### Math and Chess

There are 20 possible moves for White’s first move (1 ply) in chess (16 pawn moves and 4 knight moves).

There are 400 possible moves for the first two moves in chess (2 ply), with White moving first (20 possible moves), then Black moving.

There are 5,362 possible positions after three moves (3 ply), two moves for White and one move for Black). Of the 5,362 positions, 1,862 are unique.

There are 71,852 total positions after four moves (4 ply), two moves for White and two moves for Black), of which 9,825 positions are unique.

There are 822,518 possible positions after 5 moves (5 ply) , three moves for White and two moves for Black, of which 53,516 positions are unique.

There are 9,417,681 total positions after six moves (6 ply), 3 moves for White and 3 moves for Black. Of these, there are 311,642 unique positions.

There are 96,400,068 possible positions after 7 moves (7 ply), 4 moves for White and 3 moves for Black. Of these, there are 2,018,993 unique positions.

There are 988,187,354 total positions after 8 moves (8 ply),4 moves for White and 4 moves for Black. Of these, 12,150,635 positions are unique.

There are 9,183,421,888 total positions after 9 moves (9 ply), 5 moves for White and 4 moves for Black. Of these, 69,284,509 positions are unique.

There are 85,375,278,064 total possible positions after 10 moves (10 ply) 5 moves for White and 5 moves for Black). Of these, 382,383,387 positions are unique.

The total number of chess positions possible is about 2×10 to the 46th power.

There are 92 different ways to placing 8 queens on a chess board so that no two queens attack each other. There are 12 different ways to placing 8 queens on a chess board that are unique (excluding rotations and mirror images).

There are 26,534,728,821,064 ways to conduct a knight’s tour in which a knight is placed on an empty board, moves like a knight, and visits each square exactly once.

If you take a number and double it every time for every chess square (1, 2, 4, 8, 16, 36, 64, 128, etc), by the time you get to the 64th square, the number will be 18,446,744,073,709,551,615.

There are 204 squares on a chessboard (64 1×1 squares + 49 2×2 squares + 36 3×3 squares + 25 4×4 squares + 16 5×5 squares + 9 6×6 squares + 4 7×7 squares + 1 8×8 square).

There are 1,296 rectangles on a chessboard.

There are 638 ways to put 5 queens on a chessboard so as to attack every square on a chess board.

The longest chess game theoretically possible is 5,949 moves.

There are 8 different ways to checkmate in two moves.

1.f3 e5 2.g4 Qh4 mate

1.f3 e6 2.g4 Qh4 mate

1.f4 e5 2.g4 Qh4 mate

1.f4 e6 2.g4 Qh4 mate

1.g4 e5 2.f3 Qh4 mate

1.g4 e5 2.f4 Qh4 mate

1.g4 e6 2.f3 Qh4 mate

1.g4 e6 2.f4 Qh4 mate

There are 960 ways to set up the first and last rank of a chess board. There are 4 possibilities for the first bishop (four light squares or four dark squares), 4 possibilities for the second bishop, 6 possibilities for the queen (after the bishops have been placed), 5 possibilities for the first knight (after the 2 bishops and queen are in place), 4 possibilities for the second knight (after the 2 bishops, queen, and first knight are in place), and since the knights are not distinct, divide by 2. The rook, king, and other rook occupy the remaining three squares with only 1 possibility. So (4*4*4*6*5*1*1*1)/2 = 960.

It takes a minimum of 5 queens and a maximum of 8 queens to occupy or attack every square.

It takes a minimum and maximum of 8 rooks to occupy or attack every square.

It takes a minimum of 8 bishops and a maximum of 14 bishops to occupy or attack every square.

It takes a minimum of 12 knights and a maximum of 32 knights to occupy or attack every square.

It takes a minimum of 9 kings and a maximum of 16 kings to occupy or attack every square.

Two small squares on a chess board are chosen at random. The probability that they have a common side is 1 in 18.

Probability of winning:

Rating Gap Match Win % Match Lose % Game Win % Game Draw % Game Lose %

50 57 43 35.5 43 21.5

102 64 36 46 36 18

149 70 30 55 30 15

202 76 24 64 24 12

251 81 19 71.5 19 9.5

296 85 15 77.5 15 7.5

351 89 11 83.5 11 5.5

401 92 8 88 8 4

470 95 5 92.5 5 2.5

501 96 4 94 4 2

The Shannon number, named after Dr. Claude Shannon, is an estimated lower bound on the game-tree complexity of chess. He estimated that there are 10 raised to the 120th power variations in chess. The number of possible positions is around 10 raised to the 43rd power.

Mathematicians who played or studied chess include George Airy, Adolf Anderssen, George Atwood, Charles Babbage, Nathan Divinksy, Noam Elkes, Leonhard Euler, Max Euwe, Ed Formanek, Martin Gardner, Carl Gauss, GH Hardy, David Hilbert, Charles Kalme, Emanuel Lasker, Jonathan Mestel, John Nash, John von Neumann, John Nunn, Roger Penrose, Henri Poincare, Claude Shannon, Jonathan Speelman, Duncan Suttles, Alan Turing, and Norbert Weiner.

– Bill Wall

Tags: chess and math