examday

/TMUA

Fifteen lamps are arranged in a circle and labelled $0,1,\ldots,14$. All are initially off. Pressing switch $i$ changes the states of lamps $i$, $i+1$ and $i+2$, with labels interpreted modulo $15$.

A configuration is called reachable if some finite sequence of presses produces it. For a reachable configuration $S$, let $d(S)$ be the fewest presses needed to produce it.

Let $D$ be the largest value of $d(S)$, and let $H$ be the number of reachable configurations for which $d(S)=D$.

Which pair $(D,H)$ is correct?