373062622155659107585797616686931874970164036381491924856162327083
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24075969774780698659600903247322382509271117981454345778343923721701140
34040387142730946291948768518544291460594918104272439297270660195239204
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29742915483006071654290990776376332377597123229369363319211034520828156
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92509662606254579651749991204772662937610983604733514590588466763484779
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00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000

What a number!

Here is an outline of how I derived this upper bound:

First of all, we can count that the 120-Cell has 120
1-colored immobile center pieces, 720 2-colored pieces
(120*12/2), 1200 3-colored pieces (120*30/3), and 600
4-colored pieces (120*20/4). When a dodecahedral face
rotates, there are 2 5-cycles of the 2-colored pieces,
6 5-cycles of the 3-colored pieces, and 4 5-cycles of
the 4-colored pieces. Since 5-cycles are even permutations,
all permutations of each type of piece (including facelets),
will be even.

Therefore, the 720 2-coloreds can be permuted
720!/2 ways, dividing by 2 because of the even parity.
Similarly, the 3-coloreds and 4-coloreds can be permuted
1200!/2 and 600!/2 ways, respectively. Multiplying
these three terms together, we obtain an upper bound
for the number of ways the pieces can be permuted without
regards to orientation:

(600!/2)*(1200!/2)*(720!/2)

To show this number is exact, we will have to find 3 algorithms:
one that performs a 3-cycle of any three 2-coloreds without
affecting any other 2-coloreds, a 3-cycle of any three 3-coloreds=20
without affecting any other 3-coloreds, and a 3-cycle of any
three 4-coloreds without affecting any other 4-coloreds. These
three algorithms, when combined with each other and conjugates
(setup moves), can produce any possible permutation of the
pieces.

Now for the orientations. Since the facelets also undergo
5-cycles, all orientations will be limited to even permutations
of facelets.

Therefore 719 2-coloreds can be oriented in any of 2 ways each,
but the last will be determined by the others because of the
even parity. This results in

(2^720)/2

or 2^719 ways of orienting the 2-coloreds. To show
this number is exact, we must find an algorithm that
flips two 2-colored pieces without affecting the others.

For the 3-coloreds and 4-coloreds, I followed closely
the methods of Keane and Kamack in their paper,
"The Rubik Tesseract", modifying their arguments as neccesary
to apply them to the 120-Cell.

Any 3-colored piece can be oriented in 6 different ways.
(not three, because in four dimensions we can reflect
3-colored pieces as well as twist them!) Notice that a
twist (a 3-cycle of the facelets on that piece) is an even
permutation, while a reflection (a 2-cycle of two of the
facelets on that piece) is an odd permutation. Since the
total parity of all of the 3-coloreds must be even,
the first 1199 3-coloreds can be oriented in 6 ways each,
while the last can be oriented in only 3. (If the first
1199 3-coloreds total to an even permutation, the last
3-colored must be one of the 3 even twists, while if
they total to an odd permutation, the last 3-colored
must be one of the 3 odd reflections) This gives a total
of

(6^1200)/2

or (6^1199)*3 ways of orienting the 3-colored pieces.
To show this number is exact, we must find an algorithm
that twists one 3-colored piece without affecting the others,
and an algorithm that reflects two 3-colored pieces without
affecting the others.

Finally, the toughest part. The orientation of the 4-coloreds
required me to generalize the group theory based solution for
the corners of the 3^4 cube by Keane and Kamack to the
120-Cell.

In their paper, Keane and Kamack first describe that any
particular 4-colored piece can never be in an odd permutation
of its facelets, because the 4-coloreds split into two
different groups: the even and odd permutations, which
happen to be mirror images of each other. This rule applies
equally well to the 120-Cell.

Hence, each 4-colored piece can only be oriented in 12 ways.
(4!/2) There is an additional constraint. In their paper,
Keane and Kamack show that the orientations of 4-colored
pieces in 4D space, which form the alternating group A4,
can be divided into three sets. They describe orientations
using cycle notation of the four faces, labeled a, b, c,
and d. The subgroup N consists of the identity permutation,
(ab)(cd), (ac)(bd), and (ad)(bc). The subgroup S consists
of (abc), (adb), (acd), and (bdc). The subgroup Z
consists of (acb), abd), (adc), and (bcd).

The authors then show that the group of these three subgroups
is isomorphic to (the same as) the group of residue classes,
mod 3, with N as the identity. This means that we can assign
the number 0 to N, the number 1 to S, and the number -1 to Z,
and adding these numbers mod 3 is the same as taking the product
of elements of these three subgroups.

Notice that this entire argument applies equally well to
the 120-Cell as to the tesseract. Now the only thing left
to show is that the sum of the orientations of the 4-coloreds
(counting 0 for an orientation in N, etc.) mod 3 is always the
same, whether to the tesseract or the 120-Cell.

The orientations can be defined by assigning, to each 4-colored
piece, a letter to each facelet and each position of each
facelet. Then each orientation can be described by a 4-letter
string (e.g. ABCD) relative to the position it is occupying.

When pieces or cubies rotate in a cycle, their facelets undergo
n seperate cycles if they have n facelets. The important thing
is that there are always n disjoint cycles. In the tesseract,
every corner rotation boils down to four 4-cycles of facelets
for each cycle of four cubies. In the 120-Cell, it is the same
except it is four 5-cycles. If we can show that in
cycles of any length, the sum of the orientations of the pieces
does not change, we will have proved this for both the tesseract
and the 120-Cell.

Consider four 2-cycles:

ABCD 1
ABCD 2

Each row represents a 4-colored piece. The actual 4-cycles
are vertical in direction. For example:

ABCD 1
CDAB 2

This means that facelet A goes where facelet C was, etc.
In this example, piece 1 performed an N-twist. Now notice
that since we are dealing with 4-cycles, the facelets of
piece 2 must return to the original positions of the facelets
of piece 1. Therefore, piece 2 also performed an N-twist.
It can be checked that if piece 1 performs a Z-twist, piece
2 performs an S-twist, and if piece 1 performs an S-twist,
piece 2 performs a Z-twist. Therefore, the sum of the values
does not change. (N=3D0, Z=3D-1, S=3D1)

Now we can do a proof by induction to show that any four
cycles has this property. Assume that 4 k-cycles always sum
to the same amount:

ABCD
.
.
.
ABCD (not neccesarily this orientation)

Adding another piece, we are in the same position as before.
If the next-to-last piece is an N-twist when moving to the
position of the last piece, then the last piece must also
be an N-twist when moving to the position of the first piece
(since the ends of the cycles must return to their original
locations) Also, a Z-twist implies an S-twist and
an S-twist implies a Z-twist.

Therefore, no matter what the length of the cycles, the
sum of the values of the orientations never change. This
means it is true for both the tesseract and the 120-Cell.

So now we know the sums of the orientations, mod 3, never
change in the 120-Cell. Since an N-twist is 0, we can have
an isolated N-twist without affecting any other pieces.
The value S - Z must therefore be congruent to 0, mod 3.

This means that the first 599 4-colored pieces can
each be in any of 12 orientations. If the value of orientations
up to that point is 0, the remaining value must be an N-twist.
If it is 1, the remaining value must be a Z-twist, and if
it is -1, the remaing value must be an S-twist. In each case
there are four possible orientations left for the last
4-colored piece. Therefore, the upper bound for the orientations
of the 4-colored pieces is

(12^600)/3

or (12^599)*4. To prove this is exact, we must find algorithms
that show that any N-twist can be performed without affecting
the rest of the pieces, and algorithms showing that any Z-twist
can be performed along with any S-twist without affecting any
other pieces.

