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A couple problems I've noted for FTM:
When N = 3: your lower bound gives 20.25 when we know it's actually 20.
When N = 3: your possible twists is 24 when I count 18.
When N = 4: your possible twists is 192 when I count 184.
I'm wondering if your FTM lower bound equation is a bit high.mainly because
it beat my lower bound :-)
3^4: your FTM lower bound is 60.75, mine is 56.
--
Andy
From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf
Of Andrey
Sent: Wednesday, July 06, 2011 11:27
To: 4D_Cubing@yahoogroups.com
Subject: [MC4D] God's number for 3^N
My estimates show that lower counting limit L for God's number for 3^N is
2/9*N*3^N for QFTM (as implemented in MC5D and MC7D - with 2*N*(N-1)*(N-2)
possible twists) and 3/4*3^N for FTM (where any twist of face is counted as
1, so we have N!*2^(N-1) possible twists). Actual God's number is probably
between L and 2*L.
By the way, if we take puzzle 2*1^N (with only one twisting face), its God's
number in QFTM is N. But counting limit gives something like
N*(log(2*N)/(2*log(N)) that is N/2*(1+o(N)). So lower limit is almost the
half of the actual number.
Andrey
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osoft-com:office:office" xmlns:w=3D"urn:schemas-microsoft-com:office:word" =
xmlns:m=3D"http://schemas.microsoft.com/office/2004/12/omml" xmlns=3D"http:=
//www.w3.org/TR/REC-html40">
oNormal>color:#1F497D'>A couple problems I've noted for FTM:
<=
span style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F=
497D'>When N =3D 3: your possible twists is 24 when I count 18.
/o:p>
amily:"Calibri","sans-serif";color:#1F497D'>When N =3D 4: your possib=
le twists is 192 when I count 184.
l>#1F497D'>
nt-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>I'm wonder=
ing if your FTM lower bound equation is a bit high…mainly because it =
beat my lower bound :-)
mal>r:#1F497D'>3^4: your FTM lower bound is 60.75, mine is 56.
:"Calibri","sans-serif";color:#1F497D'>
=3DMsoNormal>rif";color:#1F497D'>--
e=3D'font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>And=
y
;font-family:"Calibri","sans-serif";color:#1F497D'>
ri","sans-serif";color:#1F497D'>
=
12.0pt'>My estimates show that lower counting limit L for God's number for =
3^N is 2/9*N*3^N for QFTM (as implemented in MC5D and MC7D - with 2*N*(N-1)=
*(N-2) possible twists) and 3/4*3^N for FTM (where any twist of face is cou=
nted as 1, so we have N!*2^(N-1) possible twists). Actual God's number is p=
robably between L and 2*L.
By the way, if we take puzzle 2*1^N (with onl=
y one twisting face), its God's number in QFTM is N. But counting limit giv=
es something like
N*(log(2*N)/(2*log(N)) that is N/2*(1+o(N)). So lower =
limit is almost the half of the actual number.
Andrey
Andrew,
You are right - I forgot to mention a couple of things:
- formula works only for N>=3D5 (where we haven't additional invariants for=
corners orientations)
- and even in that cases it's only asymptotic: (2/9+o(1))*N*3^N and (3/4+o(=
1))*3^N. I'm not sure in constant 3/4: for N=3D170 computations give someth=
ing like 0.739.. and this coefficient slowly decreases). Maximal coefficien=
t for QFTM is 0.222674992 for N=3D71, and for FTM - 0.7642 for N=3D17.
Andrey
--- In 4D_Cubing@yahoogroups.com, "Andrew Gould"
>
> A couple problems I've noted for FTM:
>=20
> When N =3D 3: your lower bound gives 20.25 when we know it's actually 20=
.
>=20
> When N =3D 3: your possible twists is 24 when I count 18.
>=20
> When N =3D 4: your possible twists is 192 when I count 184.
