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 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 ______
Show Hint
Write out the bulb states as a row of 0s and 1s and apply the two rules step by step. Watch for the row where applying the rules again gives back the same row.
Step 1: Understand the question:
Seven bulbs sit in a row and only the middle bulb, bulb 4, starts ON, so the initial row reads OFF OFF OFF ON OFF OFF OFF.
Every step, an OFF bulb with exactly one ON neighbor turns ON, an ON bulb with both neighbors ON turns OFF, and every other bulb stays the same.
End bulbs, bulb 1 and bulb 7, have only one neighbor each, so they can never be turned off by the both neighbors ON rule.
Step 2: Confirm the given steps.
Using 1 for ON and 0 for OFF, Step 0 is 0 0 0 1 0 0 0. Bulbs 3 and 5 each get exactly one ON neighbor, bulb 4, so they turn ON, giving Step 1 as 0 0 1 1 1 0 0, which matches the figure.
Applying the rules again gives Step 2 as 0 1 1 0 1 1 0, since bulb 4 now has both neighbors ON and turns OFF while bulbs 2 and 6 each gain exactly one ON neighbor, which also matches the figure.
Step 3: Continue tracing forward to Step 5.
From 0 1 1 0 1 1 0, bulbs 1 and 7 each get exactly one ON neighbor and turn ON, giving Step 3 as 1 1 1 0 1 1 1.
From Step 3, bulbs 2 and 6 now have both neighbors ON and turn OFF, while bulbs 1 and 7 stay ON, giving Step 4 as 1 0 1 0 1 0 1.
From Step 4, every ON bulb has zero ON neighbors and every OFF bulb has two ON neighbors, so nothing meets either rule and Step 5 stays 1 0 1 0 1 0 1.
Step 4: Recognize the pattern stops changing.
Since Step 5 came out identical to Step 4, the row has reached a fixed pattern that will not change again, so Step 6, Step 7, and Step 8 all stay at 1 0 1 0 1 0 1.
Step 5: Check option (A) 5.
Counting the row 1 0 1 0 1 0 1 gives four bulbs ON, not five, so this option is wrong.
Step 6: Check option (B) 4.
The row 1 0 1 0 1 0 1 has ON bulbs at positions 1, 3, 5, and 7, which is four bulbs, so this option is correct.
Step 7: Check option (C) 3.
Three is actually the count of OFF bulbs in the fixed row, not the count of ON bulbs, so this option is wrong.
Step 8: Check option (D) 0.
Once the pattern reaches 1 0 1 0 1 0 1 no bulb ever meets the turn off condition again, so the bulbs never all go OFF, and this option is wrong.
Final Answer:
The row settles into a fixed pattern with four bulbs ON from Step 4 onward.
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