When we multiply these figures with the ones for permutations,
we arrive at the final answer. I sincerely hope that I did not
bother anyone by writing this very long post. I just wanted to
share my results with those that are interested.

I am not worried about finding all of the algorithms to make
this figure exact. They will probably arise naturally as we
play with Roice's program.=20

I want to once again thank the members of this group for
helping me, and for putting up with my long posts!

Best Regards,

David

--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> Awesome, I didn't want to ask directly, but I'm excited to hear of=20
your
> interest in looking into the problem of the number of permutations!=20=20
I
> wonder if this particular calculation has ever been done before.=20=20
Also, I've
> seen and used big number calculators on the web before, so I also=20
had to
> wonder how easy it would be to find one that would be able to show=20
all the
> digits in the final answer to this problem. I hope one is out there.
>=20
> cya,
> Roice

What a number indeed!

So let me get this straight. If you imagine all the particles in the
universe, and then imagine that each one really consists of another
entire universe, and for each particle in those universes, another
universe, and so on ten times, you would still not have enough particles
so that each one could represent one unique state of this puzzle? OK, I
suppose that counts as a big number. :-)

BTW, don't worry about the length of your posts David. It's easy enough
for anyone who's not interested to just delete them. Any subject even
remotely on-topic should be fair game. Even if the posts become too
frequent for some people, they can choose to get daily digests or even
no email at all and just read the messages on the web site when they
feel like it.

-Melinda

David Smith wrote:
> Hi everyone,
>
> I have just finished calculating an upper bound for the number
> of permutations of the 120-Cell. To establish it is the exact answer
> (which I am virtually certain of), we will have to find a number
> of algorithms which I will describe in my explanation.
>
> The calculation was only difficult in one aspect, namely the
> possible orientations of the corners. Here is the upper bound:
>
> (600!/2)*(1200!/2)*(720!/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)
>
> To answer Roice's question, I did find an arbitrary-precision
> calculator (Googol+, trial version) that displays the entire
> number in its full glory:
>
> 23435018363697222779126210606140343600982219866708667227704291465940007
> 37743198001537086016413748065359228217622633869330769129523601891497799
> 90823414733250819032377663096727895392891107724676361939174468537213471
> 84699260131924584724938945790242680862147295113762851571432130901040238
> 96149551266842769465158629370618815995041314328829732432057176063611161
> 23422302770133676753359134856348612503635674252607065815753807941966366
> 98057536512196715919594180779891338303538085006270891583549467992567391
> 85180535778985103137974951114346934416286264525322532242698044327455362
> 45594013789336900464999769975314632465421639791307831594564938014806846
> 43170816415770010483963217284920963354265992129473309221874222731561178
> 16542353296798264919536869373254412130565886195935986667368898349834998
> 21329543130389608025077440970771216857898162084209764122888617411552472
> 35969553038560987694512525780640891845410615026444483377179855326132898
> 75234261959552618641928258383934632570287455387991780347467121010722113
> 61928836844443707162412473785444967682885281547729595860180055748863425
> 82987146883285105106538113133816701062677558383952546932927579065352378
> 01699938857635611816907866063280477566711511600651402621287007177419657
> 47137395706297269591116929204261763967322064643743204180740840609622274
> 50477533328851963152796037024975768039218238701900252954269938177351575
> 02677389416404108346187189424180896325665818763923753999882873138580846
> 77228966566309226326618668840915289587325249776450099751298794240127365
> 82338830099863762586940626070992713912334648392737597361950653986530252
> 30327207836333881250393819594360829900032177090494274930448392889865527
> 16614065052187986459391365691818487107035801789611790557625739664732160
> 65851938727927023709296602229334790711981400149187774330090686038188693
> 91261606235286494542181170865114851849953373717543236061723434256780261
> 70547107650201457180103595669061215322965129698879686174938686442023883
> 00748340309101390837462739717879861015966407725537701315760595302551716
> 71986415959726651855198833270638521668923163972072488967633862968020464
> 59857154591489337994834077319666200102594473324166310981151194815089205
> 57151182894539191982468293894472660275087710846277134524865818266133054
> 42334380903271185037626999712118120957184637997454373033536708899691932
> 67133888402013848180589600375369501541798982850283307728992349369110105
> 46520467826426004880473115209607600645972315535718172987982451474844986
> 38207939550826131991321502334364118044702920268720341762396367094618866
> 13506333873119893045317942691097910138170606688929064386560230196639558
> 14850013029742851991570001253742052023463664478943640221712458729805575
> 31060005263175107421358143058444429499655242088655206273565212573195537
> 60877333581750403698478423610827287681768029874613544014049713469388295
> 71527311448542611407166314396015388432540041818749117638737494542689341
> 11177171793223712397899145235623177290619123784817857187575278090251642
> 14840994381181018155477448811016038175506185165844643664291009318841540
> 44262379730778251303396955855695117379535034691606466408011495289375314
> 79718119837237414217144612432890362908806009419290522751980178308626097
> 82557352902277742867710678069168798437309117460470080742233210420451923
> 51892222110034306764675636566213792404505572865844730850823909858221035
> 85434125488705753096939302756128054596976405648147286686571256428614441
> 70150339281973567744826847917503807378348535977716210565251241466128121
> 27489744510755607030104354706835298463258585993214964912284096487358233
> 65511645083418224779444054273548541176504221269485074946310915002488718
> 06278368621631798236853973136155878954455605260743873955312576387057492
> 91464434261509507888671511637033790932545587351156241243113539812047803
> 32568991232059877343852662991568615775896662637181748468409947337080580
> 40572024450182896514659885925637625707421001762988412973222775700881854
> 24686720928281296902412091261171945310151555961276870291126372512436270
> 06952830074271589382737534210078713204605113743280142889267219197170354
> 89582163680862223974028925034235784987192176322008136686679929878224156
> 10403264150753927641888742398755843764458442462577801213007109287401551
> 75841278684502282053918624209180100149726514306210673113609117091246135
> 65223035254373475279222721857792173328260547996939875237576837159815118
> 68437995628528234082912911481312451144148067422323644080656449490476178
> 69974972208660720805312515123592330897469991534040275425615810166559862
> 90453613626399014292854794414276344511148330328685197189376868495672292
> 92219730863040706516534661225522594916193135982360752630702024083643694
> 98309370698498093084824000430278614244379613737044583505562135097992678
> 05260822568001712636170958894341958120566294218192269192589063843887982
> 76399691863692115560126140715610109114003149849449412445547918396053785
> 42102203322863117771608181202015567452798664995683183676667131261081040
> 30610697346947842941118989099929505001072885907143020380705671971970771
> 74641197649400613289762440747999594711818677478380093322693390443497615
> 06790858102509872412149022901579978879595015323736540446504645407248271
> 24442974862512599608887589752218559193144931596281284315382618742792620
> 66616881593787942961115669105927538622586908510205223079160329890976613
> 24318437454227007437361053657522104635365530904941109094426111379946913
> 71854373062622155659107585797616686931874970164036381491924856162327083
> 87215983278489287122517383893445550987886905214362677925682743059318092
> 90715982378923238819174377671246115948128333247228319128499096287090364
> 06418925007261761200623236257957747401814104819201322380783299932882694
> 97705069648412898606682892058216414535137703172325349226036435233500060
> 08811101917211049364489819890827973553466812312700794247970136249599713
> 68830975248367892523082396738072274831267273049793679458450960225575330
> 90840325055925127369491405078011569600959829055592354981900265212992888
> 74537523085955044911868546931655826676111114140917917721449373043059908
> 24075969774780698659600903247322382509271117981454345778343923721701140
> 34040387142730946291948768518544291460594918104272439297270660195239204
> 69851212038726474485921192066725395225842350618752505691550098017532445
> 29742915483006071654290990776376332377597123229369363319211034520828156
> 16383626599751692734054125142693424208441259140739967321942103460390485
> 73512549204538199361441602981588922796564372729802637150967463996220269
> 92509662606254579651749991204772662937610983604733514590588466763484779
> 75336521786978901110936704729127455396942554264720541493172351367586278
> 52118009553781736752460941012653895714556808971968820220233708185524289
> 26324734529251413367934964381909880343066993726638347012446562279909471
> 00665871099287936575368913555297252102692185719691751527961183922455231
> 72371787084227168597930766375481134730976264840880937562376002100356262
> 35107696623982892463214959186113390887996406714662349935032809366747705
> 65928739057221107528446966775483155772142995330294320084551917275407602
> 87830218813852897349462068161826723496382625465167461901840049431850524
> 89018407136301332421978685188429040584176333209422917640992438642326814
> 48471988797114061727140678598275664209888253021405519815632168746508451
> 06268660985414101246860290152701992710734632469139060273152361598118124
> 26751417487110100479881455904096718779749277515897027333976945734065218
> 13427043475236100473238801150143720068671986307968791851735260237830993
> 59283951655340545557118534217560207955199404424096337283839953943573882
> 77230454238664055061747285720327136114251359554586129278357158728391085
> 51846977682603655222729137145829356583863818649600000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 00000000000000000000000000000000000000000000000000000000000000000000000
> 000000000000000000000000000000000
>
> What a number!
[...]