>=20
>=20=20
>=20
> I'm wondering if your FTM lower bound equation is a bit high.mainly becau=
se
> it beat my lower bound :-)=20=20
>=20
> 3^4: your FTM lower bound is 60.75, mine is 56.
>=20
>=20=20
>=20
> --
>=20
> Andy
>=20
>=20=20
>=20
>=20=20
>=20
> From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Beh=
alf
> Of Andrey
> Sent: Wednesday, July 06, 2011 11:27
> To: 4D_Cubing@yahoogroups.com
> Subject: [MC4D] God's number for 3^N
>=20
>=20=20
>=20
>=20=20=20
>=20
> My estimates show that lower counting limit L for God's number for 3^N is
> 2/9*N*3^N for QFTM (as implemented in MC5D and MC7D - with 2*N*(N-1)*(N-2=
)
> possible twists) and 3/4*3^N for FTM (where any twist of face is counted =
as
> 1, so we have N!*2^(N-1) possible twists). Actual God's number is probabl=
y
> between L and 2*L.
> By the way, if we take puzzle 2*1^N (with only one twisting face), its Go=
d's
> number in QFTM is N. But counting limit gives something like
> N*(log(2*N)/(2*log(N)) that is N/2*(1+o(N)). So lower limit is almost the
> half of the actual number.
>=20
> Andrey
>
I just understood that we want not maximal, but minimal values of coefficie=
nts to use the formula. They are for N=3D5: 0.70295 for FTM and 0.22201 for=
QFTM.
Andrey
--- In 4D_Cubing@yahoogroups.com, "Andrey"
>
> Andrew,
> You are right - I forgot to mention a couple of things:
> - formula works only for N>=3D5 (where we haven't additional invariants f=
or corners orientations)
> - and even in that cases it's only asymptotic: (2/9+o(1))*N*3^N and (3/4+=
o(1))*3^N. I'm not sure in constant 3/4: for N=3D170 computations give some=
thing like 0.739.. and this coefficient slowly decreases. Maximal coefficie=
nt for QFTM is 0.222674992 for N=3D71, and for FTM - 0.7642 for N=3D17.
>=20
> Andrey
>=20
>=20
> --- In 4D_Cubing@yahoogroups.com, "Andrew Gould"
> >
> > A couple problems I've noted for FTM:
> >=20
> > When N =3D 3: your lower bound gives 20.25 when we know it's actually =
20.
> >=20
> > When N =3D 3: your possible twists is 24 when I count 18.
> >=20
> > When N =3D 4: your possible twists is 192 when I count 184.
> >=20
> >=20=20
> >=20
> > I'm wondering if your FTM lower bound equation is a bit high.mainly bec=
ause
> > it beat my lower bound :-)=20=20
> >=20
> > 3^4: your FTM lower bound is 60.75, mine is 56.
> >=20
> >=20=20
> >=20
> > --
> >=20
> > Andy
> >=20
> >=20=20
> >=20
> >=20=20
> >=20
> > From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On B=
ehalf
> > Of Andrey
> > Sent: Wednesday, July 06, 2011 11:27
> > To: 4D_Cubing@yahoogroups.com
> > Subject: [MC4D] God's number for 3^N
> >=20
> >=20=20
> >=20
> >=20=20=20
> >=20
> > My estimates show that lower counting limit L for God's number for 3^N =
is
> > 2/9*N*3^N for QFTM (as implemented in MC5D and MC7D - with 2*N*(N-1)*(N=
-2)
> > possible twists) and 3/4*3^N for FTM (where any twist of face is counte=
d as
> > 1, so we have N!*2^(N-1) possible twists). Actual God's number is proba=
bly
> > between L and 2*L.
> > By the way, if we take puzzle 2*1^N (with only one twisting face), its =
God's
> > number in QFTM is N. But counting limit gives something like
> > N*(log(2*N)/(2*log(N)) that is N/2*(1+o(N)). So lower limit is almost t=
he
> > half of the actual number.
> >=20
> > Andrey
> >
>