%I #21 Mar 15 2018 18:25:02
%S 20,20,19,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Minimal number of possible moves at the nth ply of a chess game, excluding positions where no move is possible.
%C Given the 75moves rule, any chess game, and thus this sequence, is finite.
%C The definition of this sequence excludes positions with no possible move, such as checkmate and stalemate positions, and other cases which would end the game, e.g., via draw by impossibility of checkmate, 5fold repetition, or the 75move rule.
%C Is there any further term different from 1? The first terms a(4), a(5), ... equal to 1 correspond to a specific configuration which can appear at ply 4 but as well one or a considerable number of moves later, see the Example section for details. After that, it is quite probable that other similar positions can be constructed in which again a(n) = 1. Towards the end of the longest possible game(s), one may expect very little material around, probably only the two kings plus one other material to avoid draw by impossibility of checkmate. It would require a deeper study of this context to prove or disprove that the penultimate position would always allow more than one move for the player(s). In any event, it seems quite out of reach to compute the exact index where this would occur. [Comment revised following comments by _François Labelle_ and _Rick L. Shepherd_, Nov 30 2016]
%H <a href="/index/Ch#chess">OEIS index to sequences related to chess</a>.
%e In the initial position of the chess game, each player has 20 possible moves (16 pawn moves and 4 knight moves), and the first (half)move made by White does not affect the 20 possibilities Black will have thereafter.
%e At its second move, i.e., ply 3 of the game, White may have only 19 possible moves, if he started with either a2a3 or f2f3 or h2h3 as first move.
%e If the first three halfmoves are 1. e3, f6; 2. Qh5+, then Black has only one possible move, 2. ... g6, whence a(4) = 1.
%e Similarly, a(5) = 1 corresponds to the only possible move of White in the symmetric position (apart from one additional halfmove made earlier by White).
%e A position with essentially the same configuration may occur one or more moves later, if the other earlier moves of both players do not change the relevant part of the configuration in a significant way. For example, if the game goes 2. a3 a6, before 3. Qh5+, or: 3. a3 a5, 4. Qh5+, or: 4. Ra2 Ra7, 5. Qh5+ etc. This leads to many subsequent terms a(6,7,8,9,...) = 1.
%e From a given number of halfmoves on, it will also be possible to reach other configurations in which either player has only one possible move for similar reasons, and these configurations can usually also be "delayed" by several moves. This extends further the number of consecutive 1's in this sequence.
%Y Cf. A278830 (maximal number of possible moves at ply n), A278832 (maximal material difference at ply n).
%K nonn,less,fini
%O 1,1
%A _M. F. Hasler_, Nov 29 2016
