Switch-Setting Games Torsten Muetze

Content Introduction General theory Case study

Switch-Setting Games Is an initial light pattern solvable? Is there more than one solution? Which solution needs a minimal number of switching operations? Introduction

Mathematical Model D: The 4-tuple (G,S,L,z 0 ) is a switch-setting problem if G is a symmetric, bipartite graph with an induced partition of its vertices into the sets S and L and z 0 : L  {0,1}. S L G z0z0 General theory

Solution Behavior s1s1 s2s2 s3s3 A L,S s1s1 s2s2 s3s3 l1l1 111 l2l2 110 l3l3 101 l4l4 011 A = A S,L A L,S 0 0 z0z z 0 + A L,S  s 2 z0z l3l3 110l4l4 011l2l2 111l1l1 s3s3 s2s2 s1s1 A L,S A L,S T = A S,L sequence of switches s‘ 0,s‘ 1,...,s‘ k  S z 0 + A L,S  s‘ 0 + A L,S  s‘ A L,S  s‘ k = 0 (mod 2) z 0 + A L,S (  s‘ 0 +  s‘  s‘ k ) = 0 (mod 2) =: x  {0,1} S A L,S x = z 0 (mod 2) 5 5 l1l1 l2l2 l3l3 l4l4 solvability: ker(A L,S T ) (A L,S T = A S,L ) all solutions: ker(A L,S ) General theory

Parity Domination T: X4S is an even L-dominating set 5 x  ker(A L,S ). A S,L z = 0 (mod 2) Z4L is an even S-dominating set 5 z  ker(A S,L ). D: Even L-dominating set X4S : 5 X  N(l) has even cardinality for all l  L. X S L A L,S x = 0 (mod 2) Even S-dominating set Z 4 L : 5 Z  N(s) has even cardinality for all s  S. Z S L General theory

Z1Z1 S L Z2Z2 S LL Z1+Z2Z1+Z2 S Main Theorem T: Let P=(G,S,L,z 0 ) be a switch-setting problem and A the adjacency matrix of G. The following statements are equivalent: (i) P has a solution. (ii) A L,S x = z 0 (mod 2) has a solution. (iii) For all z  ker(A S,L ) the relation z T z 0 = 0 (mod 2) holds. (iv) For all z from a basis of ker(A S,L ) the relation z T z 0 = 0 (mod 2) holds. (v) The number of lit lamps on every even S-dominating set is even. (vi) The number of lit lamps on every even S-dominating set from a basis of the set of all even S-dominating sets is even. General theory

Rules Case study Switching operation: toggle all lamps on either a row, a column a diagonal or an antidiagonal m n Formal definition of the underlying graph G m,n = (V,E) |S| = 3(m+n)-2 |L| = mn G 2,2 X={s *2, s *1, s *1 } 4 S Z={l 1,1, l 2,1 } 4 L

Even S-dominating sets of G m,n Case study D: A circle C i,j is a subset of L, defined for all i  {1,2,...,m-3} and j  {1,2,...,n-3} by C i,j := {l i,j+1, l i,j+2, l i+1,j, l i+1,j+3, l i+2,j, l i+2,j+3, l i+3,j+1,l i+3,j+2 }. C 3,2 C 1,3 T: The set of circles is a basis for the set of even S-dominating sets. C 2,2 + C 3,3 C 2,2 + C 2,3 + C 2,4 + C 4,1 + C 4,2 + C 4,4 Conclusion: a light pattern is solvable, iff the number of lit lamps on every circle is even. not solvable solvable

Case study A basis for the even L-dominating sets of G 4,4 A basis for the even L-dominating sets of G m,n (min(m,n)  4) Problem X X1‘X1‘X2‘X2‘ Possible Solutions 14 X + X1‘X + X1‘X + X2‘X + X2‘ 11 Minimal solution? For min(m,n)R4 there are always 2 7 =128 even L-dominating sets Even L-dominating sets of G m,n

Summary General theory for the mathematical treatment of switch-setting games Interesting and fruitful relations between concepts from graph theory and linear algebra Graphically aesthetic interpretations

References [1] K. Sutner. Linear cellular automata and the garden-of-eden. Math. Intelligencer, 11:49-53, [2] J. Goldwasser, W. Klostermeyer, and H. Ware. Fibonacci polynomials and parity domination in grid graphs. Graphs Combin., 18: , [3] D. Pelletier. Merlin‘s Magic Square. Amer. Math. Monthly, 94: , [4] M. Anderson and T. Feil. Turning lights out with linear algebra. Math. Magazine, 71: , [5] T. Muetze. Generalized switch-setting problems. Preprint.