Author: morphy
Date: 08:08:32 08/27/04
Go up one level in this thread
On August 27, 2004 at 10:53:51, Reinhard Scharnagl wrote:
>On August 27, 2004 at 10:45:51, Álvaro Begué wrote:
>
>>On August 27, 2004 at 10:40:30, morphy wrote:
>>
>>>I extract this from a my actual work:
>>>----------------
>>>[...]
>>>Now we can calculate all possible dispositions of white and black pawns on the
>>>chessboard:
>>>> combAllP(48);
>>> 49095494654933238
>
>>I only get 29019905518636890. Am I doing something wrong?
>
>You have done only 48!/(32!*8!*8!) thus (8+8) combinations.
>
>Reinhard.
I'm sure, I calcultate all (nPW+nPB) combinations. I like your work in your site
but your coclusions seems to me forced and artificial,unnatural. I'm sorry but
i'm working on a most mathematically strict and complete work...but I can't show
it also because it's incomplete. for the present question read all the
motivation (extracted from a 'Maple 9.5' work)
------------------
How to calculate all possible dispositions of all pieces on the chessboard?
Start to calculate all possible dispositions of white pawns on theirs 48
possible squares.
> combPW:= (S,nPW) -> combinat[numbcomb](S,nPW);
combPW := (S, nPW) -> combinat[numbcomb](S, nPW)
Where 'combPW' count the number of combinations of 'nPW' white pawns in 'S'
squares. The reason is obvious from the same definition of number of
combinations of 'nPW' elements on 'S' places. In chess S=48 and nPW is a number
of white pawns from 1 to 8.
To add all possible disposition of black pawns, add for each of such combination
all the possible disposition of black pawns on a number of squares decreased by
number of white pawns on the chessboard.
> combPWB:= (S,nPW,nPB) -> combPW(S,nPW)*combinat[numbcomb](S-nPW,nPB);
combPWB := (S, nPW, nPB) -> combPW(S, nPW) combinat[numbcomb](S - nPW, nPB)
Since combinat[numbcomb](n,m) = n!/m!/(n-m)! and, because the way to calculate
the combinations is the same, combinat[numbcomb](S-nPW,nPB) = combPW(S-nPW,nPB),
we have:
> combPWB:= (S,nPW,nPB) -> S!/nPW!/nPB!/(S-nPW-nPB)!;
combPWB := (S, nPW, nPB) ->
factorial(S)
------------------------------------------------------
factorial(nPW) factorial(nPB) factorial(S - nPW - nPB)
So to calcultate all possible dispositions of white and black pawns we add all
terms above-written for all nPW and all nPB, both from 1 to 8.
> combAllP:= S -> sum(sum(combPWB(S,nPW,nPB),nPW=1..8),nPB=1..8);
8 / 8 \
----- |----- |
\ | \ |
) | ) |
combAllP := S -> / | / combPWB(S, nPW, nPB)|
----- |----- |
nPB = 1\nPW = 1 /
> combAllP:= S -> sum(sum(S!/nPW!/nPB!/(S-nPW-nPB)!,nPW=1..8),nPB=1..8);
8 / 8
----- |-----
\ | \
) | )
combAllP := S -> / | /
----- |-----
nPB = 1\nPW = 1
\
|
|
factorial(S) |
------------------------------------------------------|
factorial(nPW) factorial(nPB) factorial(S - nPW - nPB)|
/
Now we can calculate all possible dispositions of white and black pawns on the
chessboard:
> combAllP(48);
49095494654933238
> evalf(%);
16
4.909549465 10
This is a significant result because we know how many possible 'pawns
structures' we can have: above to fifty millions of billions of pawns structure.
Assuming 1KB of data for 'Matrix of Values' for each pawn structure, we need to
store all this 'Matrix of Values' for all pawn structures about a million of
Hard Disks with a capacity of 50TB, not impossible for actual century.
-------------
I hope to see you soon, Mr. Scharnagl
Regards
Giulio
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