Thread: "Further correction of erroneous order-2 Klein's Quartic permutation count"

From: David Smith <djs314djs314@yahoo.com>
Date: Sun, 1 May 2011 08:42:23 -0700 (PDT)
Subject: Re: [MC4D] Further correction of erroneous order-2 Klein's Quartic permutation count








=C2=A0



=20=20


=20=20=20=20
=20=20=20=20=20=20
=20=20=20=20=20=20
Hello all,



Aha! I did make a mistake, as I suspected I might have. It's been too lon=
g since I counted permutations. :) My previous answer was off by a factor =
of 2; the 2 in the denominator should not have been there. I made the clas=
sic mistake of taking even permutations into account when there are identic=
ally colored pieces (in this case, stickers). Naturally, with identically =
colored pieces parity does not matter, as even and odd permutations are ide=
ntical.



Here is the correct formula and number of permutations:



209!/(((7!)^29)*(6!)) =3D



298225837482940676509184100728720811004423111510479240177876435252200583634=
046383785012187494718519005671580310160782720965515620763618140216746589818=
598244298634557645722755900360127929767933245249492609612953156836861423588=
4567287678429277548260303941234016334643200000000000000000000



My apologies for not catching this sooner; I've been very busy today with e=
mails. I am virtually certain that this count is correct, as I just sent R=
ay a complete derivation of it (which is how I discovered my error).



Thanks to everyone for your patience, and I'll keep you informed of my prog=
ress.



All the best,

David





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

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


=20



=20=20




--0-434752769-1304264543=:70816
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">Hi everyone,

Argh!  Iit turns out=
there was a second mistake. :(  This one is especially embarrassing.&=
nbsp; I apologize for these corrections, and in the future will be much mor=
e careful before posting my results.  It's been way too long since I c=
alculated permutations.  This stuff is about 100,000 times simpler tha=
n my n^d Rubik's cube formula, which makes these corrections especially fru=
strating and embarrassing.  Hopefully I do not cast doubt on the valid=
ity of my Rubik's cube formulas, which I checked for months, but I would un=
derstand if anyone questions them at this point.

I simply miscounted=
the number of different colors in the puzzle! I was using a count of 30 co=
lors, when in fact the correct number is 24.  The really funny thing i=
s that MagicTile *tells you* how many colors are in each puzzle, right next=
to
the puzzle name!  So you can see my embarrassment. :(  Here is t=
he (possibly?) correct number of permutations:

167!/(((7!)^23)*(6!))=
=3D

14255999370045115741835726463321297234734952725080156378526476=
35847
690582530108702510010741240927006490999290073130235405319596424682=
3
6192752417277237699906089211520063444550594267719476697558665102816>164056186968

If there are any more errors in this very simple calcu=
lation, I think I'm going to quit working on these!  Just kidding.&nbs=
p; Not to sound arrogant (in fact I intend to convey the opposite) but this=
stuff is child's play compared to my Rubik's cube formulas. (Sigh.)
>All the best,
David

--- On Sat, 4/30/11, djs314djs314 <=
djs314djs314@yahoo.com>
wrote:
ft: 2px solid rgb(16, 16, 255); margin-left: 5px; padding-left: 5px;">
F=
rom: djs314djs314 <djs314djs314@yahoo.com>
Subject: [MC4D]
Correction of erroneous order-2 Klein's Quartic permutation count
To: 4=
D_Cubing@yahoogroups.com
Date: Saturday, April 30, 2011, 9:32 PM

=






 




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

Hello all,



Aha! I did make a mistake, as I suspected I might have. It's been too lon=
g since I counted permutations. :) My previous answer was off by a factor =
of 2; the 2 in the denominator should not have been there. I made the clas=
sic mistake of taking even permutations into account when there are identic=
ally colored pieces (in this case, stickers). Naturally, with identically =
colored pieces parity does not matter, as even and odd permutations are ide=
ntical.



