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: 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 ______
To solve this problem, we need to follow the rules provided for changing the state of the bulbs over multiple steps. Let's analyze it step-by-step.
Initial Configuration:
Let the state of the bulbs be represented by a list where 1 is ON and 0 is OFF. The initial configuration, as given, appears as:
\([1, 1, 0, 0, 1, 1, 1]\)
Using the rules:
- Rule 1: Any OFF bulb with exactly one ON neighbor turns ON.
- Rule 2: Any ON bulb with both neighbors ON turns OFF.
- Other bulbs remain unchanged.
Step-by-step Transition:
Step 1:
- For bulb 3 (OFF), it has one ON neighbor (bulb 2). It turns ON.
- For bulb 5 (ON), both neighbors (bulbs 4 and 6) are ON. It turns OFF.
New configuration after Step 1: \([1, 1, 1, 0, 0, 1, 1]\)
Step 2:
- For bulb 4 (OFF), it has exactly one ON neighbor (bulb 5 was OFF). It turns ON.
- For bulb 6 (ON), both neighbors (bulbs 5 and 7) are OFF. So, it remains ON.
New configuration after Step 2: \([1, 1, 1, 1, 0, 1, 1]\)
Continuing similarly, we perform this process up to Step 8, applying the rules repeatedly:
\([1, 1, 1, 1, 0, 1, 1] \rightarrow [1, 1, 1, 0, 1, 1, 1] \rightarrow [1, 1, 0, 1, 1, 0, 1]\) ... and so on.
Finally, after Step 8, the bulb configuration is determined to be:
\([0, 1, 1, 0, 1, 0, 1]\)
Counting the number of ON bulbs at Step 8 gives us 4.
Thus, the number of bulbs that are ON at the end of Step 8 is 4.
Important Questions from Logical Deduction
- The following observation is made about the scores obtained by 100 students in an exam:
'For each student, there exists another student in the class such that their scores are at most ten marks away.'
If the above statement is false, which one of the following statements is necessarily true? - Consider the following two phonological rules:
Rule 1: Vowel Epenthesis of [ɪ] after sibilant-ending stems
Rule 2: Progressive Voicing Assimilation of the plural affix
Which ONE of the following rule ordering relations apply in the case of regular English pluralization as in ‘horse’ – ‘horses’? - Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you", Ramya told Josephine. Consider a five-digit number PQRST that has distinct digits P, Q, R, S and T, and satisfies the following conditions:
$P < Q$
$S > P > T$
$R < T$
If integers 1 through 5 are used to construct such a number, the value of P is:Residency is a famous housing complex with many well-established individuals among its residents. A recent survey conducted among the residents of the complex revealed that all of those residents who are well established in their respective fields happen to be academicians. The survey also revealed that most of these academicians are authors of some best-selling books.
Based only on the information provided above, which one of the following statements can be logically inferred with certainty?
