cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A278830 Maximal number of possible moves at the n-th ply of a chess game.

Original entry on oeis.org

20, 20, 31, 32, 46, 48, 52, 55, 61, 63
Offset: 1

Views

Author

M. F. Hasler, Nov 29 2016

Keywords

Comments

In view of the 75-moves rule any chess game is finite.
The largest value appearing in the sequence is thought to be 218. - François Labelle, Dec 01 2016

Examples

			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.
At its second move, i.e., ply 3 of the game, White may have as much as 31 possible moves, if White started with e2-e4 as first move, and Black offered a pawn with d7-d5 or f7-f5.

Links

Crossrefs

Cf. A278831: minimal number of possible moves at the n-th ply.
Cf. A278832: maximal material difference at the end of the n-th ply.
Cf. A083276.

Extensions

a(3)-a(4) corrected and a(5)-a(10) from François Labelle, Nov 29 2016

A278831 Minimal number of possible moves at the n-th ply of a chess game, excluding positions where no move is possible.

Original entry on oeis.org

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

M. F. Hasler, Nov 29 2016

Keywords

Comments

Given the 75-moves rule, any chess game, and thus this sequence, is finite.
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, 5-fold repetition, or the 75-move rule.
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]

Examples

			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.
At its second move, i.e., ply 3 of the game, White may have only 19 possible moves, if he started with either a2-a3 or f2-f3 or h2-h3 as first move.
If the first three half-moves are 1. e3, f6; 2. Qh5+, then Black has only one possible move, 2. ... g6, whence a(4) = 1.
Similarly, a(5) = 1 corresponds to the only possible move of White in the symmetric position (apart from one additional half-move made earlier by White).
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.
From a given number of half-moves 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.

Links

Crossrefs

Cf. A278830 (maximal number of possible moves at ply n), A278832 (maximal material difference at ply n).

A380577 a(n) is the number of distinct compositions of chess pieces with a collective material value of n that one color in a game can have, where 0 <= n <= 103.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 7, 7, 9, 13, 14, 17, 22, 24, 28, 35, 38, 41, 52, 54, 59, 72, 73, 79, 95, 95, 101, 117, 120, 122, 144, 139, 146, 166, 159, 165, 186, 174, 184, 195, 189, 199, 204, 197, 201, 208, 204, 194, 206, 194, 193, 195, 182, 182, 178, 177, 159, 177, 142, 154, 137, 145, 122, 135, 103, 121, 96, 104, 85, 96, 71, 77, 63, 73, 52, 60, 45, 48, 40, 41, 31, 39, 23, 26, 23, 22, 18, 18, 11, 15, 8, 10, 9, 6, 5, 4, 2, 5, 1, 1, 2, 0, 0, 1
Offset: 0

Views

Author

Felix Huber, Mar 30 2025

Keywords

Comments

The following (standard) values of the chess pieces are used here: pawn (P): 1, bishop (B): 3, knight (K): 3, rook (R): 5, queen (Q): 9. The King is always considered to be included and has the value 0.
The game begins with the piece numbers P = 8, B = 2, K = 2, R = 2 and Q = 1 and with the collective material value of 1*8 + 3*2 + 3*2 + 5*2 + 9*1 = 39. Pieces can be lost and pawns can be converted into one of the other four piece types. Within these rules (see also ranges and inequalities in the Maple program), a(n) is the number of nonnegative integer solutions to 1*P + 3*B + 3*K + 5*R + 9*Q = n.
The smallest collective material value of a chess piece composition is 0 (king alone), the largest 1*0 + 3*2 + 3*2 + 5*2 + 9*9 = 103 (all pawns converted into queens, no piece lost). Therefore, the definition range for n is restricted to 0 <= n <= 103 and this sequence is finite by definition.
There are a total of Sum_{n, n=0..103} a(n) = 8694 distinct compositions of chess pieces that one color in a game can have.

Examples

			a(5) = 4 because exactly 4 possible chess piece compositions (P, B, K, R, Q) satisfy 1*P + 3*B + 3*K + 5*R + 9*Q = 5: (0, 0, 0, 1, 0), (2, 0, 1, 0, 0), (2, 1, 0, 0, 0), (5, 0, 0, 0, 0).
a(97) = 5 because exactly 5 possible chess piece compositions (P, B, K, R, Q) satisfy 1*P + 3*B + 3*K + 5*R + 9*Q = 97: (0, 0, 2, 2, 9), (0, 1, 1, 2, 9), (0, 2, 0, 1, 9), (0, 2, 3, 2, 8), (0, 3, 2, 2, 8).

Links

Crossrefs

Cf. A278832, A378248.

Programs

  • Maple
    A380577:=proc(n)
    local P,B,K,R,Q,a;
    a:=0;
    for P from 0 to 8 do
        for B from 0 to 10 do
          for K from 0 to 10 do
	          for R from 0 to 10 do
	            for Q from 0 to 9 do
	              if P+3*B+3*K+5*R+9*Q=n and P+B<=10 and P+K<=10 and P+R<=10 and P+Q<=9 and P+B+K<=12 and P+B+R<=12 and P+B+Q<=11 and P+K+R<=12 and P+K+Q<=11 and P+R+Q<=11 and P+B+K+R<=14 and P+B+K+Q<=13 and P+B+R+Q<=13 and P+K+R+Q<=13 and P+B+K+R+Q<=15 then
	                 a:=a+1
	              fi
	            od
	          od
	        od
       od
    od;
    return a
end proc;
seq(A380577(n),n=0..103);

Formula

a(n) <= A378248(n).
Showing 1-3 of 3 results.

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