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From: Guy Del Mistro <gdm@msi.com>
Subject: Re: How many bits are needed ; stronger attempt
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Bert Dobbelaere wrote:
> 2) The other bits :  (153)
>    Facts :
>      -The number of remaining pieces varies between 0 and 30
>      -Once a pawn has promoted, the number of non-king pieces is limited
>       to 29.
>      -The number of pieces that are not king and not pawn is limited to
>       26 (try this out!)
>      -A pawn can only occupy 48 positions.
>      -Two pieces can not share the same position (trivial)
>    To calculate the number of possibilities there are for the remaining
>    pieces, we need some arithmetic :
> 
>             Def :    C(n,m) =  n! / (m! (n-m)!)
> 
>    We make the sum of the possibilities for a given number of pawns n,
>    with n between 0 and 16.
> 
>    The pawns can occupy 48 positions, so the number of their positions
>    is  C(48,n). They can be white or black, so the total number of pawn
>    possibilities with n pawns equals
>          C(48,n) * 2^n.
>    The other pieces can be encoded with three bits each, once you know
>    their position : they can be white or black ; queen, knight,
>    bishop or rook.
> 
>    The number of positions a single piece can occupy is now 62-n.
>    The number of (non king/non pawn) pieces can vary between 0 and K.
>       K=max(26,29-n) when n<16 or K=14 when n=16.
> 
>    So the number of possibilities they have left is
>          Sum[j= 0..K] {  C(62-n,j) * 8^j  }
> 
>    We have now for a given number n of pawns a total number of
>    possibilities
>        C(48,n) * 2^n * [ Sum[j= 0..K] {  C(62-n,j) * 8^j } ]
> 
>    Taking the sum of this result for all numbers of pawns (0..16),
>      ( I did it in Excel)
>    we find a number of
>             9,749e45 combinations.
>    each of which can be stored in 153 bits.
> 

Sorry but this is BS for reasons I've given in other postings. Just
try it. Encode a board (chosen at random) this way then try to decode
it. As I said before you CANNOT mix bases this way, ie. saying
48*47*46... = X = 2^y = y bits because this has no decryption without
specifying each component with respect to a proper base.

Guy
