How many different chess960 positions exist after one move by each side?

Assume that chess960 has 960 different starting positions, including the starting position for standard chess.

These are the letters used to represent the chess pieces.

• K = King
• Q = Queen
• R = Rook
• B = Bishop
• N = Knight

In some positions it is possible to castle on the first move.

• Castling is possible on the first move with a king on f1 and a rook on g1.
• Castling is possible on the first move with a king on d1 and a rook on c1.

White always has 18, 19, 20 or 21 possible first moves. Black and white have the same number of possible first moves. There are 20 possible first moves in standard chess.

• White has 18 different first moves if both white knights start in the corner, and castling is not possible on move 1.
• White has 19 different first moves if both white knights start in the corner, and castling is possible on move 1.
• White has 19 different first moves if one of two white knights start in the corner, and castling is not possible on move 1.
• White has 20 different first moves if one of two white knights start in the corner, and castling is possible on move 1.
• White has 20 different first moves if no white knights start in the corner, and castling is not possible on move 1.
• White has 21 different first moves if no white knights start in the corner, and castling is possible on move 1.

Square the number of possible first moves for white to compute the number of different possible positions resulting from that starting position, after one move by white and one move by black. In standard chess white has 20 possible first moves, and there are 20*20=400 possible positions after one move by each side.

• 18*18 = 324
• 19*19 = 361
• 20*20 = 400
• 21*21 = 441

Separate the starting positions into 12 cases to determine the number of possible first moves in each position.

• 19 first moves in 8 positions with: N on a1, N on h1, K on f1, R on g1
• 20 first moves in 20 positions with: N on a1, no N on h1, K on f1, R on g1
• 20 first moves in 28 positions with: no N on a1, N on h1, K on f1, R on g1
• 21 first moves in 34 positions with: no N on a1, no N on h1, K on f1, R on g1
• 19 first moves in 6 positions with: N on a1, N on h1, K on d1, R on c1
• 20 first moves in 22 positions with: N on a1, no N on h1, K on d1, R on c1
• 20 first moves in 16 positions with: no N on a1, N on h1, K on d1, R on c1
• 21 first moves in 28 positions with: no N on a1, no N on h1, K on d1, R on c1
• 18 first moves in 22 positions with: N on a1, N on h1, castling is not possible on move 1
• 19 first moves in 162 positions with: N on a1, no N on h1, castling is not possible on move 1
• 19 first moves in 160 positions with: no N on a1, N on h1, castling is not possible on move 1
• 20 first moves in 454 positions with: no N on a1, no N on h1, castling is not possible on move 1

This accounts for all 960 starting positions.

8 + 20 + 28 + 34 + 6 + 22 + 16 + 28 + 22 + 162 + 160 + 454 = 960

There are 18,882 possible chess960 positions after one move by white, using all 960 starting positions.

(8*19) + (20*20) + (28*20) + (34*21) + (6*19) + (22*20) + (16*20) + (28*21) + (22*18) + (162*19) + (160*19) + (454*20)

= [(22)*18] + [(8+6+162+160)*19] + [(20+28+22+16+454)*20] + [(34+28)*21]

= 18,882

There are 371,766 possible chess960 positions after one move by white and one move by black, using all 960 starting positions.

(8*19*19) + (20*20*20) + (28*20*20) + (34*21*21) + (6*19*19) + (22*20*20) + (16*20*20) + (28*21*21) + (22*18*18) + (162*19*19) + (160*19*19) + (454*20*20)

= [(22)*18*18] + [(8+6+162+160)*19*19] + [(20+28+22+16+454)*20*20] + [(34+28)*21*21]

= 371,766