Note on the Lamp Lighting Problem

Authors:
Henrik Eriksson;Kimmo Eriksson;Jonas Sjöstrand
Affiliations:
NADA, Kungl Tekniska Högskolan, SE-100 44, Stockholm, Swedenf1henrik@nada.kth.sef1;IMA, Mälardalens Högskola, Box 883, SE-721 23, Västerås, Swedenf2kimmo.eriksson@mdh.sef2;NADA, Kungl Tekniska Högskolan, SE-100 44, Stockholm, Swedenf3jonass@nada.kth.sef3
Venue:
Advances in Applied Mathematics
Year:
2001

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Abstract

We answer some questions concerning the so-called @s-game of Sutner [Linear cellular automata and the Garden of Eden, Math. Intelligencer11 (1989), 49-53]. It is played on a graph where each vertex has a lamp, the light of which is toggled by pressing any vertex with an edge directed to the lamp. For example, we show that every configuration of lamps can be lit if and only if the number of complete matchings in the graph is odd. In the special case of an orthogonal grid one gets a criterion for whether the number of monomer-dimer tilings of an mxn grid is odd or even.