[seqfan] Re: the revolver sequence
Tanya Khovanova
mathoflove-seqfan at yahoo.com
Fri Sep 25 03:41:10 CEST 2009
Sorry for the typo. Yes, of course, I meant that it is better to continue not to spin in the original revolver problem.
--- On Thu, 9/24/09, Andrew Weimholt <andrew at weimholt.com> wrote:
> From: Andrew Weimholt <andrew at weimholt.com>
> Subject: Re: [seqfan] Re: the revolver sequence
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Cc: mathoflove-seqfan at yahoo.com
> Date: Thursday, September 24, 2009, 2:04 PM
> Tanya,
>
> For the case of bullets in 2 adjacent chambers, it should
> be better
> NOT to spin after the first shot. If you survive the first
> shot, then
> there is a 1/4 chance that the next chamber is loaded,
> which is less
> than the 1/3 chance of fatality after an additional spin.
>
> It seems some people want to claim that since one empty
> chamber is
> taken out of the picture after the first shot, that the
> odds of the
> next shot being fatal are 2/5 instead of 1/4, but this does
> not make
> any sense. If we label the bullets A and B, and we put B
> immediately
> after A in the revolver, then if the first shot is an empty
> chamber,
> it is impossible for the next shot to be bullet B. Bullet B
> must
> therefore also be taken out of the picture.
>
> If the 2 bullets are NOT adjacent, then 2/5 is the correct
> value for
> the odds of the second shot (without a spin) being fatal.
> In this
> case, it is better to spin.
>
> If the 2 bullets are placed randomly, and we do not know
> whether or
> not they are adjacent, then the odds are 7/20 that the
> second shot
> (without spnning) will be fatal, and in this case it is
> also better to
> spin.
>
> Andrew
>
>
>
>
>
> On 9/24/09, Tanya Khovanova <mathoflove-seqfan at yahoo.com>
> wrote:
> > Thank you Max,
> >
> > As always great coding!
> >
> > My definition of this sequence should be
> clarified:
> > 1. The shooter spins before the first shot.
> > 2. I do not count the case when the revolver is
> completely loaded.
> >
> > My goal was to reproduce the case in the famous
> problem with 6 chambers and 2 adjacent bullets, as it is for
> many surprising that it is better to spin after the first
> shot.
> >
> > Tanya
> >
> >
> > --- On Wed, 9/23/09, Max Alekseyev <maxale at gmail.com>
> wrote:
> >
> > > From: Max Alekseyev <maxale at gmail.com>
> > > Subject: [seqfan] Re: the revolver
> sequence
> > > To: "Sequence Fanatics Discussion list"
> <seqfan at list.seqfan.eu>
> >
> > > Date: Wednesday, September 23, 2009, 11:12 PM
> >
> > > Thanks for explicit example - I've
> > > found a typo in my code.
> > >
> > > Here is a formula and first 50 terms of
> your sequence:
> > >
> > > { a(n) =
> sum(z=0,n,sum(r=1,min(ceil(z-z^2/n)-1,n-z),
> > > sumdiv(gcd([n,z,r]),d, eulerphi(d) *
> binomial(z/d - 1, r/d
> > > - 1) *
> > > binomial((n-z)/d - 1, r/d - 1) )/r )) }
> > >
> > > ? vector(50,n,a(n))
> > > %2 = [0, 0, 0, 0, 2, 3, 4, 5, 18, 43, 60,
> 84, 294, 472,
> > > 724, 1077,
> > > 3504, 5807, 10396, 15944, 43664, 84308,
> 137004, 217728,
> > > 582966,
> > > 1134304, 1883360, 3225812, 8964134,
> 15461472, 27796942,
> > > 45814765,
> > > 123307634, 233218013, 396304692, 663041846,
> 1754000726,
> > > 3300299482,
> > > 5679338110, 9942238090, 24989323012,
> 47334027696,
> > > 85892508384,
> > > 146496196464, 359597562016, 690754558978,
> 1256104705128,
> > > 2163505451106, 5289200995480,
> 10102216102758]
> > >
> > > Regards,
> > > Max
> > >
> > > On Wed, Sep 23, 2009 at 10:25 PM, Tanya
> Khovanova
> > > <mathoflove-seqfan at yahoo.com>
> > > wrote:
> > > > This is my list for n=9:
> > > >
> > > > {{0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0,
> 0, 0, 0, 0, 1, 1,
> > > 1}, {0, 0, 0,
> > > > 0, 0, 1, 1, 1, 1}, {0, 0, 0, 0,
> 1, 0, 1, 1, 1},
> > > {0, 0, 0, 0, 1, 1,
> > > > 0, 1, 1}, {0, 0, 0, 0, 1, 1, 1,
> 0, 1}, {0, 0, 0,
> > > 1, 0, 0, 1, 1,
> > > > 1}, {0, 0, 0, 1, 1, 0, 0, 1, 1},
> {0, 0, 0, 1, 1,
> > > 1, 0, 0, 1}, {0, 0,
> > > > 0, 0, 1, 1, 1, 1, 1},
> {0, 0, 0, 1, 0,
> > > 1, 1, 1, 1}, {0, 0, 0, 1, 1,
> > > > 0, 1, 1, 1}, {0, 0, 0, 1, 1, 1,
> 0, 1, 1}, {0, 0,
> > > 0, 1, 1, 1, 1, 0,
> > > > 1}, {0, 0, 1, 0, 0, 1, 1, 1, 1},
> {0, 0, 1, 1, 0,
> > > 0, 1, 1, 1}, {0, 0,
> > > > 0, 1, 1, 1, 1, 1, 1},
> {0, 0, 1, 1, 1,
> > > 1, 1, 1, 1}}
> > > >
> > > > --- On Wed, 9/23/09, Max Alekseyev
> <maxale at gmail.com>
> > > wrote:
> > > >
> > > >> From: Max Alekseyev <maxale at gmail.com>
> > > >> Subject: [seqfan] Re: the revolver
> sequence
> > > >> To: "Sequence Fanatics Discussion
> list" <seqfan at list.seqfan.eu>
> > > >> Date: Wednesday, September 23,
> 2009, 6:58 PM
> > > >> Hi Tanya,
> > > >>
> > > >> I've got different values starting
> with a(9) = 13,
> > > a(10) =
> > > >> 32 etc.
