The bulb gets hot slowly when turned on. Turn on the switch #$1$ for 10 minutes, turn it off and turn on the switch #$2$ and get into the room. There are $3$ possible cases.

• If the bulb is on, then switch #$2$ controls the bulb.
• If the bulb is off, but hot, then switch #$1$ controls the bulb.
• If the bulb is off and cool, then switch #$3$ controls the bulb.

Thus, we can infer the switch that controls the bulb in one entry.

---

Follow up question

What if there were $4$ switches?

Follow up answer

Assuming that modern bulbs heat up slowly, we can infer this in one entry. Turn on the switch #$1$ and #$2$ keeping others off. Wait for 10 minutes. Turn off #$2$, turn on #$3$, and quickly enter the room. Following are the possible states and the corresponding switch that controls the bulb.

State		Inferred Switch
on and hot	$\rightarrow$	1
off but hot	$\rightarrow$	2
on but cool	$\rightarrow$	3
off and cool	$\rightarrow$	4

The following table shows this information in a grid format:

	On	Off
Hot	1	2
Cold	3	4