A278832 Maximal material difference at the end of the n-th ply of a chess game.

0, 0, 3, 3, 10, 10, 15, 15, 24, 24

Offset: 1

Author

M. F. Hasler, Nov 29 2016

Keywords

• nonn
• hard
• more
• fini

Comments

This sequence uses values of 1, 3, 3, 5 and 9 for a pawn, knight, bishop, rook and queen. The terms give the maximum possible difference of White's material minus Black's material at the n-th ply, i.e., after n half-moves.

I conjecture that, unless Black is forced to capture a white piece in all of the maximizing positions, every other term will be equal to the preceding one, a(2n) = a(2n-1).

The sequence is bounded from above by the theoretical maximum of 8 + 4*3 + 2* 5 + 9 = 39, the total value of all of one player's material, plus 8*8 = 64 more points in case all pawns of the "winning side" can be promoted to queens.

Several variants of this sequence are possible. For example, every other term could give the best possible value for Black, in signed or in absolute value.

From Michael S. Branicky, Dec 29 2022: (Start)

a(7) = 15 can be achieved with 1. e4 h5 2. Qxh5 e6 3. QxR Qh4 4. QxQ;

a(9) = 24, with 1. d4 e5 2. dxe5 Nf6 3. exf6 Be7 4. fxe7 f6 5. exd8=Q+.

Continuing the latter, ... Kf7 6. QxR f5 7. QxB Na6 8. QxR Nb8 9. QxN we see that a(11) >= 29, a(13) >= 32, a(15) >= 37, and a(17) >= 40. (End)

Examples

			In the first two half-moves no material can be captured. At its second move, i.e., ply 3, White has a few possibilities of capturing a black knight, e.g., 1. d3 Nh6 2. Bxh6; which yields a material difference of +3 for White. With 5 plies available, White should instead aim to capture a black pawn and Black's queen, as in 1. d3 g5 2. Bxg5 e5 3. Bxd8. This would yield an advantage of 1 + 9 = 10 for White.
		

Links

• OEIS index to sequences related to chess.

Crossrefs

Cf. A278830, A278831 (maximal / minimal number of possible moves at the n-th ply).

Programs

PARI

/* For illustrative purpose only: yields correct values at least up to n = 6, but too slow for larger n; en-passant, castling and illegal moves (when king in check) are not handled correctly */ {A278832(n,B=concat([B=digits(211107889e8\9*10^32),-B[9..16],-B[1..8],1]),M=concat(vector(64,F,if( B[F]*B[65]>0,Vec(moveGen[abs(B[F])](B,F-1)),[]))))=vecmax(apply(if(n>1,m-> A278832(n-1,makeMove(B,m))-VALUE[VAL_OFFSET+B[m%64+1]], m->-VALUE[B[m%64+1]+VAL_OFFSET]),M))};
   makeMove(B,m)={B[m%64+1]=B[m\64+1];B[m\64+1]=0;B[65]*=-1;B};
   VALUE=[0,-9,-3,-3,-5,-1,0,1,5,3,3,9,0]; VAL_OFFSET=7;
   KING=setunion(ROOK=[-8,-1,1,8], BISHOP=[-9,-7,7,9]);
   moveGen=[pawn(B,F,s8=B[65]<<3,L=List(),F8=F%8,F65=F*65)={F8>0 && B[F+s8]*s8<0 && listput(L,s8-1+F65); F8<7 && B[F+s8+2]*s8<0 && listput(L,s8+1+F65); B[1+F+s8]==0 && listput(L,s8+F65) && F\8==6^(s8<0) && listput(L,s8<<1+F65); L},\
   rook(B,F,d=ROOK,L=List(),T)={ for(i=1,#d,T=F; while(T%8*2!=(d[i]+9)%8*7 && T\8*2!=(d[i]+9)\8*7 && B[1+T+=d[i]]*B[65]<=0,listput(L,T+64*F); B[1+T] && break));L}, knight(B,F)=king(B,F,[-17,-15,-10,-6,6,10,15,17]), bishop(B,F)=rook(B,F,BISHOP), queen(B,F)=rook(B,F,KING), king(B,F,d=KING,L=List(),T)={for(i=1,#d,(T=F+d[i])>=0 && T<64 && (d[i]+2)%8 + F%8 > 1 && (d[i]+2)%8 + F%8<10 && B[1+T]*B[65]<=0 && listput(L,T+64*F));L}]

Extensions

a(3)-a(4) corrected and a(7)-a(10) from François Labelle, Nov 29 2016