[Retros] More than 100 positions leading to the foolish mate. Yefim 10/09/2004

[TregerYefim at aol.com]

   Hello, from Yefim.
   My position with only one e.p. move is:
6b1/ppp1p2p/3pPk1P/3PpPpK/6P1/8/8/8 w - - 0 1
  Interesting observations: mathematically a requirement about last Black 
move to be a pawn move is not necessary as long we have the main condition: only 
one move is e.p. The given position in a graph has only one inward edge and 
one outward edge. Further, this e.p. move does not lead to final position as in 
two previous examples. Let me popularly explain a difference between final 
positions and not final ones.
  The key idea is a local degree for a position in a graph what is a sum of 
all indegree positions (leading to the given one) and outdegree (leading from 
the given one) positions.
  Examples of two other wonderful moves. If you have castle (in this moment) 
you have minimum 4 outward edges (4 moves). If you have a promotion move the 
minimal local outdegree for the position is also 4.
   It is a pity that computer programs do not recognize some simple 
properties for 
positions. But there is a deep reason for that: e.p. properties, castle 
properties (if castle is possible only in future) and some others are mostly game 
properties.
  The funny example: a foolish mate: 1.f3 e6 2.g4 Qh4#  Computer (WinBoard) 
gives final position as: 
   rnb1kbnr/pppp1ppp/4p3/8/6Pq/5P2/PPPPP2P/RNBQKBNR w KQkq - 0 3
  with all possible castles!?
  For all of us: No final position contains any castle rights! Local 
Outdegree for them is 0. 
  Now, my question about last mate position: what is local degree for it? Or: 
how many positions lead to this final one? (final position does not need any 
number in a game, mate or stalemate is always superior). My answer is 112. 
  Why: wait for the next teaching lesson...
  Yefim (still writing a book about Math and Chess) 10/09/2004.
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