1 / 12

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.

jade-gamble
Download Presentation

Switch-Setting Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related