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Switch-Setting Games

Switch-Setting Games. Torsten Muetze. Content. Introduction General theory Case study. Introduction. Switch-Setting Games. Is an initial light pattern solvable? Is there more than one solution? Which solution needs a minimal number of switching operations?. General theory.

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Switch-Setting Games

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  1. Switch-Setting Games Torsten Muetze

  2. Content • Introduction • General theory • Case study

  3. Introduction Switch-Setting Games • Is an initial light pattern solvable? • Is there more than one solution? • Which solution needs a minimal numberof switching operations?

  4. General theory Mathematical Model S G L z0 • D: The 4-tuple (G,S,L,z0) 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 z0: L{0,1}.

  5. z0 + AL,Ss2 z0 AL,S s1 s2 s3 0 l1 1 1 1 1 1 l2 1 1 0 0 l3 1 0 1 1 1 l4 0 1 1 1 0 sequence of switches s‘0,s‘1,...,s‘k  S z0 + AL,Ss‘0+ AL,Ss‘1 + ... + AL,Ss‘k = 0 (mod 2) 5 0 AS,L z0 + AL,S (s‘0+ s‘1 + ... + s‘k) = 0 (mod 2) =: x  {0,1}S A = AL,S 0 5 AL,S x = z0 (mod 2) General theory Solution Behavior s1 s2 s3 l1 l2 l3 l4 AL,ST = AS,L solvability: ker(AL,ST) (AL,ST = AS,L) all solutions: ker(AL,S)

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

  7. S S S L L L Z1 Z2 Z1+Z2 General theory Main Theorem • T: Let P=(G,S,L,z0) be a switch-setting problem and A the adjacency matrix of G. The following statements are equivalent: (i) P has a solution. (ii) AL,S x = z0 (mod 2) has a solution. (iii) For all z ker(AS,L) the relation zTz0 = 0 (mod 2) holds. (iv) For all z from a basis of ker(AS,L) the relation zTz0 = 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.

  8. Formal definition of the underlying graph Gm,n = (V,E) |S| = 3(m+n)-2 |L| = mn G2,2 Z={l1,1, l2,1}4 L X={s*2, s*1, s*1}4S Case study n Rules • Switching operation: toggle alllamps on either a row, a columna diagonal or an antidiagonal m

  9. C3,2 C1,3 C2,2+C3,3 C2,2+C2,3+C2,4+C4,1+C4,2+C4,4 • D: A circle Ci,j is a subset of L, defined for all i  {1,2,...,m-3} andj  {1,2,...,n-3} by Ci,j := {li,j+1, li,j+2, li+1,j, li+1,j+3, li+2,j, li+2,j+3, li+3,j+1,li+3,j+2}. • T: The set of circles is a basis for the set of even S-dominating sets. • Conclusion: a light pattern is solvable, iff the number of lit lamps onevery circle is even. Case study Even S-dominating sets of Gm,n solvable solvable not solvable

  10. A basis for the even L-dominating sets of G4,4 • A basis for the even L-dominating sets of Gm,n (min(m,n)4) X1‘ X2‘ Problem Possible Solutions X X+X1‘ X+X2‘ Minimalsolution? 14 14 11 Case study Even L-dominating sets of Gm,n • For min(m,n)R4 there are always 27=128 even L-dominating sets

  11. 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

  12. References • [1] K. Sutner. Linear cellular automata and the garden-of-eden. Math. Intelligencer, 11:49-53, 1989. • [2] J. Goldwasser, W. Klostermeyer, and H. Ware. Fibonacci polynomials and parity domination in grid graphs. Graphs Combin., 18:271-283, 2002. • [3] D. Pelletier. Merlin‘s Magic Square. Amer. Math. Monthly, 94:143-150, 1987. • [4] M. Anderson and T. Feil. Turning lights out with linear algebra. Math. Magazine, 71:300-303, 1998. • [5] T. Muetze. Generalized switch-setting problems. Preprint.

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