That was a great analogy!

I did a quick calculation - the number is approximately
2.3 x 10^8240, therefore not even one hundred layers of
universes would be enough!!!

I realized right after I submitted my post that I made
a minor error. When I said:

> To show this number is exact, we will have to find 3 algorithms:
> one that performs a 3-cycle of any three 2-coloreds without
> affecting any other 2-coloreds, a 3-cycle of any three 3-coloreds
> without affecting any other 3-coloreds, and a 3-cycle of any
> three 4-coloreds without affecting any other 4-coloreds. These
> three algorithms, when combined with each other and conjugates
> (setup moves), can produce any possible permutation of the
> pieces.

I should have said:

> To show this number is exact, we will have to find 3 algorithms:
> one that performs a 3-cycle of any three 2-coloreds without
> affecting any other pieces, a 3-cycle of any three 3-coloreds
> without affecting any other pieces, and a 3-cycle of any
> three 4-coloreds without affecting any other pieces. These
> three algorithms, when combined with each other and conjugates
> (setup moves), can produce any possible permutation of the
> pieces.

where each algorithm performs its task without affecting
any other pieces, instead of any other pieces with the same
number of colors. But this error is minor, and does not
affect the final answer!

Also, I won't worry about the length of my posts anymore!
Roice mentioned an interest in my general algorithm that
performs a 3-cycle of any three pieces on any sized 4D
cube, so I will post it soon, although I do think it is
relatively simple. However, I think similar techniques
could be used for the pieces of the 120-cell, which
would help validate the number I calculated. Also,
I looked up the section in "The Rubik Tesseract" in
Appendix A on the algorithms they discovered to validate
their calculation of the 3^4 cube. They managed to
obtain all of the required algorithms for the 4-coloreds
we would need to find (20 different algorithms!)
using a single pair of twists! I believe similar methods
could be used on the 120-cell.

-David

--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>
> What a number indeed!
>=20
> So let me get this straight. If you imagine all the particles in=20
the=20
> universe, and then imagine that each one really consists of=20
another=20
> entire universe, and for each particle in those universes, another=20
> universe, and so on ten times, you would still not have enough=20
particles=20
> so that each one could represent one unique state of this puzzle?=20
OK, I=20
> suppose that counts as a big number. :-)
>=20
> BTW, don't worry about the length of your posts David. It's easy=20
enough=20
> for anyone who's not interested to just delete them. Any subject=20
even=20
> remotely on-topic should be fair game. Even if the posts become=20
too=20
> frequent for some people, they can choose to get daily digests or=20
even=20
> no email at all and just read the messages on the web site when=20
they=20
> feel like it.
>=20
> -Melinda

------=_Part_4080_4947200.1210173370749
Content-Type: text/plain; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
Content-Disposition: inline

You guys will likely find this article interesting and a fun read:

http://www.scottaaronson.com/writings/bignumbers.html

Charlie shared it with me some time ago and it discusses truly unbelievably
unimaginably big numbers (not that the number David presented doesn't fall
into this category ;))

David, thank you *so much* for your writeup on this!! I fixed my incorrect
listing of the number of pieces on the site (I apparently accidentally
multiplied by 2 instead of dividing by 2 when calculating the number of 2C
pieces, which is also why my quick permutation estimate was so far off).
For the total I wrote there, I also included the 0C (120-cell shaped)
interior and hidden piece of the puzzle.

Roice

On Wed, May 7, 2008 at 1:43 AM, Melinda Green
wrote:

