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

TregerYefim at aol.com TregerYefim at aol.com
Sat Oct 9 23:42:30 EDT 2004

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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