Wiring switches to more light bulbs

TL;DR

This paper investigates the minimum number of bulbs that can be lit regardless of button presses in a system where each button toggles multiple bulbs, providing sharp bounds for cases with up to 4 or 5 wires per switch and an asymptotic bound of 1/2.

Contribution

It extends previous work by establishing sharp lower bounds for systems with up to 5 wires per switch and derives the asymptotic bound for any number of wires.

Findings

Sharp lower bounds for up to 4 or 5 wires per switch.

Asymptotic bound of 1/2 for arbitrary wires per switch.

Method to determine the minimum lit bulbs regardless of button actions.

Abstract

Given $n$ buttons and $n$ bulbs so that the $i$th button toggles the $i$th bulb and perhaps some other bulbs, we compute the sharp lower bound on the number of bulbs that can be lit regardless of the action of the buttons. In the previous article we dealt with the case where each button affects at most 2 or 3 bulbs. In the present article we give sharp lower bounds for up to 4 or 5 wires per switch, and we show that the sharp asymptotic bound for an arbitrary number of wires is $\frac12$. (Even if you've found their buttons, you can please no more than half the people all the time!)