> > > >> Could you please double check your
> values or lists
> > > all 18
> > > >> ways of
> > > >> loading a revolver with n=9
> chambers?
> > > >>
> > > >> Thanks,
> > > >> Max
> > > >>
> > > >> On Wed, Sep 23, 2009 at 5:06 PM,
> Tanya Khovanova
> > > >> <mathoflove-seqfan at yahoo.com>
> > > >> wrote:
> > > >> > Hello all,
> > > >> >
> > > >> > I just submitted a sequence
> related to the
> > > revolver
> > > >> puzzle:
> > > >> > http://blog.tanyakhovanova.com/?p=174
> > > >> >
> > > >> > I generalized it to revolvers
> having n
> > > chambers.
> > > >> >
> > > >> > I think there should be more
> sequences: for
> > > example,
> > > >> the one when it is preferable to
> spin and the one
> > > when the
> > > >> chances are the same.
> > > >> >
> > > >> > Also, my programs are usually
> not very
> > > efficient. It
> > > >> will be nice if someone can add
> more terms to
> > > this
> > > >> sequence.
> > > >> >
> > > >> > %I A165646
> > > >> > %S A165646
> > > 0,0,0,0,2,3,4,5,18,43,60,84,294,472,724
> > > >> > %N A165646 The revolver
> sequence.
> > > >> > %C A165646 a(n) is in how
> many ways you can
> > > load a
> > > >> revolver with n chambers, so that
> if a person
> > > survives after
> > > >> the first shot, he/she has better
> chances of
> > > survival for
> > > >> the second shot if the shooter
> continues rather
> > > than spins.
> > > >> a(n) <= A000031(n) as
> A000031(n) counts the
> > > number of
> > > >> possible revolver loadings.
> > > >> > %H A165646 Tanya Khovanova,
> <a href="http://blog.tanyakhovanova.com/?p=174">Heard on the
> > > >> Street</a>
> > > >> > %e A165646 Based on the
> famous interview
> > > question
> > > >> where the revolver has six
> chambers and the
> > > shooter loads
> > > >> two adjacent bullets. As the
> answer to this
> > > question is to
> > > >> continue, a(6) must be at least
> 1.
> > > >> > %t A165646 <<
> Combinatorica` colors[x_]
> > > :=
> > > >> Table[PadLeft[Table[1, {n, i}],
> x], {i, 0, x}]
> > > >> continueBetter[list_] := (len =
> Length[list]; c0 =
> > > 0; c00 =
> > > >> 0; rotateN = 0; While[rotateN <
> len, newList =
> > > >> RotateLeft[list, rotateN];
> If[newList[[1]] == 0,
> > > c0++;
> > > >> If[newList[[2]] == 0, c00++]];
> rotateN++]; If[c0
> > > > 0,
> > > >> c00/c0 > c0/len, False])
> continueNum[num_] :=
> > > (neckNum =
> > > >> Length[colors[num]]; ans = 0;
> count = 1;
> > > While[count <=
> > > >> neckNum, ans = ans +
> > > Length[Select[ListNecklaces[num,
> > > >> colors[num][[count]], Cyclic],
> continueBetter[#]
> > > &]];
> > > >> count++]; ans)
> Table[continueNum[n], {n, 2, 15}]
> > > >> > %Y A165646 A000031
> > > >> > %K A165646 more,nonn
> > > >> > %O A165646 1,5
> > > >> > %A A165646 Tanya Khovanova
> > > (tanyakh(AT)yahoo.com), Sep
> > > >> 23 2009
> > > >> >
> > > >> >
> > > >> >
> > > >> >
> > >
> _______________________________________________
> > > >> >
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> > > >> >
> > > >>
> > > >>
> > > >>
> _______________________________________________
> > > >>
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> > > >>
> > > >
> > > >
> > > >
> _______________________________________________
> > > >
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> > >
> > >
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