> What a number indeed!
>
> So let me get this straight. If you imagine all the particles in the
> universe, and then imagine that each one really consists of another
> entire universe, and for each particle in those universes, another
> universe, and so on ten times, you would still not have enough particles
> so that each one could represent one unique state of this puzzle? OK, I
> suppose that counts as a big number. :-)
>
> BTW, don't worry about the length of your posts David. It's easy enough
> for anyone who's not interested to just delete them. Any subject even
> remotely on-topic should be fair game. Even if the posts become too
> frequent for some people, they can choose to get daily digests or even
> no email at all and just read the messages on the web site when they
> feel like it.
>
> -Melinda
>
>
> David Smith wrote:
> > Hi everyone,
> >
> > I have just finished calculating an upper bound for the number
> > of permutations of the 120-Cell. To establish it is the exact answer
> > (which I am virtually certain of), we will have to find a number
> > of algorithms which I will describe in my explanation.
> >
> > The calculation was only difficult in one aspect, namely the
> > possible orientations of the corners. Here is the upper bound:
> >
> > (600!/2)*(1200!/2)*(720!/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)
> >
> > To answer Roice's question, I did find an arbitrary-precision
> > calculator (Googol+, trial version) that displays the entire
> > number in its full glory:
> >
> > 23435018363697222779126210606140343600982219866708667227704291465940007
> > 37743198001537086016413748065359228217622633869330769129523601891497799
> > 90823414733250819032377663096727895392891107724676361939174468537213471
> > 84699260131924584724938945790242680862147295113762851571432130901040238
> > 96149551266842769465158629370618815995041314328829732432057176063611161
> > 23422302770133676753359134856348612503635674252607065815753807941966366
> > 98057536512196715919594180779891338303538085006270891583549467992567391
> > 85180535778985103137974951114346934416286264525322532242698044327455362
> > 45594013789336900464999769975314632465421639791307831594564938014806846
> > 43170816415770010483963217284920963354265992129473309221874222731561178
> > 16542353296798264919536869373254412130565886195935986667368898349834998
> > 21329543130389608025077440970771216857898162084209764122888617411552472
> > 35969553038560987694512525780640891845410615026444483377179855326132898
> > 75234261959552618641928258383934632570287455387991780347467121010722113
> > 61928836844443707162412473785444967682885281547729595860180055748863425
> > 82987146883285105106538113133816701062677558383952546932927579065352378
> > 01699938857635611816907866063280477566711511600651402621287007177419657
> > 47137395706297269591116929204261763967322064643743204180740840609622274
> > 50477533328851963152796037024975768039218238701900252954269938177351575
> > 02677389416404108346187189424180896325665818763923753999882873138580846
> > 77228966566309226326618668840915289587325249776450099751298794240127365
> > 82338830099863762586940626070992713912334648392737597361950653986530252
> > 30327207836333881250393819594360829900032177090494274930448392889865527
> > 16614065052187986459391365691818487107035801789611790557625739664732160
> > 65851938727927023709296602229334790711981400149187774330090686038188693
> > 91261606235286494542181170865114851849953373717543236061723434256780261
> > 70547107650201457180103595669061215322965129698879686174938686442023883
> > 00748340309101390837462739717879861015966407725537701315760595302551716
> > 71986415959726651855198833270638521668923163972072488967633862968020464
> > 59857154591489337994834077319666200102594473324166310981151194815089205
> > 57151182894539191982468293894472660275087710846277134524865818266133054
> > 42334380903271185037626999712118120957184637997454373033536708899691932
> > 67133888402013848180589600375369501541798982850283307728992349369110105
> > 46520467826426004880473115209607600645972315535718172987982451474844986
> > 38207939550826131991321502334364118044702920268720341762396367094618866
> > 13506333873119893045317942691097910138170606688929064386560230196639558
> > 14850013029742851991570001253742052023463664478943640221712458729805575
> > 31060005263175107421358143058444429499655242088655206273565212573195537
> > 60877333581750403698478423610827287681768029874613544014049713469388295
> > 71527311448542611407166314396015388432540041818749117638737494542689341
> > 11177171793223712397899145235623177290619123784817857187575278090251642
> > 14840994381181018155477448811016038175506185165844643664291009318841540
> > 44262379730778251303396955855695117379535034691606466408011495289375314
> > 79718119837237414217144612432890362908806009419290522751980178308626097
> > 82557352902277742867710678069168798437309117460470080742233210420451923
> > 51892222110034306764675636566213792404505572865844730850823909858221035
> > 85434125488705753096939302756128054596976405648147286686571256428614441
> > 70150339281973567744826847917503807378348535977716210565251241466128121
> > 27489744510755607030104354706835298463258585993214964912284096487358233
> > 65511645083418224779444054273548541176504221269485074946310915002488718
> > 06278368621631798236853973136155878954455605260743873955312576387057492
> > 91464434261509507888671511637033790932545587351156241243113539812047803
> > 32568991232059877343852662991568615775896662637181748468409947337080580
> > 40572024450182896514659885925637625707421001762988412973222775700881854
> > 24686720928281296902412091261171945310151555961276870291126372512436270
> > 06952830074271589382737534210078713204605113743280142889267219197170354
> > 89582163680862223974028925034235784987192176322008136686679929878224156
> > 10403264150753927641888742398755843764458442462577801213007109287401551
> > 75841278684502282053918624209180100149726514306210673113609117091246135
> > 65223035254373475279222721857792173328260547996939875237576837159815118
> > 68437995628528234082912911481312451144148067422323644080656449490476178
> > 69974972208660720805312515123592330897469991534040275425615810166559862
> > 90453613626399014292854794414276344511148330328685197189376868495672292
> > 92219730863040706516534661225522594916193135982360752630702024083643694
> > 98309370698498093084824000430278614244379613737044583505562135097992678
> > 05260822568001712636170958894341958120566294218192269192589063843887982
> > 76399691863692115560126140715610109114003149849449412445547918396053785
> > 42102203322863117771608181202015567452798664995683183676667131261081040
> > 30610697346947842941118989099929505001072885907143020380705671971970771
> > 74641197649400613289762440747999594711818677478380093322693390443497615
> > 06790858102509872412149022901579978879595015323736540446504645407248271
> > 24442974862512599608887589752218559193144931596281284315382618742792620
> > 66616881593787942961115669105927538622586908510205223079160329890976613
> > 24318437454227007437361053657522104635365530904941109094426111379946913
> > 71854373062622155659107585797616686931874970164036381491924856162327083
> > 87215983278489287122517383893445550987886905214362677925682743059318092
> > 90715982378923238819174377671246115948128333247228319128499096287090364
> > 06418925007261761200623236257957747401814104819201322380783299932882694
> > 97705069648412898606682892058216414535137703172325349226036435233500060
> > 08811101917211049364489819890827973553466812312700794247970136249599713
> > 68830975248367892523082396738072274831267273049793679458450960225575330
> > 90840325055925127369491405078011569600959829055592354981900265212992888
> > 74537523085955044911868546931655826676111114140917917721449373043059908
> > 24075969774780698659600903247322382509271117981454345778343923721701140
> > 34040387142730946291948768518544291460594918104272439297270660195239204
> > 69851212038726474485921192066725395225842350618752505691550098017532445
> > 29742915483006071654290990776376332377597123229369363319211034520828156
> > 16383626599751692734054125142693424208441259140739967321942103460390485
> > 73512549204538199361441602981588922796564372729802637150967463996220269
> > 92509662606254579651749991204772662937610983604733514590588466763484779
> > 75336521786978901110936704729127455396942554264720541493172351367586278
> > 52118009553781736752460941012653895714556808971968820220233708185524289
> > 26324734529251413367934964381909880343066993726638347012446562279909471
> > 00665871099287936575368913555297252102692185719691751527961183922455231
> > 72371787084227168597930766375481134730976264840880937562376002100356262
> > 35107696623982892463214959186113390887996406714662349935032809366747705
> > 65928739057221107528446966775483155772142995330294320084551917275407602
> > 87830218813852897349462068161826723496382625465167461901840049431850524
> > 89018407136301332421978685188429040584176333209422917640992438642326814
> > 48471988797114061727140678598275664209888253021405519815632168746508451
> > 06268660985414101246860290152701992710734632469139060273152361598118124
> > 26751417487110100479881455904096718779749277515897027333976945734065218
> > 13427043475236100473238801150143720068671986307968791851735260237830993
> > 59283951655340545557118534217560207955199404424096337283839953943573882
> > 77230454238664055061747285720327136114251359554586129278357158728391085
> > 51846977682603655222729137145829356583863818649600000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 00000000000000000000000000000000000000000000000000000000000000000000000
> > 000000000000000000000000000000000
> >
> > What a number!
> [...]
>
>
>

------=_Part_4080_4947200.1210173370749
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable
Content-Disposition: inline

You guys will likely find this article interesting and a fun read:v>

http=
://www.scottaaronson.com/writings/bignumbers.html

Charlie shared it with me some time ago and it discusses truly unbelie=
vably unimaginably big numbers (not that the number David presented do=
esn't fall into this category ;))

David, thank you so much for your writeup on this!!&n=
bsp; I fixed my incorrect listing of the number of pieces on the site =
(I apparently accidentally multiplied by 2 instead of dividing by 2 when ca=
lculating the number of 2C pieces, which is also why my quick permutation e=
stimate was so far off). For the total I wrote there, I also included=
the 0C (120-cell shaped) interior and hidden piece of the puzzle.v>

Roice

On Wed, May 7, 2008 at 1:43 AM, Melinda Green &l=
t;melinda@superliminal.com&=
gt; wrote:

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

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

What a number indeed!

So let me get this straight. If you imagine=
all the particles in the
universe, and then imagine that each one real=
ly consists of another
entire universe, and for each particle in those =
universes, another

universe, and so on ten times, you would still not have enough particles r>so that each one could represent one unique state of this puzzle? OK, I <=
br>suppose that counts as a big number. :-)

BTW, don't worry abo=
ut the length of your posts David. It's easy enough

for anyone who's not interested to just delete them. Any subject even <=
br>remotely on-topic should be fair game. Even if the posts become too
=
frequent for some people, they can choose to get daily digests or even

no email at all and just read the messages on the web site when they
fe=
el like it.

