Non-Attacking Chess Pieces: The Dance of Bishops Thomas Zaslavsky - PDF document

Non-Attacking Chess Pieces: The Dance of Bishops Thomas Zaslavsky Binghamton University (State University of New York) Joint with Seth Chaiken and Christopher R.H. Hanusa Outline 1. Chess Problems: Non-Attacking Pieces 2. Largely Czech Numbers and Formulas 3. Riders 4. Configurations and Inside-Out Polytopes 5. The Bishop Equations 6. Signed Graphs 7. Signed Graphs to the Rescue

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 2 Non-Attacking Chess Pieces The Dance of Bishops 1. Chess Problems: Non-Attacking Pieces · · · · · · · · · � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � B B B · · · · · · · · · � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � · · · · · · · · · � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � · · · · · · · · · B · · · · · · · · · · · · · · · · · · B B · · · · · · · · · · · · · · · · · · B · · · · · · · · · 7 non-attacking bishops on an 8 × 8 board. Question 1 : How many bishops can you get onto the board? Question 2 : Given q bishops, how many ways can you place them on the board so none attacks any other?

Non-Attacking Chess Pieces The Dance of Bishops 3 2. Numbers and Formulas from Our Czech Mate Vacl´ av Kotˇ eˇ sovec (Czech Republic) has a book of chess problems and so- lutions [5], called Chess and Mathematics . Some of the numbers: N B ( q ; n ) := the number of ways to place q non-attacking bishops on an n × n board. n = 1 2 3 4 5 6 7 8 Period Denom q = 1 1 4 9 16 25 36 49 64 1 1 2 0 4 26 92 240 520 994 1736 1 1 3 0 0 26 232 1124 3896 10894 26192 2 2 4 0 0 8 260 2728 16428 70792 242856 2 2 5 0 0 0 112 3368 39680 282248 1444928 2 2 6 0 0 0 16 1960 53744 692320 5599888 2 2 N Q ( q ; n ) := the number of ways to place q non-attacking queens on an n × n board. n = 1 2 3 4 5 6 7 8 Period Denom q = 1 1 4 9 16 25 36 49 64 1 1 2 0 0 8 44 140 340 700 1288 1 1 3 0 0 0 24 204 1024 3628 10320 2 2 4 0 0 0 2 82 982 7002 34568 6 6 5 0 0 0 0 10 248 4618 46736 60 ?? 6 0 0 0 0 0 4 832 22708 840 ?? 7 0 0 0 0 0 0 40 3192 360360 ??

4 Non-Attacking Chess Pieces The Dance of Bishops N B (1; n ) = n 2 . N B (2; n ) = n 3 n 3 − 4 n 2 + 3 n − 2 � � . 6 N B (3; n ) = 4 n 6 − 16 n 5 + 30 n 4 − 40 n 3 + 32 n 2 − 16 n + 3 − ( − 1) n 1 8 . 24 N B (4; n ) = ( n − 2)(15 n 7 − 90 n 6 + 260 n 5 − 524 n 4 + 727 n 3 − 646 n 2 + 393 n − 90) 360 − ( − 1) n ( n − 2) 2 . 8 N B (5; n ) = 6 n 9 − 68 n 8 + 354 n 7 − 1180 n 6 + 2870 n 5 � � ( n − 2) − 5284 n 4 + 6697 n 3 − 6018 n 2 + 3558 n − 810 720 − ( − 1) n ( n − 2)(3 n 3 − 22 n 2 + 58 n − 54) . 48 N B (6; n ) = ( n − 1)( n − 3)(126 n 10 − 2016 n 9 + 14868 n 8 − 69244 n 7 + 234017 n 6  − 607984 n 5 + 1211879 n 4 − 1797328 n 3 + 1953593 n 2 − 1550820 n + 722925)        90720     if n is odd ,   n ( n − 2)(126 n 10 − 2268 n 9 + 18774 n 8 − 97216 n 7 + 361165 n 6   − 1029454 n 5 + 2283178 n 4 − 3841960 n 3 + 4676932 n 2 − 3808152 n + 1640160)       90720     if n is even . 

Non-Attacking Chess Pieces The Dance of Bishops 5 3. Riders Rider : Moves any distance in specified directions (forward and back). The move is specified by an integral vector ( m 1 , m 2 ) ∈ R 2 in the direction of the line. Bishop : (1 , 1) , (1 , − 1). Queen : (1 , 0) , (0 , 1) , (1 , 1) , (1 , − 1). Nightrider : (1 , 2) , (2 , 1) , (1 , − 2) , (2 , − 1). Configuration : A point z = ( z 1 , z 2 , . . . , z q ) ∈ ( R 2 ) q = R 2 q where z i = ( x i , y i ) , which describes the locations of all the bishops (or queens, or . . . ). Constraints : The equations that correspond to attacking positions: z j − z i ∈ a line of attack , or in a formula: z j − z i ⊥ m for some move vector m = ( m 1 , m 2 ) , or, m 2 ( x j − x i ) = m 1 ( y j − y i ) .

