Switch-Setting GamesTorsten 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,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}.

S

L

G

z0

General theory

Solution Behaviors1 s2 s3 AL,S s1 s2 s3

l1 1 1 1

l2 1 1 0

l3 1 0 1

l4 0 1 1

A =

AS,L

AL,S

0

0

z0

0

1

1

0

z 0 + A L,S

s 2

z0

0

1

1

0

101l3

110l4

011l2

111l1

s3s2s1AL,S

1

1

0

1

AL,ST = AS,L

sequence of switches s‘0,s‘1,...,s‘k Sz0 + AL,S s‘0

+ AL,S s‘1 + ... + AL,S s‘k

= 0 (mod 2)

z0 + AL,S (s‘0 + s‘1

+ ... + s‘k) = 0 (mod 2)

=: x {0,1}S

AL,S x = z0 (mod 2)

5

5

l1 l2 l3 l4

solvability: ker(AL,ST) (AL,S

T = AS,L)all solutions: ker(AL,S)

General theory

Parity Domination

T: X4S is an even L-dominating set 5 x ker(AL,S).

AS,L z = 0 (mod 2)

Z4L is an even S-dominating set 5 z ker(AS,L).

D: Even L-dominating set X4S :5 XN(l) has even cardinality for all l L.

X S

L

AL,S x = 0 (mod 2)

Even S-dominating set Z4L :5 ZN(s) has even cardinality for all s S.

Z

S

L

General theory

Z1

S

L

Z2

S

L L

Z1+Z2

S

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.

General theory

Rules

Case study

Switching operation: toggle alllamps on either a row, a columna diagonal or an antidiagonal

m

n

Formal definition of the underlying graph Gm,n = (V,E)

|S| = 3(m+n)-2

|L| = mn

G2,2

X={s*2, s*1, s*1}4SZ={l1,1, l2,1}4 L

Even S-dominating sets of Gm,n

Case study

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}.

C3,2 C1,3

T: The set of circles is a basis for the set of even S-dominating sets.

C2,2 + C3,3 C2,2 + C2,3 + C2,4 + C4,1 + C4,2 + C4,4

Conclusion: a light pattern is solvable, iff the number of lit lamps onevery circle is even.

not solvable

solvablesolvable

Case study

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)

ProblemX

X1‘ X2‘

Possible Solutions

14

X + X1‘ X + X2‘

14 11Minimalsolution?

For min(m,n)R4 there are always 27=128 even L-dominating sets

Even L-dominating sets of Gm,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, 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.