Wiring switches to more light bulbs
Stephen M. Buckley, Anthony G. O'Farrell
TL;DR
This work analyzes the minimum number of bulbs that can be lit regardless of how $n$ switches are wired to affect up to $m$ bulbs each, by formulating the problem over binary matrices and defining $\mu(n,m)$ and related quantities. It establishes sharp lower bounds and asymptotics, showing $\mu(n, m) \sim \frac{2}{3}n$ for $m=2$, $\sim \frac{4}{7}n$ for $m=4,5$, and a universal $\lim_{n\to\infty} \mu(n)/n = \tfrac{1}{2}$, with $\mu(n)/\nu(n) \to 1$. The paper connects optimal wirings to Sylvester–Hadamard matrices and Hadamard codes, introduces an explicit upper bound $U(n,m)$ governed by a meta-Fibonacci sequence, and proves that $\mu(n,m)=U(n,m)$ in several key regimes, including all $m\le 5$ and certain near-power-of-two cases. It also develops a robust toolkit—pivoting, edge functions, forward-invariant subgraphs, and tower constructions—that supports a detailed structural understanding of optimal wirings and raises open questions about the general behavior as $m$ grows. The results have implications for the design of wiring patterns and illuminate deep connections between combinatorial optimization, coding theory, and binary linear algebra.
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!)