-Melinda=20

David Smith wrote:
> Hi everyone,
&g=
t;
> I have just finished calculating an upper bound for the numberr>> of permutations of the 120-Cell. To establish it is the exact answer=

> (which I am virtually certain of), we will have to find a number
&g=
t; of algorithms which I will describe in my explanation.
>
> T=
he calculation was only difficult in one aspect, namely the
> possibl=
e orientations of the corners. Here is the upper bound:

>
> (600!/2)*(1200!/2)*(720!/2)*((2^720)/2)*((6^1200)/2)*((12^600)=
/3)
>
> To answer Roice's question, I did find an arbitrary=
-precision
> calculator (Googol+, trial version) that displays the e=
ntire

> number in its full glory:
>
> 2343501836369722277912621060=
6140343600982219866708667227704291465940007
> 37743198001537086016413=
748065359228217622633869330769129523601891497799
> 908234147332508190=
32377663096727895392891107724676361939174468537213471

> 8469926013192458472493894579024268086214729511376285157143213090104023=
8
> 96149551266842769465158629370618815995041314328829732432057176063=
611161
> 234223027701336767533591348563486125036356742526070658157538=
07941966366

> 9805753651219671591959418077989133830353808500627089158354946799256739=
1
> 85180535778985103137974951114346934416286264525322532242698044327=
455362
> 455940137893369004649997699753146324654216397913078315945649=
38014806846

> 4317081641577001048396321728492096335426599212947330922187422273156117=
8
> 16542353296798264919536869373254412130565886195935986667368898349=
834998
> 213295431303896080250774409707712168578981620842097641228886=
17411552472

> 3596955303856098769451252578064089184541061502644448337717985532613289=
8
> 75234261959552618641928258383934632570287455387991780347467121010=
722113
> 619288368444437071624124737854449676828852815477295958601800=
55748863425

> 8298714688328510510653811313381670106267755838395254693292757906535237=
8
> 01699938857635611816907866063280477566711511600651402621287007177=
419657
> 471373957062972695911169292042617639673220646437432041807408=
40609622274

> 5047753332885196315279603702497576803921823870190025295426993817735157=
5
> 02677389416404108346187189424180896325665818763923753999882873138=
580846
> 772289665663092263266186688409152895873252497764500997512987=
94240127365

> 8233883009986376258694062607099271391233464839273759736195065398653025=
2
> 30327207836333881250393819594360829900032177090494274930448392889=
865527
> 166140650521879864593913656918184871070358017896117905576257=
39664732160

> 6585193872792702370929660222933479071198140014918777433009068603818869=
3
> 91261606235286494542181170865114851849953373717543236061723434256=
780261
> 705471076502014571801035956690612153229651296988796861749386=
86442023883

> 0074834030910139083746273971787986101596640772553770131576059530255171=
6
> 71986415959726651855198833270638521668923163972072488967633862968=
020464
> 598571545914893379948340773196662001025944733241663109811511=
94815089205

> 5715118289453919198246829389447266027508771084627713452486581826613305=
4
> 42334380903271185037626999712118120957184637997454373033536708899=
691932
> 671338884020138481805896003753695015417989828502833077289923=
49369110105

> 4652046782642600488047311520960760064597231553571817298798245147484498=
6
> 38207939550826131991321502334364118044702920268720341762396367094=
618866
> 135063338731198930453179426910979101381706066889290643865602=
30196639558

> 1485001302974285199157000125374205202346366447894364022171245872980557=
5
> 31060005263175107421358143058444429499655242088655206273565212573=
195537
> 608773335817504036984784236108272876817680298746135440140497=
13469388295

> 7152731144854261140716631439601538843254004181874911763873749454268934=
1
> 11177171793223712397899145235623177290619123784817857187575278090=
251642
> 148409943811810181554774488110160381755061851658446436642910=
09318841540

> 4426237973077825130339695585569511737953503469160646640801149528937531=
4
> 79718119837237414217144612432890362908806009419290522751980178308=
626097
> 825573529022777428677106780691687984373091174604700807422332=
10420451923

> 5189222211003430676467563656621379240450557286584473085082390985822103=
5
> 85434125488705753096939302756128054596976405648147286686571256428=
614441
> 701503392819735677448268479175038073783485359777162105652512=
41466128121

> 2748974451075560703010435470683529846325858599321496491228409648735823=
3
> 65511645083418224779444054273548541176504221269485074946310915002=
488718
> 062783686216317982368539731361558789544556052607438739553125=
76387057492

> 9146443426150950788867151163703379093254558735115624124311353981204780=
3
> 32568991232059877343852662991568615775896662637181748468409947337=
080580
> 405720244501828965146598859256376257074210017629884129732227=
75700881854

> 2468672092828129690241209126117194531015155596127687029112637251243627=
0
> 06952830074271589382737534210078713204605113743280142889267219197=
170354
> 895821636808622239740289250342357849871921763220081366866799=
29878224156

> 1040326415075392764188874239875584376445844246257780121300710928740155=
1
> 75841278684502282053918624209180100149726514306210673113609117091=
246135
> 652230352543734752792227218577921733282605479969398752375768=
37159815118

> 6843799562852823408291291148131245114414806742232364408065644949047617=
8
> 69974972208660720805312515123592330897469991534040275425615810166=
559862
> 904536136263990142928547944142763445111483303286851971893768=
68495672292

> 9221973086304070651653466122552259491619313598236075263070202408364369=
4
> 98309370698498093084824000430278614244379613737044583505562135097=
992678
> 052608225680017126361709588943419581205662942181922691925890=
63843887982

> 7639969186369211556012614071561010911400314984944941244554791839605378=
5
> 42102203322863117771608181202015567452798664995683183676667131261=
081040
> 306106973469478429411189890999295050010728859071430203807056=
71971970771

> 7464119764940061328976244074799959471181867747838009332269339044349761=
5
> 06790858102509872412149022901579978879595015323736540446504645407=
248271
> 244429748625125996088875897522185591931449315962812843153826=
18742792620

> 6661688159378794296111566910592753862258690851020522307916032989097661=
3
> 24318437454227007437361053657522104635365530904941109094426111379=
946913
> 718543730626221556591075857976166869318749701640363814919248=
56162327083

> 8721598327848928712251738389344555098788690521436267792568274305931809=
2
> 90715982378923238819174377671246115948128333247228319128499096287=
090364
> 064189250072617612006232362579577474018141048192013223807832=
99932882694

> 9770506964841289860668289205821641453513770317232534922603643523350006=
0
> 08811101917211049364489819890827973553466812312700794247970136249=
599713
> 688309752483678925230823967380722748312672730497936794584509=
60225575330

> 9084032505592512736949140507801156960095982905559235498190026521299288=
8
> 74537523085955044911868546931655826676111114140917917721449373043=
059908
> 240759697747806986596009032473223825092711179814543457783439=
23721701140

> 3404038714273094629194876851854429146059491810427243929727066019523920=
4
> 69851212038726474485921192066725395225842350618752505691550098017=
532445
> 297429154830060716542909907763763323775971232293693633192110=
34520828156

> 1638362659975169273405412514269342420844125914073996732194210346039048=
5
> 73512549204538199361441602981588922796564372729802637150967463996=
220269
> 925096626062545796517499912047726629376109836047335145905884=
66763484779

