Consider a linear arrangement of seven bulbs, each of which can be in the ON…
2026
Consider a linear arrangement of seven bulbs, each of which can be in the ON or OFF states. The initial configuration of the bulbs is shown in the figure. In every Step, the states of the bulbs are changed 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 state of bulbs at the end of Step 1 and Step 2 are also shown in the figure.
The number of bulbs which are ON at the end of Step 8 is ______
Answer: B. 4 — Encode OFF as 0 and ON as 1. Initial: 0001000. Step 1: bulbs 3 and 5 (OFF with exactly one ON neighbor) turn ON, giving 0011100. Step 2: bulbs 2 and 6 turn…
- A.
5
- B.
4
- C.
3
- D.
0
Attempted by 8 students.
Show answer & explanation
Correct answer: B
Encode OFF as 0 and ON as 1. Initial: 0001000.
Step 1: bulbs 3 and 5 (OFF with exactly one ON neighbor) turn ON, giving 0011100.
Step 2: bulbs 2 and 6 turn ON; bulb 4 (ON with both neighbors ON) turns OFF, giving 0110110.
Step 3: bulbs 1 and 7 (end bulbs with their single neighbor ON) turn ON; bulb 4 has two ON neighbors so it stays OFF, giving 1110111.
Step 4: bulbs 2 and 6 (ON with both neighbors ON) turn OFF, giving 1010101.
Step 5: in 1010101 no OFF bulb has exactly one ON neighbor (each has two or zero) and no ON bulb has both neighbors ON, so the state is unchanged. The configuration is a fixed point from Step 4 onward.
Therefore at the end of Step 8 the state is 1010101, with 4 bulbs ON.