Non-Attacking Chess Pieces: The Bishop

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1 Non-Attacking Chess Pieces: The Bishop Thomas Zaslavsky Binghamton University (State University of New York) C R Rao Advanced Institute of Mathematics, Statistics and Computer Science 9 July 010 Joint work with Seth Chaiken and Christopher R.H. Hanusa Outline 1. Chess Problems: Non-Attacking Pieces. 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 Bishop 9 July 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 : Given q bishops, how many ways can you place them on the board so none attacks any other?

3 Non-Attacking Chess Pieces: The Bishop 9 July Largely Czech Numbers and Formulas Vacláv Kotěšovec (Czech Republic) has a book of chess problems and solutions [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 = Period Denom q = N Q (q; n) := the number of ways to place q non-attacking queens on an n n board. n = Period Denom q = ?? ?? ??

4 4 Non-Attacking Chess Pieces: The Bishop 9 July 010 N B (1; n) = n. N B (; n) = n ( 3n 3 4n + 3n ). 6 (n 1)(n 5 6n 4 + 9n 3 11n + 5n 3) N B (3; n) = 1 n(n )(n 4 4n 3 + 7n 6n + 4) 1 if n is odd, if n is even. N B (4; n) = (n 1)(n )(15n 6 75n n 4 339n n 58n + 180) 360 n(n )(15n 6 90n n 4 54n n 646n + 348) 360 if n is odd, if n is even. N B (5; n) = (n 1)(n )(n 3)(3n 7 n n 5 04n n 3 464n n 70) 360 if n is odd, n(n )(3n 8 34n n 6 590n n 4 59n n 844n ) 360 if n is even. N B (6; n) = (n 1)(n 3)(16n n n n n n n n n n + 795) if n is odd, 9070 n(n )(16n 10 68n n n n n n n n n ) if n is even. 9070

5 Non-Attacking Chess Pieces: The Bishop 9 July Riders Rider: Moves any distance in specified directions (forward and back). The move is specified by an integral vector (m 1, m ) R in the direction of the line. Bishop: (1, 1), (1, 1). Queen: (1, 0), (0, 1), (1, 1), (1, 1). Nightrider: (1, ), (, 1), (1, ), (, 1). Configuration: A point z = (z 1, z,..., z q ) (R ) q = R 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: or in a formula: or, z j z i a line of attack, z j z i m for some move vector m = (m 1, m ), m (x j x i ) = m 1 (y j y i ).

6 6 Non-Attacking Chess Pieces: The Bishop 9 July Configurations and Inside-Out Polytopes The board. A square board with squares { (x, y) : x, y {1,,..., n} } = {1,,..., 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) (,0) (3,0) (4,0) (5,0) (0,1) (0,) (0,3) (0,4) (0,5) (1,1) (1,) (1,3) (1,4) (1,5) (,1) (,) (,3) (,4) (,5) (3,1) (3,) (3,3) (3,4) (3,5) (4,1) (4,) (4,3) (4,4) (4,5) (5,1) (5,) (5,3) (5,4) The dot picture in Z. The border points are hollow. (0, 0) (1, 0) (, 0) (3, 0) (4, 0) (5, 0) (5,5) (0, 1) (1, 1) (, 1) (3, 1) (4, 1) (5, 1) (0, ) (1, ) (, ) (3, ) (4, ) (5, ) (0, 3) (1, 3) (, 3) (3, 3) (4, 3) (5, 3) (0, 4) (1, 4) (, 4) (3, 4) (4, 4) (5, 4) (0, 5) (1, 5) (, 5) (3, 5) (4, 5) (5, 5)

