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
- 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. In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of P, Q, and R ?
- Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.
Column-I Column-II P. This house is in a mess. 1. Alright, I won't bring it up during our conversations. Q. I am not happy with the marks given to me. 2. Well, you can easily look it up. R. Politics is a subject I avoid talking about. 3. No problem, let me clear it up for you. S. I don't know what this word means. 4. Don't worry, I will take it up with your teacher.
Identify the option that has the correct match between Column-I and Column-II. - In the following truth table, what does X stand for?
P Q X 1 1 1 1 0 0 0 1 0 0 0 1 A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K).
Which one of the following options displays the color codes that are consistent with the color model?
