[Retros] Little less than a million positions leading to the first one!? Yefim 10/09/2004

[TregerYefim at aol.com]

  Hello, from Yefim.
  There are 112 positions leading to the final one (after 1.f3 e6 2.g4 Qh4#): 

rnb1kbnr/pppp1ppp/4p3/8/6Pq/5P2/PPPPP2P/RNBQKBNR w 
because Black Queen may be on d8; e7; f6; g5, h3; h5; h6 and each particular
configuration may have 16 different combinations of castle what constitutes a 
Position (7*16=112). It does not mean that the move Qh4 leads to 16 different 
positions. Final position is always the unique one, it cannot be repeated. 
  In general, there are positions which can be repeated and positions that 
cannot.
  Only the first ones may be repeated 3 times and give a reason to claim a 
draw. Only such positions forms circles (or contours) in a graph, when one can 
pass from the given position to itself. An example of position which cannot be 
repeated (besides final one) is a position with e.p. move. 
   This is my second simple math lesson. To make teaching more enjoyable I 
will ask you some interesting question and give another interesting 
information/problem.
  My next question concerns the very first position (lets call it the 
Original Position).
  Here it is:
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq
   What is local degree for it? It is simple so I answer :).
  Indegree=4 (Black moves: Na6-b8; Nc6-b8; Nf6-g8; Nh6-g8) Outdegree=20 (20 
moves: 16 by Pawns and 4 by Knights). Total (local degree) =4+20=24.
    But the next question and answer amazes many of us: 
  From how many positions can we pass (not necessarily directly) to the 
Original one?
  My book gives an exact answer; but I want you give an approximate estimate.
  Hint: it is more than 1000. Variants for you: A: 1000-10000; 
B:10000-100000; C: more than 100000.
     Yefim.
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