# "Lamp-switch set-up number" of $n$ Answer

Hello dear visitor to our network We will offer you a solution to this question "Lamp-switch set-up number" of $n$ ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

## "Lamp-switch set-up number" of $n$

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 on "$$e$$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 example, $$\big\{\{k\}: k\in \{1,\ldots,n\}\big\}$$ 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?

we will offer you the solution to "Lamp-switch set-up number" of $n$ question via our network which brings all the answers from multiple reliable sources.