[Retros] Who can build a set of 256 positions with the given property, YefimT 09/09/2004

[TregerYefim at aol.com]

    Hello from YefimT.
   Last time I gave 9 positions to check them on legality. 
rB2k1B1/1Bp5/ppn1p3/1B3p2/2p2p2/1P2P3/8/R3K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1P3/1B1prp2/2pP1P2/1P2N3/8/R3K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1p3/1B1pRp2/2pP1p2/1P2P3/8/R3K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1P3/1B1prp2/2pP1P2/1P2N3/2P5/Rb2K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1n3/1B1pRp2/2pP1P2/1P2P3/2P5/Rb2K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1n3/1B1pRp2/2pP1p2/1P2P3/2P5/RB2K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1P3/1B1prp2/2pP1P2/PP2N3/2P5/Rbb1K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1n3/1B1pRp2/2pP1P2/PP2P3/2P5/Rbb1K1b1 w Qq - 0 1
rB2k1B1/1Bp5/ppn1n3/1B1pRp2/2Pp1p2/PP2P3/2P5/RBb1K1b1 w Qq - 0 1

   Thank you (Ryan M.) for the answer, I am glad that they are OK. 
 Now I want to explain an idea.
  In each position both sides have only those moving pieces (no pawn moves, 
captures, and "castle" moves), which have exactly two possibilities. 
  In the first one White can move his 4 Bishops, each of them
having two possibilities (Bg8, for example, may move only g8-h7 and cannot 
capture and check because it breaks castle right or "integrity" of all set).
  There is the simpler illustration (remove all knights and WQueen, all 
castle rights lost):
r1bqkb1r/pppppppp/8/8/8/8/PPPPPPPP/R1B1KB1R w  - 0 1
In this position we have two rooks from each side and WKing, each of them can 
have only one move in all positions of a type (of course, when there is a 
required turn's move). Clearly, there are 32 positions (32=2^5).
  In the first position of my list also 32 positions. Only difference is that 
a composition of pieces is "4+1" (White - 4 pieces; Black - 1 piece).
   I want (for the book) to illustrate notions of binary system, graphs and 
hypercubes.
   If we add some previous and intermediate positions (which I did not give 
because they are obvious in sense of legality), for example:
2bqkb1r/pppppppp/8/8/8/8/PPPPPPPP/2BQKB1R w  - 0 1
 Here are 4 positions; 4=2^2; ...
r1b1kb1r/pppppppp/8/8/8/8/PPPPPPPP/R1B1KB1R w  - 0 1
 Here are 64 positions: 64=2^6. 
  We got all degrees of two: 4;8;16;32;64;128;256... 
   Note, if the main condition is observed then amounts of positions are the 
same upon different schemes of compositions (you may believe that graphs will 
be almost the same, only orientation of edges changes).

 Now, the task for you (in my opinion, many of you are good composers and, 
especially, are good retroexperts). 
I could not build legal positions (types) with more than 8 pieces in total; 
and for 8 pieces I could not build the scheme of "7+1" (in the last position 
White has 6 pieces and Black has 2 pieces).
  Can you build some of them? 
  If you still have questions about this task, please feel free asking me. 
  Thank you in advance. Yefim T. 09/09/2004
 
   
 

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