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 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 ______

• A. 5
• B. 4
• C. 3
• D. 0

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.

Answer 1

The Correct Option is B

Step 1: Track the pattern using a table instead of writing each row in words:
We list bulbs 1 to 7 across the top and fill in a 1 for ON and a 0 for OFF at each step.

Step	B1	B2	B3	B4	B5	B6	B7
0	0	0	0	1	0	0	0
1	0	0	1	1	1	0	0
2	0	1	1	0	1	1	0
3	1	1	1	0	1	1	1
4	1	0	1	0	1	0	1
5	1	0	1	0	1	0	1

Step 2: Notice the table stops changing after Step 4.
Row 5 is identical to row 4, so applying the rules again cannot change anything from here on, meaning every step from Step 4 onward, including Step 8, is 1 0 1 0 1 0 1.

Step 3: Count the ON bulbs in the fixed row.
Bulbs 1, 3, 5, and 7 hold a 1, which are four bulbs, and bulbs 2, 4, and 6 hold a 0, which are three bulbs, and $4 + 3 = 7$ checks out against the total of seven bulbs.

Step 4: Check option (A) 5.
The table shows only four 1's in the fixed row, not five, so this option is wrong.

Step 5: Check option (B) 4.
The table gives exactly four 1's in the fixed row, so this option is correct.

Step 6: Check option (C) 3.
Three is the count of OFF bulbs in the fixed row, not the count of ON bulbs, so this option is wrong.

Step 7: Check option (D) 0.
The table never reaches an all OFF row, since the fixed row keeps four bulbs ON forever, so this option is wrong.

Final Answer:
Reading straight off the table, Step 8 has four bulbs ON. \[ \boxed{4} \]