Re: Combinatorica questions!!!
- To: mathgroup at smc.vnet.net
- Subject: [mg20645] Re: [mg20499] Combinatorica questions!!!
- From: kewjoi at hixnet.co.za (Kew Joinery)
- Date: Sun, 7 Nov 1999 02:09:52 -0500
- References: <7vogo7$n55$10@dragonfly.wolfram.com>
- Sender: owner-wri-mathgroup at wolfram.com
Hello,
The case has been solved perfectly well. Can you do so for slightly different
task:
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.
> >
> >
> >
> >