7 Non-Attacking Chess Pieces: The Bishop 9 July The cube. Reduce (x, y) to 1 n + 1 (x, y) [0, 1]. The position of a piece becomes z i = (x i, y i ) (0, 1) 1 n + 1 Z. The configuration becomes z = (z 1,..., z q ) (0, 1) q 1 n + 1 Zq, a 1 n+1 -fractional point in the open cube ( [0, 1] q). ( 0 5, 0 5 ) (1 5, 0 5 ) ( 5, 0 5 ) (3 5, 0 5 ) (4 5, 0 5 ) (5 5, 0 5 ) ( 0 5, 1 5 ) (1 5, 1 5 ) ( 5, 1 5 ) (3 5, 1 5 ) (4 5, 1 5 ) (5 5, 1 5 ) ( 0 5, 5 ) (1 5, 5 ) ( 5, 5 ) (3 5, 5 ) (4 5, 5 ) (5 5, 5 ) ( 0 5, 3 5 ) (1 5, 3 5 ) ( 5, 3 5 ) (3 5, 3 5 ) (4 5, 3 5 ) (5 5, 3 5 ) ( 0 5, 4 5 ) (1 5, 4 5 ) ( 5, 4 5 ) (3 5, 4 5 ) (4 5, 4 5 ) (5 5, 4 5 ) ( 0 5, 5 5 ) (1 5, 5 5 ) ( 5, 5 5 ) (3 5, 5 5 ) (4 5, 5 5 ) (5 5, 5 5 ) 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 q given by the bishop equations.

8 8 Non-Attacking Chess Pieces: The Bishop 9 July 010 Summary: We have a convex polytope P = [0, 1] q, and a set H = {h + ij, h ij } of forbidden hyperplanes, and we want the number of ways to pick z [ P 1 n+1 Zq ] \ [ H ]. Inside-Out Polytopes. (P, H) is an inside-out polytope. We want E P,H(n + 1) := the number of points in [ ] [ ] P 1 n+1 Zq \ H. Inside-out Ehrhart theory (Beck & Zaslavsky 005, based on Ehrhart and Macdonald) says that E P,H(n + 1) is a quasipolynomial function of n + 1, for n + 1 Z >0. Vertex of (P, H): A point in P determined by the intersection of hyperplanes in H and facets of P. Quasipolynomial: A cyclically repeating series of polynomials, c d (n)n d + c d 1 (n)n d c 1 (n)n + c 0 (n), where the c i are periodic functions of n that depend on n mod p for some p Z >0. The smallest p is called the period of the quasipolynomial. Lemma 4.1 ([1]). If P has rational vertices and the hyperplanes in H are given by an integral linear equation, then: (a) E P,H(n + 1) is a quasipolynomial function of n. (b) Its degree is d = dim P, and its leading term is vol(p )n d. (c) Its period is a factor of the least common denominator of all coordinates of vertices of (P, H). Define N R (q; n) := the number of ways to place q non-attacking R-pieces on an n n board. Theorem 4.. For a rider chess piece R, N R (q; n) is a quasipolynomial function of n, for each fixed q > 0; the leading term of each polynomial is 1 q! nq. Agrees with Kotěšovec s formulas!

9 Non-Attacking Chess Pieces: The Bishop 9 July The chess problem. What is the quasipolyomial for q bishops (or queens, or... )? We have qp undetermined coefficients. The actual numbers N B (q; n) for 1 n pq will determine the whole thing. Aye, there s the rub. Two rubs: (1) We don t know p. () It may be impossible to compute enough values of N B (q; n). Finding a small upper bound on the period is not so easy. Lemma 4.1(c) says: The period is a factor of the gcd of the denominators of the vertices of (P, H). Good, if we can find the denominator. The Bishop Solution. Theorem 4.3. The bishop quasipolynomial N B (q; n) has period at most. Theorem 4.3 is an immediate corollary of Lemma 6.1, which bounds the denominator.

