**Motivation.** The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way.


**Question.** Let $n\in\mathbb{N}$ be a positive integer and let $\{1,\ldots,n\}$ represent $n$ lamps, each of which is in exactly one of the states OFF or ON. Let $E\subseteq {\cal P}(\{1,\ldots,n\})$. For every $e\in E$ we have an "*$e$-button*", such that if that button is pressed, every element of $e$ switches its state (either from OFF to ON, or vice versa). We say that $E$ is *state-complete* if the following condition holds:



> 
> If all lamps are OFF and if $k\in \{1,\ldots, n\}$ is given, there is a finite button-sequence $e\_1, \ldots, e\_m \in E$ such that after all the buttons have been pressed, lamp $k$ is ON and all other lamps are OFF.
> 
> 
> 


For instance, $\big\{\{k\}: k\in \{1,\ldots,n\}\big\}$ is state-complete, and $\big\{\{1,\ldots,n\}\big\}$ is not state-complete for $n\geq 2$.


Given $n\in \mathbb{N}$, what is the least cardinality that a state-complete set $E\subseteq {\cal P}(\{1,\ldots,n\})$ can have?



"Lamp-switch set-up number" of $n$
This problem is a sort of generalization of the game 'Lights out'.


There are a few paper's e.g. by M. Zaidenberg
(for example "Periodic harmonic functions on lattices and points count in positive characteristic.")
on the underlying mathematics.