> 7533652178697890111093670472912745539694255426472054149317235136758627=
8
> 52118009553781736752460941012653895714556808971968820220233708185=
524289
> 263247345292514133679349643819098803430669937266383470124465=
62279909471

> 0066587109928793657536891355529725210269218571969175152796118392245523=
1
> 72371787084227168597930766375481134730976264840880937562376002100=
356262
> 351076966239828924632149591861133908879964067146623499350328=
09366747705

> 6592873905722110752844696677548315577214299533029432008455191727540760=
2
> 87830218813852897349462068161826723496382625465167461901840049431=
850524
> 890184071363013324219786851884290405841763332094229176409924=
38642326814

> 4847198879711406172714067859827566420988825302140551981563216874650845=
1
> 06268660985414101246860290152701992710734632469139060273152361598=
118124
> 267514174871101004798814559040967187797492775158970273339769=
45734065218

> 1342704347523610047323880115014372006867198630796879185173526023783099=
3
> 59283951655340545557118534217560207955199404424096337283839953943=
573882
> 772304542386640550617472857203271361142513595545861292783571=
58728391085

> 5184697768260365522272913714582935658386381864960000000000000000000000=
0
> 00000000000000000000000000000000000000000000000000000000000000000=
000000
> 000000000000000000000000000000000000000000000000000000000000=
00000000000

> 0000000000000000000000000000000000000000000000000000000000000000000000=
0
> 00000000000000000000000000000000000000000000000000000000000000000=
000000
> 000000000000000000000000000000000000000000000000000000000000=
00000000000

> 0000000000000000000000000000000000000000000000000000000000000000000000=
0
> 00000000000000000000000000000000000000000000000000000000000000000=
000000
> 000000000000000000000000000000000000000000000000000000000000=
00000000000

> 000000000000000000000000000000000
>
> What a number!
div>
[...]

lockquote>

------=_Part_4080_4947200.1210173370749--

From: "spel_werdz_rite" <spel_werdz_rite@yahoo.com>
Date: Thu, 08 May 2008 01:43:02 -0000
Subject: [MC4D] Re: Magic120Cell Realized

I can confirm that is the total permutations. I calculated this about
a year ago when I first started talking to Roice about the 120 cell
and it got deleted, I didn't have the energy to re-comma separate
8,000 digits, but I remember it slightly and the number I found
matches this one. =3D)

From: "spel_werdz_rite" <spel_werdz_rite@yahoo.com>
Date: Thu, 08 May 2008 02:19:59 -0000
Subject: [MC4D] Re: Magic120Cell Realized

Hello Roice,

I really think this is an amazing program and I will definitely
attempt to solve it, but for me, looking at 120 different colors is
just too confusing, so I thought it might be easier if each color had
a number or some type of distinguishing pattern it so there wouldn't
be so much confusion between the colors. Also, on the MagicCube5D, I
really liked the piece finder, and I think it would make the 120 Cell
much easier if there were a piece finder in it. Just a couple
suggestions, but great job creating such an awesome puzzle!

--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> The 4D analog of Megaminx is here! This has been a work in progress for
> some time really (the basics were done in 2006), but it is now evolved
> enough for you crazy solvers to have at it! I like playing with it
and it
> will be fun to figure out sequences, but I'm not imagining I'll try a
> solution myself. I do hope some of you will take it on, and that
getting it
> out now is good timing with the summer break for students and all. I'm
> really pleased with how it is turning out, and have screenshots at
the site
> to check out if you don't feel like installing...
>=20
> www.gravitation3d.com/magic120cell/
>=20
> You'll see that I made some easier (or should I say just slightly less
> ludicrous) puzzles which are only different in that they have
multiple cells
> set to the same color. These accentuate some of the 120cell
symmetries and
> could be more enjoyable simply because there are not so many colors
to deal
> with. The biggest feat is of course the full puzzle with 120 different
> colored cells, but all of them are hard and I will post the name of
anyone
> who solves any of the puzzles in a Hall of Insanity.
>=20
> I admit straight away there are still features that would be nice to
have at
> some point, and I do have my own wish list. Though I make no guarantees
> when or if I can add stuff, please feel free to make suggestions or
requests
> you would find valuable. Bug reports will also be appreciated.
>=20
> I hope you all really enjoy it. Good Luck!
>=20
> Roice
>=20
> P.S. This is what I alluded to this morning. I haven't figured it out
> exactly yet and may not be able to, but a quick estimate shows it to
have
> well over 10^15000 (that's right, 10^15K) permutations!
>

From: "David Smith" <djs314djs314@yahoo.com>
Date: Fri, 09 May 2008 01:29:46 -0000
Subject: [MC4D] Re: Magic120Cell Realized

Roice, that was a great article! Some of those numbers make
the number I found look like nothing! Thank you again for
putting my result on your website.

spel_werdz_rite, thank you for verifying this result! I had
no idea anyone else had calculated this number.

I recently had another idea for Magic120Cell before I go
back to the n^4 cube. It seems like it will be very difficult,
but I am going to try to find the number of visually different
positions of each of the other variations of puzzles (the
2-colored, both 9-colored, and the 12-colored versions) of=20
Magic120Cell. This will involve accounting for the similarly
colored pieces (4-colored pieces with the same colors may not be=20
visually identical due to their orientation, and counting the pieces=20
will require the use of the Magic120Cell program), and the similarly
colored centers (accounting for apparently different positions
acctually being visually identical due to rotations of the entire
puzzle in 4-space; the corner orientation logic would also apply
to the centers for counting how many ways the they can be visually
identical when rotated. This would be made eaiser by imagining the
0-colored piece that Roice mentioned.) These are just a few quick=20
observations, there may be more complications I am not yet aware of.

All the best,
David

--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> You guys will likely find this article interesting and a fun read:
>=20
> http://www.scottaaronson.com/writings/bignumbers.html
>=20
> Charlie shared it with me some time ago and it discusses truly=20
unbelievably
> unimaginably big numbers (not that the number David presented=20
doesn't fall
> into this category ;))
>=20
> David, thank you *so much* for your writeup on this!! I fixed my=20
incorrect
> listing of the number of pieces on the site (I apparently=20
accidentally
> multiplied by 2 instead of dividing by 2 when calculating the=20
number of 2C
> pieces, which is also why my quick permutation estimate was so far=20
off).
> For the total I wrote there, I also included the 0C (120-cell=20
shaped)
> interior and hidden piece of the puzzle.
>=20
> Roice
>=20
>=20
> On Wed, May 7, 2008 at 1:43 AM, Melinda Green
> wrote:
>=20
> > What a number indeed!
> >
> > So let me get this straight. If you imagine all the particles in=20
the
> > universe, and then imagine that each one really consists of=20
another
> > entire universe, and for each particle in those universes,=20
another
> > universe, and so on ten times, you would still not have enough=20
particles
> > so that each one could represent one unique state of this=20
puzzle? OK, I
> > suppose that counts as a big number. :-)
> >
> > BTW, don't worry about the length of your posts David. It's easy=20
enough
> > for anyone who's not interested to just delete them. Any subject=20
even
> > remotely on-topic should be fair game. Even if the posts become=20
too
> > frequent for some people, they can choose to get daily digests=20
or even
> > no email at all and just read the messages on the web site when=20
they
> > feel like it.
> >
> > -Melinda

From: Jay Berkenbilt <ejb@ql.org>
Date: Sat, 10 May 2008 10:57:26 -0400
Subject: Re: [MC4D] Magic120Cell Realized

I have to add my voice to the rest in expression of awe at this
puzzle. It's been years since I've even done mc4d -- my life has
gotten busier. One day maybe I'll try it, and I'm sure I'll
eventually play around with it just to see what it feels like. As
with many of the other participants on this list, I have always had a
special affinity for the 120-cell. It always seemed to me that it
sort of snuck in to the regular polyhedron list, just barely fitting,
kind of like the pentagon just barely being able to be the face shape
of one of the platonic solids. :-) Do I recall correctly that this
polyhedron is its own dual?

