Re: Re: Combinatorica questions!!!

• To: mathgroup at smc.vnet.net
• Subject: [mg20720] Re: [mg20656] Re: [mg20499] Combinatorica questions!!!
• From: "Arnold Knopfmacher" <arnoldk at cam.wits.ac.za>
• Date: Wed, 10 Nov 1999 00:17:38 -0500
• Organization: MS, University of the Witwatersrand
• Sender: owner-wri-mathgroup at wolfram.com

```The solutions to the rook on the 3D chessboard by Rob Pratt
and Andrzej Kozlowski assume that the rook only moves one unit in any direction at a time.
But this is not how a rook moves, as Joinery in fact stated in the problem:
> > > Example for allowable move:
> > > Say from {0,0,0} to {0,6,0} but not {0,6,6},
Therefore the generating function should be
1/(1-x/(1-x)-y/(1-y)-z/(1-z)) and the answer is the coefficient
of x^7y^7z^7 which Mathematica gives as 75059524392.

Arnold Knopfmacher and Helmut Prodinger
Witwatersrand University

Date sent:      	Sun, 7 Nov 1999 02:09:58 -0500
From:           	Rob Pratt <rpratt at email.unc.edu>
To: mathgroup at smc.vnet.net
Subject: [mg20720]        	[mg20656] Re: [mg20499] Combinatorica questions!!!

> Equivalently,
>
> Multinomial[7,7,7]
>
> gives 399072960.
>
> The generating function is (x + y + z)^21.  The coefficient of
> x^7 y^7 z^7 is 399072960.
>
> Rob Pratt
> Department of Operations Research
> The University of North Carolina at Chapel Hill
>
> rpratt at email.unc.edu
>
> http://www.unc.edu/~rpratt/
>
> On Thu, 4 Nov 1999, Andrzej Kozlowski wrote:
>
> > It seems clear that Rob Pratt's method applies to this also: the rook has
>  to
> > make 21 moves, 7 in the x direction, 7 in the y direction and 7 in the z
> > direction. So we need to choose seven out of the 21 moves to be in the x,
> > direction and for each of such choices we need to choose 7 out of the
> > remaining 14 as moves in the y direction. The rest will automatically be
> > moves in the z direction. So the answer is:
> >
> > In[6]:=
> > Binomial[21, 7]*Binomial[14, 7]
> > Out[6]=
> > 399072960
> >
> >
> >
> > --
> > Andrzej Kozlowski
> > Toyama International University
> > JAPAN
> > http://sigma.tuins.ac.jp
> >
> >
> > > From: kewjoi at hixnet.co.za (Kew Joinery)
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> > > Reply-To: kewjoi at hixnet.co.za
> > > Date: Thu, 04 Nov 1999 10:30:35 +0200
> > > To: "mathgroup at smc.vnet.net" <mathgroup at smc.vnet.net>
> > > Cc: Rob Pratt <rpratt at email.unc.edu>, Andrzej Kozlowski <andrzej at tuins.ac.jp>
> > > Subject: [mg20720] [mg20656] Re: [mg20499] Combinatorica questions!!!
> > >
> > > Hello,
> > > The case has been solved perfectly well. Can you do so for slightly
>
> > > different
> > > Same conditions, but for 3 dimensional 8x8x8 chessboard (cube). Imagine that
> > >
> > > the rook can move not only on the surface but inside the cube too.
> > > To make it clear I will denote the start position of the rook {0,0,0}. The
> > > target
> > > is final position {7,7,7} which is the farthest opposite point. The rook can
> > > move
> > > as usual (not diagonally), the only constraint is you can move the rook
>  in
> > > increasing order of each coordinate.
> > > Example for allowable move:
> > > Say from {0,0,0} to {0,6,0} but not {0,6,6},
> > > Say from {4,3,5} to {4,7,5} but not {4,1,5}.
> > > In other words: the change of only one coordinate at a time equals one
> move of
> > > the rook, and the change could be in increasing order of each coordinate!
> > > *** The task is how many different ways (walks) does a castle have to reach
> > > from
> > > position {0,0,0} to position {7,7,7}?
> > > (**Is there a general formula or generating function for higher dimension?  )
> > > (Note: some people could find the question not relevant to the group, but this
> > > is
> > > pure mathematics and this is just the beginning of the difficult questions and
> > > answers normally are available only to research people as usual, so everyone
> > > could learn something positive).
> > >
> > > Thank you.
> > > Eugene
> > >
> > > Rob Pratt wrote:
> > >
> > >> An approach to problem 1 that is simpler than those already given is to
> > >> recognize that each path consists of a sequence of 14 moves, 7 of them
>  to
> > >> the RIGHT one space and 7 of them UP one space.  Hence a path is uniquely
> > >> determined by specifying which 7 of the 14 moves are RIGHT (the rest are
> > >> UP).  We are choosing 7 objects from among 14 positions, so the answer is
> > >>
> > >> Binomial[14,7]=3432
> > >>
> > >> Rob Pratt
> > >> Department of Operations Research
> > >> The University of North Carolina at Chapel Hill
> > >>
> > >> rpratt at email.unc.edu
> > >>
> > >> http://www.unc.edu/~rpratt/
> > >>
> > >> On Wed, 27 Oct 1999, Keren Edwards wrote:
> > >>
> > >>> Hi all!!
> > >>>
> > >>> 2 different questions:
> > >>>
> > >>> 1.    how many ways does a castle have to reach from the bottom left
> side
> > >>> corner
> > >>> of a chess board to the upper right corner of the board if he can
> > >>> move right
> > >>> and up only?
> > >>>
> > >>>
> > >>>
> > >>> 2.     you have 8 red identical balls, 9 purple identical balls and 7 white
> > >>> identical ones.
> > >>> a.  How many ways can you choose 10 balls with no matter to the
> > >>> order of the balls?
> > >>> b.  How many ways can you choose 10 balls with no matter to the
> > >>> order of the balls, if each color must
> > >>> be chosen once at least?
> > >>>
> > >>>
> > >>>
> > >>> Many thanx.
> > >>>
> > >>>
> > >>>
> > >>>
> > >
> > >
> > >
> >
> >
>
>
>

```

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