Question

Consider a linear arrangement of seven bulbs, each of which can be in the ON or OFF state. The initial configuration of the bulbs is shown below. In every step, the states of the bulbs change based on the following rules:

• Any OFF bulb with exactly one ON neighbor at the end of the previous step is turned ON.
• Any ON bulb with both neighbors ON at the end of the previous step is turned OFF.
• The state of any bulb not meeting the conditions above is left unchanged.

The states of the bulbs at the end of Step 1 and Step 2 are also shown below.

Stage	Bulb 1	Bulb 2	Bulb 3	Bulb 4	Bulb 5	Bulb 6	Bulb 7
Initial	OFF	OFF	OFF	ON	OFF	OFF	OFF
Step 1	OFF	OFF	ON	ON	ON	OFF	OFF
Step 2	OFF	ON	ON	OFF	ON	ON	OFF

The number of bulbs which are ON at the end of Step 8 is ______

• A. 5
• B. 4
• C. 3
• D. 0

Hint:

Simulate the ON/OFF rule step by step; the pattern reaches a fixed alternating state well before Step 8, so counting the ON bulbs there gives the answer directly.

Answer 1

The Correct Option is B

Step 1: Understand the update rule.
Each bulb looks only at its own previous state and the previous state of its immediate neighbors. An OFF bulb turns ON only when exactly one of its neighbors was ON. An ON bulb turns OFF only when both of its neighbors were ON. In every other case the bulb keeps its old state. The two end bulbs, bulb 1 and bulb 7, have only one neighbor each, so an end bulb can never see "both neighbors ON" and, once it turns ON, it stays ON forever.
Step 2: Confirm the rule against the given data.
The initial row is OFF, OFF, OFF, ON, OFF, OFF, OFF, with only bulb 4 ON. Bulb 3 has one ON neighbor, bulb 4, so it turns ON. Bulb 5 has one ON neighbor, bulb 4, so it turns ON. Bulb 4 itself has both neighbors OFF, so it is left unchanged and stays ON. Every other bulb has zero ON neighbors and stays OFF. This gives OFF, OFF, ON, ON, ON, OFF, OFF, matching the given Step 1 row, so the rule is confirmed.
Step 3: Advance from Step 1 to Step 2.
Step 1 is OFF, OFF, ON, ON, ON, OFF, OFF. Bulb 2 has one ON neighbor, bulb 3, so it turns ON. Bulb 4 has both neighbors ON, bulb 3 and bulb 5, so it turns OFF. Bulb 6 has one ON neighbor, bulb 5, so it turns ON. Bulbs 3 and 5 each have one ON neighbor and one OFF neighbor, so they stay ON. This gives OFF, ON, ON, OFF, ON, ON, OFF, which matches the given Step 2 row.
Step 4: Advance from Step 2 to Step 3.
Step 2 is OFF, ON, ON, OFF, ON, ON, OFF. Bulb 1 has one ON neighbor, bulb 2, so it turns ON, and being an end bulb it now stays ON forever. Bulb 7 similarly turns ON and stays ON forever. Bulb 4 has both neighbors ON, bulb 3 and bulb 5, so this is not "exactly one" and bulb 4 stays OFF. Bulbs 2, 3, 5, 6 each already ON have only one ON neighbor, so they stay ON. Step 3 becomes ON, ON, ON, OFF, ON, ON, ON, with 6 bulbs ON.
Step 5: Advance from Step 3 to Step 4.
Step 3 is ON, ON, ON, OFF, ON, ON, ON. Bulb 2 has both neighbors ON, bulb 1 and bulb 3, so it turns OFF. Bulb 6 has both neighbors ON, bulb 5 and bulb 7, so it turns OFF. Bulb 3 has one ON neighbor, bulb 2, and one OFF neighbor, bulb 4, so it stays ON. Bulb 5 similarly stays ON. Bulb 4 has both neighbors ON, bulb 3 and bulb 5, so it is not "exactly one" and stays OFF. Bulbs 1 and 7 stay ON as end bulbs. Step 4 becomes ON, OFF, ON, OFF, ON, OFF, ON, with 4 bulbs ON.
Step 6: Check whether Step 4 is a stable configuration.
Apply the rule to ON, OFF, ON, OFF, ON, OFF, ON again. Bulb 2 (OFF) has both neighbors ON, bulb 1 and bulb 3, so it is not "exactly one" and stays OFF. Bulb 3 (ON) has both neighbors OFF, bulb 2 and bulb 4, so it stays ON. Bulb 4 (OFF) has both neighbors ON, so it stays OFF. Bulb 5 (ON) has both neighbors OFF, so it stays ON. Bulb 6 (OFF) has both neighbors ON, so it stays OFF. Bulbs 1 and 7 stay ON. The result is exactly ON, OFF, ON, OFF, ON, OFF, ON again, identical to Step 4. So this alternating pattern is a fixed point of the rule: once reached, it never changes again.
Step 7: Final Answer.
Since the pattern became fixed at Step 4 and stays unchanged for every later step, Step 5, Step 6, Step 7, and Step 8 are all identical to Step 4, namely ON, OFF, ON, OFF, ON, OFF, ON. The number of bulbs ON is 4 (bulbs 1, 3, 5, and 7).
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