> Honestly, the reason I wasn't planning on working through a solution
> was that I am a bit scared of the sheer number of pieces! I just
> finished up the final parts that I felt were needed for it to be
> solvable today, and I actually haven't even figured out a single
> sequence yet. So as of this evening, I only have the thoughts about
> it we've discussed in the past, which is that it will be easier in
> some ways than MC4D because of the larger space to sequester pieces,
> but that it will be a big effort in time. Also, I think I am ready
> for a bit of a rest and was too excited to share to let it sit on a
> shelf. Sarah will be happy to get my attention back now too since
> I've been spending a lot of time on it lately :)

My recollection of solving the megaminx is that you can do all but the
last few steps as localized solutions. Each twist affects such a
small number of pieces that the constraints don't play a big role
until the end. It seems that each twist would necessarily alter
pieces on the 12 adjacent cells.

I don't find it surprising that five random twists would result in
some interacting pieces. The first twist affects pieces on 12 of the
120 cells, not including the cell twisted. In order for the second
twist to not interact with any pieces, it must be on a cell that is
neither any of the 12 affected faces nor adjacent to any of them
(except that it could be another twist of the first face). I'm not
sure how many cells that is. If you managed to get one, there are
even fewer places for the third twist. It seems to me that the number
of twists after which there is some guaranteed interaction must be
very small....maybe three or four? I could probably work it out, but
I imagine others on this list could do it faster. My "math chops" may
be good compared to the general population, but not compared to many
of the readers of this list. :-)

Anyway, the 120 cell puzzle is a work of beauty!

--Jay

From: "David Barr" <david20708@gmail.com>
Date: Tue, 13 May 2008 10:22:51 -0700
Subject: Re: [MC4D] Re: Magic120Cell Realized

On Tue, May 6, 2008 at 10:24 PM, David Smith wrote:
> To answer Roice's question, I did find an arbitrary-precision
> calculator (Googol+, trial version) that displays the entire
> number in its full glory:
>
> 23435018363697222779126210606140343600982219866708667227704291465940007
> 37743198001537086016413748065359228217622633869330769129523601891497799
> 90823414733250819032377663096727895392891107724676361939174468537213471

The Unix program "bc" can also compute this value:

bc << EOF
define f (x) {
if (x <= 1) return (1);
return (f(x-1) * x);
}
(f(600)/2)*(f(1200)/2)*(f(720)/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)
EOF

A Windows version of "bc" is available with Cygwin (www.cygwin.com).

From: Jay Berkenbilt <ejb@ql.org>
Date: Tue, 13 May 2008 22:11:02 -0400
Subject: Re: [MC4D] Magic120Cell Realized

Mark Oram wrote:

> I fear the dual of the dodecahedron is in fact the icosohedron; while
> the cube and octahedron are similarly dual to each other. The
> tetrahedron IS its own dual however: possibly this is where your
> recollection came from?

My whole thing was about the 120-cell, not the dodecahedron. Reading
my original post, I see I used the word "polyhedron" all over the
place when I meant polytope or polychoron. I know that the
icosohedron and dodecahedron are duals, but I was just not remembering
whether the 120-cell had a dual.

I realize now that the 120-cell's dual is the 600-cell and that it's
the 24-cell with octohedron cells that's self-dual. (I looked on
wikipedia.) I remembered that one of the six platonic 4-topes was
self-dual.

It's been too many years since I've really played with these. :-)

--Jay

>
> --- On Sat, 10/5/08, Jay Berkenbilt wrote:
>
> From: Jay Berkenbilt
> Subject: Re: [MC4D] Magic120Cell Realized
> To: 4D_Cubing@yahoogroups.com
> Date: Saturday, 10 May, 2008, 3:57 PM
>
> I have to add my voice to the rest in expression of awe at this
> puzzle. It's been years since I've even done mc4d -- my life has
> gotten busier. One day maybe I'll try it, and I'm sure I'll
> eventually play around with it just to see what it feels like. As
> with many of the other participants on this list, I have always
> had a
> special affinity for the 120-cell. It always seemed to me that it
> sort of snuck in to the regular polyhedron list, just barely
> fitting,
> kind of like the pentagon just barely being able to be the face
> shape
> of one of the platonic solids. :-) Do I recall correctly that this
> polyhedron is its own dual?
>
> > Honestly, the reason I wasn't planning on working through a
> solution
> > was that I am a bit scared of the sheer number of pieces! I just
> > finished up the final parts that I felt were needed for it to be
> > solvable today, and I actually haven't even figured out a single
> > sequence yet. So as of this evening, I only have the thoughts
> about
> > it we've discussed in the past, which is that it will be easier
> in
> > some ways than MC4D because of the larger space to sequester
> pieces,
> > but that it will be a big effort in time. Also, I think I am
> ready
> > for a bit of a rest and was too excited to share to let it sit
> on a
> > shelf. Sarah will be happy to get my attention back now too
> since
> > I've been spending a lot of time on it lately :)
>
> My recollection of solving the megaminx is that you can do all but
> the
> last few steps as localized solutions. Each twist affects such a
> small number of pieces that the constraints don't play a big role
> until the end. It seems that each twist would necessarily alter
> pieces on the 12 adjacent cells.
>
> I don't find it surprising that five random twists would result in
> some interacting pieces. The first twist affects pieces on 12 of
> the
> 120 cells, not including the cell twisted. In order for the second
> twist to not interact with any pieces, it must be on a cell that
> is
> neither any of the 12 affected faces nor adjacent to any of them
> (except that it could be another twist of the first face). I'm not
> sure how many cells that is. If you managed to get one, there are
> even fewer places for the third twist. It seems to me that the
> number
> of twists after which there is some guaranteed interaction must be
> very small....maybe three or four? I could probably work it out,
> but
> I imagine others on this list could do it faster. My "math chops"
> may
> be good compared to the general population, but not compared to
> many
> of the readers of this list. :-)
>
> Anyway, the 120 cell puzzle is a work of beauty!
>
> --Jay
>
> =E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=
=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=
=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81
> Sent from Yahoo! Mail.
> A Smarter Email.
>
>=20

From: Mark Oram <markoram109@yahoo.co.uk>
Date: Wed, 14 May 2008 15:52:28 +0000 (GMT)
Subject: Re: [MC4D] Magic120Cell Realized

<=
/table>

Sent from href=3D"http://us.rd.yahoo.com/mailuk/taglines/isp/control/*http://us.rd.ya=
hoo.com/evt=3D52418/*http://uk.docs.yahoo.com/nowyoucan.html" target=3D_bla=
nk>Yahoo! Mail.

A Smarter Email.

From: David Smith <djs314djs314@yahoo.com>
Date: Thu, 15 May 2008 17:48:56 -0700 (PDT)
Subject: Re: [MC4D] Re: Magic120Cell Realized

--0-930228213-1210898936=:89726
Content-Type: text/plain; charset=iso-8859-1
Content-Transfer-Encoding: 8bit

Hi David,

Thanks for letting me know about bc! I actually already had two
Computer Algebra Systems on my computer (GAP and MuPad, which
is commercial!) but I couldn't get either of them to display the entire number,
let me copy the number, write the number to a file, or even print the
number!