Here is the correct formula and number of permutations:



209!/(((7!)^29)*(6!)) =3D



298225837482940676509184100728720811004423111510479240177876435252200583634=
046383785012187494718519005671580310160782720965515620763618140216746589818=
598244298634557645722755900360127929767933245249492609612953156836861423588=
4567287678429277548260303941234016334643200000000000000000000



My apologies for not catching this sooner; I've been very busy today with e=
mails. I am virtually certain that this count is correct, as I just sent R=
ay a complete derivation of it (which is how I discovered my error).



Thanks to everyone for your patience, and I'll keep you informed of my prog=
ress.



All the best,

David






=20=20=20=20=20



=20



--0-434752769-1304264543=:70816--




From: David Smith <djs314djs314@yahoo.com>
Date: Sun, 1 May 2011 09:50:03 -0700 (PDT)
Subject: Re: [MC4D] Further correction of erroneous order-2 Klein's Quartic permutation count








=C2=A0



=20=20


=20=20=20=20
=20=20=20=20=20=20
=20=20=20=20=20=20
Hi everyone,

Argh!=C2=A0 Iit turns out there was a second mistake. :(=C2=A0 This one is =
especially embarrassing.=C2=A0 I apologize for these corrections, and in th=
e future will be much more careful before posting my results.=C2=A0 It's be=
en way too long since I calculated permutations.=C2=A0 This stuff is about =
100,000 times simpler than my n^d Rubik's cube formula, which makes these c=
orrections especially frustrating and embarrassing.=C2=A0 Hopefully I do no=
t cast doubt on the validity of my Rubik's cube formulas, which I checked f=
or months, but I would understand if anyone questions them at this point.

I simply miscounted the number of different colors in the puzzle! I was usi=
ng a count of 30 colors, when in fact the correct number is 24.=C2=A0 The r=
eally funny thing is that MagicTile *tells you* how many colors are in each=
puzzle, right next to
the puzzle name!=C2=A0 So you can see my embarrassment. :(=C2=A0 Here is t=
he (possibly?) correct number of permutations:

167!/(((7!)^23)*(6!)) =3D=20

1425599937004511574183572646332129723473495272508015637852647635847
6905825301087025100107412409270064909992900731302354053195964246823
6192752417277237699906089211520063444550594267719476697558665102816
164056186968

If there are any more errors in this very simple calculation, I think I'm g=
oing to quit working on these!=C2=A0 Just kidding.=C2=A0 Not to sound arrog=
ant (in fact I intend to convey the opposite) but this stuff is child's pla=
y compared to my Rubik's cube formulas. (Sigh.)

All the best,
David

--- On Sat, 4/30/11, djs314djs314 wrote:

From: djs314djs314
Subject: [MC4D]
Correction of erroneous order-2 Klein's Quartic permutation count
To: 4D_Cubing@yahoogroups.com
Date: Saturday, April 30, 2011, 9:32 PM







=C2=A0



=20=20=20=20
=20=20=20=20=20=20
=20=20=20=20=20=20
Hello all,



Aha! I did make a mistake, as I suspected I might have. It's been too lon=
g since I counted permutations. :) My previous answer was off by a factor =
of 2; the 2 in the denominator should not have been there. I made the clas=
sic mistake of taking even permutations into account when there are identic=
ally colored pieces (in this case, stickers). Naturally, with identically =
colored pieces parity does not matter, as even and odd permutations are ide=
ntical.



Here is the correct formula and number of permutations:



209!/(((7!)^29)*(6!)) =3D



298225837482940676509184100728720811004423111510479240177876435252200583634=
046383785012187494718519005671580310160782720965515620763618140216746589818=
598244298634557645722755900360127929767933245249492609612953156836861423588=
4567287678429277548260303941234016334643200000000000000000000



My apologies for not catching this sooner; I've been very busy today with e=
mails. I am virtually certain that this count is correct, as I just sent R=
ay a complete derivation of it (which is how I discovered my error).



Thanks to everyone for your patience, and I'll keep you informed of my prog=
ress.



