Chess960 starting position

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A Chess960 starting position is one of 960 possible initial game positions in the chess variant Chess960. The special arrangement of pieces on the players' first ranks is selected randomly before play according to Chess960 rules, and can be generated either by a computer program, or using dice, coin, cards, etc.


Starting position requirements[edit]

White pawns are placed on the second rank as in standard chess. All remaining white pieces are placed randomly on the first rank, with a couple restrictions:

  • The bishops must be placed on opposite-color squares.
  • The king must be placed on a square between the rooks.

Black's pieces are placed equal-and-opposite to White's pieces. For example, if the white king is randomly determined to start on f1, then the black king is placed on f8. (The king never starts on the a - or h -files, since this would leave no space for a rook.)

Methods for creating starting positions[edit]

There are several procedures for generating random starting positions with equal probability.

Single die method[edit]

Dr Ingo Althöfer
a b c d e f g h
8
Chessboard480.svg
a8 black knight
b8 black bishop
c8 black knight
d8 black rook
e8 black bishop
f8 black queen
g8 black king
h8 black rook
a7 black pawn
b7 black pawn
c7 black pawn
d7 black pawn
e7 black pawn
f7 black pawn
g7 black pawn
h7 black pawn
a2 white pawn
b2 white pawn
c2 white pawn
d2 white pawn
e2 white pawn
f2 white pawn
g2 white pawn
h2 white pawn
a1 white knight
b1 white bishop
c1 white knight
d1 white rook
e1 white bishop
f1 white queen
g1 white king
h1 white rook
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
Chess960 starting position generated by the following die rolls:
  • 3 (bishop on e1)
  • 5 (skip and reroll)
  • 1 (bishop on b1)
  • 4 (queen on f1)
  • 6 (skip and reroll)
  • 2 (knight on c1)
  • 1 (knight on a1)
  • king on g1; rooks on d1 and h1

A common method for selecting a starting position is one proposed by Ingo Althöfer in 1998, which requires only a single cube die.[1][2][3] The position of White's pieces is determined as follows:

  1. Roll the die and place a bishop on the black square indicated by the die, counting from the left, a through h.
    Thus, 1 indicates the first black square from the left (a1), 2 indicates the second black square (c1), 3 indicates the third (e1), and 4 the fourth (g1). Since there are no fifth or sixth positions, reroll a 5 or 6 until another number shows.
  2. Roll the die and place a bishop on the white square indicated.
    1 indicates b1, 2 indicates d1, and so on. Reroll a 5 or 6.
  3. Roll the die and place the queen on the first empty position indicated, always skipping filled positions.
    Thus, 1 is the first (leftmost) empty square, while 6 is the sixth (rightmost) empty square.
  4. Roll the die and place a knight on the empty position indicated. Reroll a 6.
  5. Roll the die and place a knight on the empty position indicated. Reroll a 5 or 6.

This leaves three empty squares. Place the king on the middle empty square, and the rooks on the remaining two squares. Place the white and black pawns on their usual squares, and Black's first-row pieces to exactly mirror White's. (So, Black should have on a8 the same piece type White has on a1.)

The above procedure uses an average of 6.7 die rolls. Note that one of the random positions (rolled by 23323 or 23342) is the standard chess starting position, at which point a standard chess game ensues.

Optimization[edit]

On average, the single die procedure uses 6.7 die rolls. An optimal procedure would use on average log(960)/log(6) = 3.83 die rolls. It is straightforward to reduce the average number of dice rolls to 6.2. Instead of rerolling 5s and 6s until something else comes up, reroll once and use the following scheme:

  • 5 followed by 1 to 31
  • 5 followed by 4 to 62
  • 6 followed by 1 to 33
  • 6 followed by 4 to 64

Polyhedral dice method[edit]

With polyhedral dice shaped like each of the five Platonic solids, one never needs to reroll any dice.

