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 > > > 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. > > >>> > > >>> > > >>> > > >>> > > > > > > > > > > > > > > > >