All the best,

David





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



=20




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

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


=20



=20=20




--0-1352969901-1304268603=:24576
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">Hi everyone,

There was a third, major =
mistake.  It seems like I need to take a break, at least from posting =
erroneous results.  This is very frustrating, and certainly not as eas=
y as I thought it was.  I didn't realize this more subtle error (than =
the other two) until I was taking a look at the order-3 puzzle.  Half =
an hour is hardly sufficient, especially when I am new to calculating permu=
tations of this type of puzzle, to have any confidence in my answers!
r>My sincerest apologies, and perhaps I'll collaborate with Roice, if he is=
able, willing, and has the time, to make sure my counts are accurate befor=
e posting any more results.

I'm sorry, and will stop flooding the me=
ssage board with results, until I am certain beyond any doubt they are corr=
ect.

All the best,
David

--- On Sun, 5/1/11, David Smit=
h
<djs314djs314@yahoo.com>
wrote:
order-left: 2px solid rgb(16, 16, 255); margin-left: 5px; padding-left: 5px=
;">
From: David Smith <djs314djs314@yahoo.com>
Subject: Re: [MC=
4D] Further correction of erroneous order-2 Klein's Quartic permutation cou=
nt
To: 4D_Cubing@yahoogroups.com
Date: Sunday, May 1, 2011, 11:42 AM<=
br>






 




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

>
Hi everyone,

Argh! =
; Iit turns out there was a second mistake. :(  This one is especially=
embarrassing.  I apologize for these corrections, and in the future w=
ill be much more careful before posting my results.  It's been way too=
long since I calculated permutations.  This stuff is about 100,000 ti=
mes simpler than my n^d Rubik's cube formula, which makes these corrections=
especially frustrating and embarrassing.  Hopefully I do not cast dou=
bt on the validity of my Rubik's cube formulas, which I checked for months,=
but I would understand if anyone questions them at this point.

I si=
mply miscounted the number of different colors in the puzzle! I was using a=
count of 30 colors, when in fact the correct number is 24.  The reall=
y funny thing is that MagicTile *tells you* how many colors are in each
puzzle, right next to
the puzzle name!  So you can see my embarrassment. :(  Here is t=
he (possibly?) correct number of permutations:

167!/(((7!)^23)*(6!))=
=3D

14255999370045115741835726463321297234734952725080156378526476=
35847
690582530108702510010741240927006490999290073130235405319596424682=
3
6192752417277237699906089211520063444550594267719476697558665102816>164056186968

If there are any more errors in this very simple calcu=
lation, I think I'm going to quit working on these!  Just kidding.&nbs=
p; Not to sound arrogant (in fact I intend to convey the opposite) but this=
stuff is child's play compared to my Rubik's cube formulas. (Sigh.)
>All the best,
David

--- On Sat, 4/30/11, djs314djs314 <=
djs314djs314@yahoo.com>
wrote:
ft: 2px solid rgb(16, 16, 255);">
From: djs314djs314 <djs314djs314@ya=
hoo.com>
Subject: [MC4D]
Correction of erroneous order-2 Klein's Quartic permutation count
To: 4=
D_Cubing@yahoogroups.com
Date: Saturday, April 30, 2011, 9:32 PM

=






 




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

Hello all,



Aha! I did make a mistake, as I suspected I might have. It's been too lon=
g since I counted permutations. :) My previous answer was off by a factor =
of 2; the 2 in the denominator should not have been there. I made the clas=
sic mistake of taking even permutations into account when there are identic=
ally colored pieces (in this case, stickers). Naturally, with identically =
colored pieces parity does not matter, as even and odd permutations are ide=
ntical.



Here is the correct formula and number of permutations:



209!/(((7!)^29)*(6!)) =3D



298225837482940676509184100728720811004423111510479240177876435252200583634=
046383785012187494718519005671580310160782720965515620763618140216746589818=
598244298634557645722755900360127929767933245249492609612953156836861423588=
4567287678429277548260303941234016334643200000000000000000000



My apologies for not catching this sooner; I've been very busy today with e=
mails. I am virtually certain that this count is correct, as I just sent R=
ay a complete derivation of it (which is how I discovered my error).



Thanks to everyone for your patience, and I'll keep you informed of my prog=
ress.



All the best,

David






=20=20=20=20=20



=20





=20=20=20=20=20



=20



--0-1352969901-1304268603=:24576--





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