A set of Platonic solids dice, from left: tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron (d12), icosahedron (d20)
a b c d e f g h
8
Chessboard480.svg
a8 black rook
b8 black king
c8 black queen
d8 black rook
e8 black knight
f8 black bishop
g8 black bishop
h8 black knight
a7 black pawn
b7 black pawn
c7 black pawn
d7 black pawn
e7 black pawn
f7 black pawn
g7 black pawn
h7 black pawn
a2 white pawn
b2 white pawn
c2 white pawn
d2 white pawn
e2 white pawn
f2 white pawn
g2 white pawn
h2 white pawn
a1 white rook
b1 white king
c1 white queen
d1 white rook
e1 white knight
f1 white bishop
g1 white bishop
h1 white knight
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
Starting position using four polyhedral dice:
  • d8=6 (bishop on f1)
  • d4=4 (bishop on g1)
  • d6=3 (queen on c1)
  • d20=20 (knights on h1 and e1)
  • king on b1; rooks on a1 and d1
a b c d e f g h
8
Chessboard480.svg
a8 black rook
b8 black king
c8 black queen
d8 black knight
e8 black rook
f8 black bishop
g8 black bishop
h8 black knight
a7 black pawn
b7 black pawn
c7 black pawn
d7 black pawn
e7 black pawn
f7 black pawn
g7 black pawn
h7 black pawn
a2 white pawn
b2 white pawn
c2 white pawn
d2 white pawn
e2 white pawn
f2 white pawn
g2 white pawn
h2 white pawn
a1 white rook
b1 white king
c1 white queen
d1 white knight
e1 white rook
f1 white bishop
g1 white bishop
h1 white knight
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
Starting position using all five polyhedrals:
  • d8, d4, and d6 (same as with four polyhedrals)
  • d20=20 (knight on h1)
  • d12=11 (knight on d1)
  • king on b1; rooks on a1 and e1

Roll all the dice in one throw and place White's pieces as follows:

  1. Place a bishop on one of the eight squares (counting from the left, a through h ) as indicated by the octahedron (d8).
  2. Place the other bishop on one of the four squares of opposite color as indicated by the tetrahedron (d4).
  3. Place the queen on one of the remaining six squares as indicated by the cube (d6).
  4. Take the value of the icosahedron (d20), subtract one, divide by four, and let x = the quotient + 1, and y = the remainder + 1. Place a knight on the xth empty square. Then place the other knight on the yth remaining empty square.
    Or alternatively (using an additional die instead of calculations): Place the first knight according to the value of the d20 die, by counting the five empty squares and looping back to the left whenever reaching the rightmost empty square. Then with four empty squares remaining, do the same for the other knight using the dodecahedron (d12) die.
  5. Place the king between the rooks on the remaining three squares.

Place the white pawns and mirror the position for Black.

Coins (binary) method[edit]

Two coins (small and large) are used to randomly generate numbers with equal probability. Tails on the smaller coin counts as 0, tails on the larger coin counts as 1, and heads on either coin counts as 2. To create numbers in the range 1 through 4, toss both coins and add their values together. To create numbers in the range 1 through 3, do the same but retoss whenever 4 is the result. To create numbers in the range 1 through 2, just toss the larger coin (tails is 1, heads is 2).

There is a way of using coins and making all starting positions equally likely. It uses a third coin for which tails counts as 0, and heads counts as 4. Tossing all three coins generates the values 1 through 8 with equal probability. The method follows the piece placements used for a die. Two coins are used for the bishops as before. Then six squares are available for the queen. All three coins are tossed and retossed until a number in the range 16 comes up. Then five squares are available for the first knight. Now the three coins should be tossed and retossed until a number in the range 15 shows up. For the other knight, only a four-way choice is needed, so a single toss of two coins suffices. The average number of tosses needed for this method is 5 + 14/15.

A similar coin-tossing method uses one coin to generate all starting positions with equal probability. Toss the coin four times and record the results. If the four coin tosses are all tails, start again. Otherwise toss the coin an additional six times and record the results. Then convert the sequence into a binary number counting heads as 0, tails as 1. The resulting number is a number between 0 and 959 that can then be converted into a starting position using the Chess960 numbering scheme. For example, if the tosses are T, H, T, T, H, H, H, H, T, T this converts to the binary number 1011000011, or 707, which in the Chess960 numbering scheme is the starting position BRKQNNRB.

See also[edit]

References[edit]

  1. ^ Althöfer, Ingo (1998). Computerschach und Spiele (CSS) (in German) (Dieter Steinwender and Frederic Friedel). 
  2. ^ Althöfer, Ingo (March 1998). van den Herik, Jaap, ed. "LIST-3-HIRN vs. Grandmaster Yusupov". International Computer Chess Association (ICCA) (Universiteit Maastricht) 21 (1): pp. 52–60. ISSN 0920-234X. 
  3. ^ Hans Bodlaender (2002-05-10). "Fischer Random Chess: Manual Procedure for Generating Piece Placements". The Chess Variant Pages. Retrieved 2013-01-26. 

External links[edit]