I later downloaded Yacas, which is an excellent Computer Algebra System
that naturally supports Windows. I used it to check the number, because
for a while I thought it might have been incorrect, but it turned out that
only the number of digits listed on Roice's page was incorrect. It is now
fixed, and I have updated the paper, which is on Roice's main page.

Sorry I didn't respond earlier; my internet connection was down. Once again,
thanks!

Best Regards,

David

David Barr wrote:
On Tue, May 6, 2008 at 10:24 PM, David Smith wrote:
> To answer Roice's question, I did find an arbitrary-precision
> calculator (Googol+, trial version) that displays the entire
> number in its full glory:
>
> 23435018363697222779126210606140343600982219866708667227704291465940007
> 37743198001537086016413748065359228217622633869330769129523601891497799
> 90823414733250819032377663096727895392891107724676361939174468537213471

The Unix program "bc" can also compute this value:

bc << EOF
define f (x) {
if (x <= 1) return (1);
return (f(x-1) * x);
}
(f(600)/2)*(f(1200)/2)*(f(720)/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)
EOF

A Windows version of "bc" is available with Cygwin (www.cygwin.com).

--0-930228213-1210898936=:89726
Content-Type: text/html; charset=iso-8859-1
Content-Transfer-Encoding: 8bit

Hi David,

Thanks for letting me know about bc! I actually already had two

Computer Algebra Systems on my computer (GAP and MuPad, which

is commercial!) but I couldn't get either of them to display the entire number,

let me copy the number, write the number to a file, or even print the

number!

I later downloaded Yacas, which is an excellent Computer Algebra System

that naturally supports Windows. I used it to check the number, because

for a while I thought it might have been incorrect, but it turned out that

only the number of digits listed on Roice's page was incorrect. It is now

fixed, and I have updated the paper, which is on Roice's main page.

Sorry I didn't respond earlier; my internet connection was down. Once again,

thanks!

Best Regards,

David

David Barr <david20708@gmail.com> wrote:

On Tue, May 6, 2008 at 10:24 PM, David Smith <djs314djs314@yahoo.com> wrote:
> To answer Roice's question, I did find an arbitrary-precision
> calculator (Googol+, trial version) that displays the entire
> number in its full glory:
>
> 23435018363697222779126210606140343600982219866708667227704291465940007
>
37743198001537086016413748065359228217622633869330769129523601891497799
> 90823414733250819032377663096727895392891107724676361939174468537213471

The Unix program "bc" can also compute this value:

bc << EOF
define f (x) {
if (x <= 1) return (1);
return (f(x-1) * x);
}
(f(600)/2)*(f(1200)/2)*(f(720)/2)*((2^720)/2)*((6^1200)/2)*((12^600)/3)
EOF

A Windows version of "bc" is available with Cygwin (www.cygwin.com).

--0-930228213-1210898936=:89726--

Return to MagicCube4D main page
Return to the Superliminal home page

ont: inherit;'>
Jay,

Thanks for the update, and for amending my original post. I certianly di=
dn't know about the duality (?) of the 24- and 120-cells; and thank-you for=
the wikipedia research and for extending to my understanding.

Mark.

--- On Wed, 14/5/08, Jay Berkenbilt <ejb@ql.org>=
wrote:

16,16,255) 2px solid">From: Jay Berkenbilt <ejb@ql.org>
Subject: R=
e: [MC4D] Magic120Cell Realized
To: 4D_Cubing@yahoogroups.com
Date: W=
ednesday, 14 May, 2008, 3:11 AM

Mark Oram <nk rel=3Dnofollow>markoram109@ yahoo.co. uk> wrote:

> I fe=
ar the dual of the dodecahedron is in fact the icosohedron; while
> t=
he cube and octahedron are similarly dual to each other. The
> tetrah=
edron IS its own dual however: possibly this is where your
> recollec=
tion came from?

My whole thing was about the 120-cell, not the dodec=
ahedron. Reading
my original post, I see I used the word "polyhedron" al=
l over the
place when I meant polytope or polychoron. I know that the>icosohedron and dodecahedron are duals, but I was just not remembering
=
whether the 120-cell had a dual.

I realize now that the 120-cell's d=
ual is the 600-cell and that it's
the 24-cell with octohedron cells that=
's self-dual. (I looked on
wikipedia.) I remembered that one of the six =
platonic 4-topes was
self-dual.

It's been too many years since
I've really played with these. :-)

--Jay

>
> --- On=
Sat, 10/5/08, Jay Berkenbilt <_blank rel=3Dnofollow>ejb@ql.org> wrote:
>
> From: Jay B=
erkenbilt <w>ejb@ql.org>
> Subject: Re: [MC4D] Magic120Cell Realized
&=
gt; To: =3Dnofollow>4D_Cubing@yahoogrou ps.com
> Date: Saturday, 10 May, =
2008, 3:57 PM
>
> I have to add my voice to the rest in express=
ion of awe at this
> puzzle. It's been years since I've even done mc4=
d -- my life has
> gotten busier. One day maybe I'll try it, and I'm =
sure I'll
> eventually play around with it just to see what it feels =
like. As
> with many of the other participants on this list, I have a=
lways
> had a
> special affinity for the 120-cell. It always se=
emed to me
that it
> sort of snuck in to the regular polyhedron list, just bare=
ly
> fitting,
> kind of like the pentagon just barely being abl=
e to be the face
> shape
> of one of the platonic solids. :-) D=
o I recall correctly that this
> polyhedron is its own dual?
><=
BR>> > Honestly, the reason I wasn't planning on working through a>> solution
> > was that I am a bit scared of the sheer number =
of pieces! I just
> > finished up the final parts that I felt were=
needed for it to be
> > solvable today, and I actually haven't ev=
en figured out a single
> > sequence yet. So as of this evening, I=
only have the thoughts
> about
> > it we've discussed in th=
e past, which is that it will be easier
> in
> > some ways t=
han MC4D because of the larger space to sequester
> pieces,
> &=
gt; but that it will be a big effort in time. Also, I think I
am
> ready
> > for a bit of a rest and was too excited to s=
hare to let it sit
> on a
> > shelf. Sarah will be happy to =
get my attention back now too
> since
> > I've been spending=
a lot of time on it lately :)
>
> My recollection of solving t=
he megaminx is that you can do all but
> the
> last few steps a=
s localized solutions. Each twist affects such a
> small number of pi=
eces that the constraints don't play a big role
> until the end. It s=
eems that each twist would necessarily alter
> pieces on the 12 adjac=
ent cells.
>
> I don't find it surprising that five random twis=
ts would result in
> some interacting pieces. The first twist affects=
pieces on 12 of
> the
> 120 cells, not including the cell twis=
ted. In order for the second
> twist to not interact with any pieces,=
it must be on a cell that
> is
> neither any of the 12
affected faces nor adjacent to any of them
> (except that it could b=
e another twist of the first face). I'm not
> sure how many cells tha=
t is. If you managed to get one, there are
> even fewer places for th=
e third twist. It seems to me that the
> number
> of twists aft=
er which there is some guaranteed interaction must be
> very small...=
.maybe three or four? I could probably work it out,
> but
> I i=
magine others on this list could do it faster. My "math chops"
> may<=
BR>> be good compared to the general population, but not compared to
=
> many
> of the readers of this list. :-)
>
> Anyway, =
the 120 cell puzzle is a work of beauty!
>
> --Jay
>
&=
gt;
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=
=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=
=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=
=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=94=81=E2=
=94=81=E2=94=81=E2=94=81=E2=94=81
> Sent from Yahoo! Mail.
> A =
Smarter Email.
>
>

Thread: "Magic120Cell Realized"