10 10 Non-Attacking Chess Pieces: The Bishop 9 July 010 Graph (N, E): Node set N = {v 1, v,..., v q }. Edge set E. 5. Signed Graphs 1-Forest: each component is a tree with one more edge. (Each component contains exactly one circle.) Signed graph Σ = (N, E, σ): Graph (N, E). Signature σ : E {+, }. Circle sign σ(c). Signed circuit: a positive circle; or a connected subgraph that contains exactly two circles, both negative. Homogeneous node: all incident edges have the same sign. Incidence matrix H(Σ): N E matrix. In column of edge e:v i v j, (a) η(v, e) = ±1 if v is an endpoint of e and = 0 if not; (b) η(v i, e)η(v j, e) = σ(e). (The column of a positive edge: one +1 and one 1, the column of a negative edge: two +1 s or two 1 s.) Lemma 5.1 ([8, Theorems 5.1(j) and 8B.1]). For a signed graph Σ: H(Σ) has full column rank iff Σ contains no signed circuit. H(Σ) has full row rank iff every component of Σ contains a negative circle. The usual hyperplane arrangement H[Σ]: Edge e:v i v j h e : x j = σ(e)x i in R q. Vector-space dual to columns of incidence matrix. (Linear dependencies are those of the incidence matrix columns.)

11 Non-Attacking Chess Pieces: The Bishop 9 July Clique graph C(Σ): Positive clique: maximal set of nodes connected by positive edges. A := {positive cliques}. Negative clique: maximal set of nodes connected by negative edges. B := {negative cliques}. Signed clique: either one. v one positive clique and one negative clique. For a signed forest with q nodes, k A + k B = q + c, where c = number of components, k A = number of positive cliques, k B = number of negative cliques. Clique graph: N(C(Σ)) = A B. An edge A k B l for each v i A k B l.

12 1 Non-Attacking Chess Pieces: The Bishop 9 July Signed Graphs to the Rescue Lemma 6.1. A point z = (z 1, z,..., z q ) R q, determined by a total of q bishop equations and fixations, is weakly half integral. Furthermore, in each z i, either both coordinates are integers or both are strict half integers. Consequently, a vertex of the bishops inside-out polytope ([0, 1] q, A B ) has each z i {0, 1} or z i = ( 1, 1 ). Proof. A vertex z is the intersection of q hyperplanes: Bishop hyperplanes, h + ij : x j y j = x i y i and h ij : x j + y j = x i + y i. Facet hyperplanes, x i = c i and y j = d j. Equations: Bishop equations (relations between coordinates). Fixations: Facet hyperplanes (fix some coordinates to chosen integers). Signed graph of z: Σ z the equations, i.e., hyperplanes h ε ij. Clique graph of z: C z := C(Σ z ), and ±C z. Meaning of a clique: A k = {v i, v j,...} = x i y i = x j y j = x i y i = a k v i A k. B l = {v i, v j,...} = x i + y i = x j + y j = x i + y i = b l v i B l. Method: (1) Convert to variables a k, b l. () Find enough fixations to determine a k, b l. (3) Solve for all x i, y j.

13 Non-Attacking Chess Pieces: The Bishop 9 July Example 6.1. A = {A 1, A, A 3 } and B = {B 1, B, B 3, B 4 }, N = {v 1,..., v 8 }: C z : A 1 B 1 v 3 v 1 v A B v 4 v 5 A 3 v 6 B 3 v 7 B 4 A suitable 1-forest Σ z ±C z (superscript x is +, y is ): Σ z ±C z v x 1 A 1 B 1 v3 x vy A B A 3 Incidence matrix (invertible): v4 x v y 5 v7 x v y 7 B 3 B 4 fixations x 1 = c 1, y = d 1, x 3 = c, x 4 = c 3, y 5 = d, x 7 = c 4, y 7 = d 3. M := H(Σ z ) = x 1 x 3 x 4 x 7 y y 5 y A 1 A A 3 B 1 B B 3 B 4.

