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In a classroom with \( n \) students, only one student has a pen, but you don't know who. How many questions do you need to ask to determine which student has the pen?

Answer :

The number of questions needed to determine which student has the pen in a classroom with n students is [tex]\( \lceil \log_2(n) \rceil \).[/tex]

To find the student with the pen, one can use a divide-and-conquer strategy similar to a binary search algorithm. This method involves asking questions that can be answered with a yes or no and that effectively halve the number of students under consideration with each question.

For example, if there are n students, one could ask, Is the student with the pen in the first half of the classroom? If the answer is yes, then the student is in one of the first [tex]\( \frac{n}{2} \)[/tex] seats; if no, then the student is in one of the last [tex]\( \frac{n}{2} \)[/tex] seats. This process is repeated, each time asking about the half of the remaining group that contains the student with the pen, until only one student is left.

The number of times this halving process can occur is given by the ceiling function [tex]\( \lceil \log_2(n) \rceil \)[/tex], which represents the smallest integer greater than or equal to [tex]\( \log_2(n) \)[/tex]. This is because each question effectively divides the search space in half, and [tex]\( \log_2(n) \)[/tex] gives the number of times n can be divided by 2 until reaching 1. Since we cannot ask a fraction of a question, we round up to the nearest whole number, hence the use of the ceiling function.

For example

If n = 4, [tex]\( \log_2(4) = 2 \)[/tex] , so [tex]\( \lceil \log_2(4) \rceil = 2 \)[/tex] questions are needed.

If n = 5, [tex]\( \log_2(5) \approx 2.32 \)[/tex] , so [tex]\( \lceil \log_2(5) \rceil = 3 \)[/tex] questions are needed.

Thus, the number of questions needed to find the student with the pen in a classroom of n students is [tex]\( \lceil \log_2(n) \rceil \)[/tex].

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Rewritten by : Brahmana

To determine which student in a classroom of n students has a pen, you may need to ask up to n - 1 questions in the worst-case scenario with a linear search strategy. However, using a binary search approach, you would need around log2(n) questions, rounded up to the nearest whole number.

The question asks about determining which student in a classroom of n students has a pen by asking questions. This is a problem of logic and can be classified under Mathematics, specifically combinatorics or problem-solving strategies. In an ideal situation, if each question asked cuts the number of possibilities in half, you will be executing a binary search.

In the worst-case scenario, you would start by asking the first student if they have the pen, then proceed to the next if the answer is no, and so on. This linear search strategy would mean you might end up asking n - 1 questions if the last student has the pen. However, if you're allowed to ask questions that divide the group into two (like 'Is the person with the pen on the left side of the room?'), you could determine who has the pen in
log2(n)
questions, rounded up to the nearest whole number, due to the nature of binary search.

Strategy 1 from the given information suggests asking one student at a time without influence from others, which would apply to the linear search method described.