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 6 Non-Attacking Chess Pieces The Dance of Bishops 4. Configurations and Inside-Out Polytopes The board. A square board with squares = { 1 , 2 , . . . , n } 2 . � � ( x, y ) : x, y ∈ { 1 , 2 , . . . , n } The board is n × n = 4 × 4 with a border, coordinates shown on the left side of each square. Note the border coordinates with 0 or n + 1, not part of the main square. · · · · · · · � � � � � � � � � � � � � � � � � � (0 , 0) (1 , 0) (2 , 0) (3 , 0) (4 , 0) (5 , 0) · · · · · · · � � � � � � (0 , 1) (1 , 1) (2 , 1) (3 , 1) (4 , 1) (5 , 1) · · · · · · · � � � � � � (0 , 2) (1 , 2) (2 , 2) (3 , 2) (4 , 2) (5 , 2) · · · · · · · � � � � � � (0 , 3) (1 , 3) (2 , 3) (3 , 3) (4 , 3) (5 , 3) · · · · · · · � � � � � � (0 , 4) (1 , 4) (2 , 4) (3 , 4) (4 , 4) (5 , 4) · · · · · · · � � � � � � (0 , 5) (1 , 5) (2 , 5) (3 , 5) (4 , 5) (5 , 5) · · · · · · · � � � � � � � � � � � � � � � � � � The dot picture in Z 2 . The border points are hollow. ◦ (0 , 0) ◦ (1 , 0) ◦ (2 , 0) ◦ (3 , 0) ◦ (4 , 0) ◦ (5 , 0) ◦ (0 , 1) • (1 , 1) • (2 , 1) • (3 , 1) • (4 , 1) ◦ (5 , 1) ◦ (0 , 2) • (1 , 2) • (2 , 2) • (3 , 2) • (4 , 2) ◦ (5 , 2) ◦ (0 , 3) • (1 , 3) • (2 , 3) • (3 , 3) • (4 , 3) ◦ (5 , 3) ◦ (0 , 4) • (1 , 4) • (2 , 4) • (3 , 4) • (4 , 4) ◦ (5 , 4) ◦ (0 , 5) ◦ (1 , 5) ◦ (2 , 5) ◦ (3 , 5) ◦ (4 , 5) ◦ (5 , 5)

Non-Attacking Chess Pieces The Dance of Bishops 7 The cube . 1 n + 1( x, y ) ∈ [0 , 1] 2 . Reduce ( x, y ) to 1 The position of a piece becomes z i = ( x i , y i ) ∈ (0 , 1) 2 ∩ n + 1 Z 2 . 1 The configuration becomes z = ( z 1 , . . . , z q ) ∈ (0 , 1) 2 q ∩ n + 1 Z 2 q , [0 , 1] 2 q � ◦ . 1 � a n +1 -fractional point in the open cube ◦ ( 0 5 , 0 ◦ ( 1 5 , 0 ◦ ( 2 5 , 0 ◦ ( 3 5 , 0 ◦ ( 4 5 , 0 ◦ ( 5 5 , 0 5 ) 5 ) 5 ) 5 ) 5 ) 5 ) ◦ ( 0 5 , 1 • ( 1 5 , 1 • ( 2 5 , 1 • ( 3 5 , 1 • ( 4 5 , 1 ◦ ( 5 5 , 1 5 ) 5 ) 5 ) 5 ) 5 ) 5 ) ◦ ( 0 5 , 2 • ( 1 5 , 2 • ( 2 5 , 2 • ( 3 5 , 2 • ( 4 5 , 2 ◦ ( 5 5 , 2 5 ) 5 ) 5 ) 5 ) 5 ) 5 ) ◦ ( 0 5 , 3 • ( 1 5 , 3 • ( 2 5 , 3 • ( 3 5 , 3 • ( 4 5 , 3 ◦ ( 5 5 , 3 5 ) 5 ) 5 ) 5 ) 5 ) 5 ) ◦ ( 0 5 , 4 • ( 1 5 , 4 • ( 2 5 , 4 • ( 3 5 , 4 • ( 4 5 , 4 ◦ ( 5 5 , 4 5 ) 5 ) 5 ) 5 ) 5 ) 5 ) ◦ ( 0 5 , 5 ◦ ( 1 5 , 5 ◦ ( 2 5 , 5 ◦ ( 3 5 , 5 ◦ ( 4 5 , 5 ◦ ( 5 5 , 5 5 ) 5 ) 5 ) 5 ) 5 ) 5 )

8 Non-Attacking Chess Pieces The Dance of Bishops The Bishop Equations. Bishops must not attack. The forbidden equations: z i / ∈ y j − y i = x j − x i and z i / ∈ y j − y i = − ( x j − x i ) . Left: The move line of slope +1. Right: The move line of slope − 1. Forbidden hyperplanes in R 2 q given by the ‘bishop equations’. Summary. We have a convex polytope P = [0 , 1] 2 q , and a set H = { h + ij , h − ij } of forbidden hyperplanes, and we want the number of ways to pick � P ◦ ∩ n +1 Z 2 q � � � � 1 z ∈ \ H .