14 14 Non-Attacking Chess Pieces: The Bishop 9 July 010 In matrix form: where c i, d j Z. Solution: M T a 1 a a 3 b 1 b b 3 b 4 = x 1 x 3 x 4 x 7 y y 5 y 7 = c 1 c c 3 c 4 d 1 d d 3, a 1 = x 1 x 3 + x 4 + y = c 1 c + c 3 + d 1, a = x 1 + x 3 + x 4 + y = c 1 + c + c 3 + d 1, a 3 = x 7 + y 7 = c 4 + d 3, b 1 = x 1 x 3 + x 4 + y = c 1 c + c 3 + d 1, b = x 1 + x 3 x 4 + y = c 1 + c c 3 + d 1, b 3 = x 1 x 3 x 4 y + y 5 = c 1 c c 3 d 1 + d, b 4 = x 7 + y 7 = c 4 + d 3, and the unfixed variables are x = a 1 b x 5 = a b 3 x 6 = a 3 b 3 y 1 = a 1 + b 1 y 3 = a + b 1 y 4 = a + b = c 1 c + c 3, = c 1 + c + c 3 + d 1 d, = c 1 + c + c 3 + c 4 + d 1 d + d 3, = c + c 3 + d 1, = c 1 + c 3 + d 1, = c 1 + c + d 1, y 6 = a 3 + b 3 = c 1 c c 3 + c 4 d 1 + d + d 3. x 6 and y 6 are the only possibly fractional coordinates; their sum is integral; therefore, either z 6 is integral or z 6 = ( 1, 1 ).

15 Non-Attacking Chess Pieces: The Bishop 9 July Lemma 6. (Hochbaum, Megiddo, Naor, and Tamir 1993). The solution of a linear system with integral constant terms, whose coefficient matrix is the transpose of a nonsingular signed-graph incidence matrix, is weakly half-integral. Proof of Theorem, concluded. [ [ ] x = H(±C y] z ) T (M 1 ) T c. d This is half integral, because (M 1 ) T [ c d is integral. ] is half integral by the lemma, and H(±C z ) T

16 16 Non-Attacking Chess Pieces: The Bishop 9 July 010 References [1] Matthias Beck and Thomas Zaslavsky, Inside-out polytopes. Adv. Math. 05 (006), no. 1, MR 007e:5017. Zbl arxiv.org math.co/ [] Seth Chaiken, Christopher R.H. Hanusa, and Thomas Zaslavsky, Mathematical analysis of a q-queens problem. In preparation. The full paper of this talk. [3] F. Harary, On the notion of balance of a signed graph. Michigan Math. J. ( ), and addendum preceding p. 1. MR 16, 733h. Zbl [4] Dorit S. Hochbaum, Nimrod Megiddo, Joseph (Seffi) Naor, and Arie Tamir, Tight bounds and - approximation algorithms for integer programs with two variables per inequality. Math. Programming Ser. B 6 (1993), Zbl [5] Václav Kotěšovec, Non-attacking chess pieces (chess and mathematics) [Šach a matematika - počty rozmístění neohrožujících se kamenu]. [Self-published online book], 010; second edition 010, URL non attacking chess pieces 010.pdf An amazing source of formulas and conjectures; no proofs. [6] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, URL njas/sequences/ Many numbers for bishops up to 6, queens up to 7, nightriders up to 3. [7] Thomas Zaslavsky, The geometry of root systems and signed graphs. Amer. Math. Monthly 88 (1981), MR 8g:0501. Zbl Hyperplanes led me to signed graphs. [8] Thomas Zaslavsky, Signed graphs. Discrete Appl. Math. 4 (198), Erratum. Discrete Appl. Math. 5 (1983), 48. MR 84e: Zbl The theory of the incidence matrix.

arxiv: v1 [math.co] 12 May 2014

arxiv: v1 [math.co] 12 May 2014 A q-queens PROBLEM V. THE BISHOPS PERIOD May 14, 014 SETH CHAIKEN, CHRISTOPHER R. H. HANUSA, AND THOMAS ZASLAVSKY arxiv:1405.3001v1 [math.co] 1 May 014 Abstract. Part I showed that the number of ways to