A battery powers three identical bulbs labeled B1, B2, and B3 through a network of ideal switches arranged as follows:
• Switch S_A lies in series with every branch that connects the battery to the bulbs.
• Downstream of S_A, the circuit splits into two parallel branches.
– Branch X contains only bulb B1 and switch S_B in series.
– Branch Y splits again: one sub-branch contains bulb B2 and switch S_C in series, while the other sub-branch contains bulb B3 with no additional switch.
All switches are initially open.
Answer the questions below about operating the switches to control which bulbs are lit.
(a) With all switches open, identify which bulb(s), if any, are lit. Explain your reasoning.
(b) Determine the minimum number of switch closures—and specify which switches must be closed—to light bulb B3 while keeping bulbs B1 and B2 unlit. Justify why no other switch operations are necessary.
(c) Starting from the state achieved in part (b), outline the shortest possible sequence of individual switch operations (closings or openings) required to light all three bulbs simultaneously. Clearly list each step in order.
(d) For each step in part (c), justify why that particular action is required, referencing the existence or elimination of closed conducting loops in the circuit.