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.
|This article uses algebraic notation to describe chess moves.|
Starting position requirements
White pawns are placed on the second rank as in standard chess. All remaining white pieces are placed randomly on the first rank, with two 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 all starting positions with equal probability
There are 5040 distinct arrangements of the pieces. Of the positions, 2880 of them have the bishops on squares of different colors, and 2160 of these have bishops on squares of the same color.
There are many procedures for creating a random starting position. The methods that are presented below fall into two general categories:
- Using n repetitions of a die or of a coin (binary). The problem is that the number 6n/10 or 2n/10 is not integer.
- Using random permutations of objects. The difficulty is that the number 5040/2880 or 5040/960 is not integer.
- 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, re-roll a 5 or 6 until another number shows.
- Roll the die and place a bishop on the white square indicated.
- 1 indicates b1, 2 indicates d1, and so on. Re-roll a 5 or 6.
- Roll the die and place the queen on the first empty position indicated, always skipping filled positions.
- Thus, 1 is the first (leftmost) empty position, while 6 is the sixth (rightmost) empty position.
- Roll the die and place a knight on the empty position indicated. Re-roll a 6.
- Roll the die and place a knight on the empty position indicated. Re-roll 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 these initial positions (rolled by 2–3–3–2–3 or 2–3–3–4–2) is the standard chess position, at which point a standard chess game ensues.
For each bishop one or two indications are required for the 4 squares of the same color. If the indication on two successive rolls is greater than 4 (4/36 cases), then with the first indication (5 or 6) the 2 squares (left or right) are selected and with second indication (5 or 6) from the 2 squares the left or the right is selected.
For the queen one indication is required.
For the first knight one indication is required. If it is 6, then the roll is repeated.
For the second knight one indication is required. If it is 5, then the relative position from the first knight is the next empty square, else if it is 6 then the next after. If the empty square does not exist, then the five empty squares are considered in circle.
Roll all the dice in one throw and place White's pieces as follows:
- Place a bishop on one of the eight squares (counting from the left, a through h ) as indicated by the octahedron (d8).
- Place the other bishop on one of the four squares of opposite colour as indicated by the tetrahedron (d4).
- Place the queen on one of the remaining six squares as indicated by the cube (d6).
- Take the number of the icosahedron (d20). Subtract one, divide by four, and let x = the quotient + 1, and y = the remainder + 1. Put a knight on the xth blank square. Then put the other knight on the yth remaining blank square.
Or as an alternative not needing calculations, place the first knight according to the d20 die, by counting the five empty squares and looping back to the left when necessary after reaching the rightmost empty square. Then with four empty squares remaining, do the same for the second knight, using the value of the dodecahedron (d12) die.
- Place the rooks and the king between the rooks on the remaining three squares.
Place the white pawns and mirror the position for Black.
It is possible more than one coin to be used after defining their priority (by size, color etc.)
The term indication is [(heads or tails) or (black or white pawn from a box)] that means (left or right) or (zero or plus) respectively.
If the available squares for the piece are 4, then with the first indication the 2 squares left or right are selected and with the second indication from the 2 squares the left or the right is selected.
For each bishop two indications are required.
For the first knight one indication required in order to select 3 left or right from the 6 remaining squares. Then 4 indications required and for each indication which is plus, is added 1, 2, 4, 8, respectively. If all 4 indications are plus this step is repeated (1/16 cases). The resulting number from this addition is divided by 5 and this yields the integer x and the remainder y. For the first knight the x+1 is selected from the 3 squares that initially selected and for the second knight the y+1 is selected from the 5 remaining squares.
For the queen two indications are required.
The remaining positions are for rook, king, rook.
Chess pieces or cards
In the last step of a method, if the king is not between the two rooks, then he is swapped with the nearest rook.
- Τhe 6 pieces without the bishops are divided randomly into two groups of 3 each.
- The first 3 pieces and a bishop are placed randomly on dark squares.
- The remaining pieces and the other bishop are placed randomly on light squares.
The math: The first step yields 6!/(3!3!) = 6*5*4/3/2 ways the pieces to be divided. It is multiplied in the second and in the third step by 4*3*2 permutations of 4 pieces. It is divided by 2 because the knights are the same. It is divided by 2 because the rooks are the same. The result is 2880. It is divided by 3 because the king must be between the rooks. The result is 960.
To be shuffled totally fair, it is noted at the bottom of eight pawns RNBQKBNR and these are shuffled instead of the pieces.
- Pawn, knight, bishop, rook, queen, king (from the smallest to the greatest) are placed randomly in a row.
The position of the first bishop is one of the 4 squares of the same color:
(The third piece from how many of its next pieces is greater ) + 1
The position of the second bishop is one of the 4 squares of the same color:
2 * (first > second) + (penultimate > last) + 1
If first > second, then from the 4 squares are selected the two on the right, else the two on the left.
If penultimate > last, then from 2 squares is selected the right, else the left.
- Then 2 knights, 2 rooks, queen, king are placed randomly on the six empty squares.
Eight white and eight black pawns (1-8) marked below are shuffled and then each random pawn is displayed in its row.
White pawns, odd numbers: 7, 3, 1, 5
White pawns, even numbers: 4, 2, 8, 6
Black pawns: 5, 7, 8, 3, 6, 4, 1, 2
The first white pawns 7 and 4 on the rows of the four white pawns are the positions for the bishops.
The corresponding two black pawns 7 and 4 are removed from the row of eight black pawns and the remaining six black pawns
5, 8, 3, 6, 1, 2 are the positions for the queen, knights, king and rooks.
If the king is not between the two rooks, then he is swapped with the nearest rook.
Also four black bishops (1, 3, 5, 7) and four white bishops (2, 4, 6, 8) and six pawns (Q, N, N, K, R, R) can be used.
A black bishop and a white bishop and afterwards the six pawns are selected randomly.
- Hans Bodlaender (2002-05-10). "Fischer Random Chess: Manual Procedure for Generating Piece Placements". The Chess Variant Pages. Retrieved 2013-01-26.