$-$When switch $1$ is closed, three lights will come on$.$

$-$If switch $2$ is then closed $($switch $1$ and $2$ are closed now$),$ two of the lights will drop out, leaving one light on$.$

$-$If switch $3$ is then closed $($switches $1,2,$and $3$ all closed at this point$),$ one of the lights that dropped out in operating in the second state above will come on$,$ thus showing two lights on$.$