DIRECTIONS for questions: Read the information given below and answer the question that follows.
Alia, Badal, Clive, Dilshan, and Ehsaan played a game in which each asks a unique question to all the others and they respond by tapping their feet, either once or twice or thrice. One tap means “Yes”, two taps mean “No”, and three taps mean “Maybe”.
A total of 40 taps were heard across the five questions. Each question received at least one “Yes”, one “No”, and one “Maybe.”
The following information is known.
1. Alia tapped a total of 6 times and received 9 taps to her question. She responded “Yes” to the questions asked by both Clive and Dilshan.
2. Dilshan and Ehsaan tapped a total of 11 and 9 times respectively. Dilshan responded “No” to Badal.
3. Badal, Dilshan, and Ehsaan received equal number of taps to their respective questions.
4. No one responded “Yes” more than twice.
5. No one’s answer to Alia’s question matched the answer that Alia gave to that person’s question. This was also true for Ehsaan.
6. Clive tapped more times in total than Badal.
What was Clive’s response to Ehsaan’s question?
Correct Answer :
No
Solution :
The correct answer is No.
Let us break down the problem step-by-step. Let the five individuals be represented by their initials: Alia (A), Badal (B), Clive (C), Dilshan (D), and Ehsaan (E). Each person asks a question to the other four and receives responses based on the number of taps: 1 tap for “Yes”, 2 taps for “No”, and 3 taps for “Maybe”.
Since each question must receive at least one “Yes” (1), one “No” (2), and one “Maybe” (3), the sum of the minimum taps for any question is:
The fourth response can be 1, 2, or 3, meaning a question can receive a total of 7, 8, or 9 taps.
Let the total taps received by the questions of A, B, C, D, and E be QA, QB, QC, QD, and QE respectively. We are given that the sum of all taps is 40.
From clue 1, Alia tapped 6 times in total and answered "Yes" (1) to Clive and Dilshan. Her remaining two taps (to Badal and Ehsaan) must sum to:
Because of clue 4 ("No one responded 'Yes' more than twice"), she cannot use another 1. Hence, she must have responded with 2 taps ("No") to both Badal and Ehsaan. Alia's responses given are therefore: to B is 2, to C is 1, to D is 1, to E is 2.
We are also told in clue 1 that Alia received 9 taps, so QA = 9. From clue 3, Badal, Dilshan, and Ehsaan received an equal number of taps, so let QB = QD = QE = x. Therefore:
Since the possible values for any question total are only 7, 8, or 9, we can test these for x. If x = 7, then QC = 10 (not possible). If x = 9, then QC = 4 (not possible). The only valid solution is x = 8, making QC = 7. Thus, the totals are: QA = 9, QB = 8, QC = 7, QD = 8, and QE = 8.
Based on these totals, we can determine the exact set of four responses each question received:
C's question (QC = 7) received 1, 1, 2, 3.
B, D, and E's questions (totals of 8) each received 1, 2, 2, 3 (since 1, 1, 3, 3 lacks a 2).
A's question (QA = 9) received 1, 2, 3, 3.
From clue 2, Dilshan tapped 11 times. He responded "No" (2) to Badal. His remaining three responses must sum to:
The only way to get a sum of 9 from three responses (where the maximum value is 3) is if all three are 3. Thus, Dilshan responded "Maybe" (3) to Alia, Clive, and Ehsaan. Dilshan's responses given are: to A is 3, to B is 2, to C is 3, to E is 3.
Now let us evaluate Ehsaan's answers. From clue 2, Ehsaan tapped a total of 9 times. Let us figure out his individual responses to A, B, C, and D.
First, consider Dilshan's question (which received 1, 2, 2, 3). We know Alia gave a 1. The remaining responses are 2, 2, 3. Clue 5 states the mismatched answer rule applies to Ehsaan: no one's answer to Ehsaan can match Ehsaan's answer to them. Since Dilshan answered Ehsaan with 3, Ehsaan cannot answer Dilshan with 3. So Ehsaan must answer Dilshan with 2.
Now Ehsaan's remaining three responses (to A, B, and C) must sum to:
Let us check the available options. Badal's question (needs 1, 2, 2, 3) already has a 2 from Alia and a 2 from Dilshan, leaving 1 and 3 for Clive and Ehsaan. So Ehsaan's response to Badal is either 1 or 3.
Clive's question (needs 1, 1, 2, 3) already has a 1 from Alia and a 3 from Dilshan, leaving 1 and 2 for Badal and Ehsaan. So Ehsaan's response to Clive is either 1 or 2.
From clue 5, no one’s answer to Alia matched Alia’s answer to them. Alia’s responses to B, C, D, and E were 2, 1, 1, and 2 respectively. We know Alia’s question received responses 1, 2, 3, 3, and Dilshan gave a 3. Since Clive cannot give a 1 (it would match Alia's 1 to him), and Badal/Ehsaan cannot give a 2 (it would match Alia's 2 to them), Clive must have given the 2. Thus, Ehsaan must have given either 1 or 3 to Alia.
So, we need Ehsaan's responses to A, B, and C to sum to 7. We know his response to A is in {1, 3}, to B is in {1, 3}, and to C is in {1, 2}. If Ehsaan's response to Clive was 2, the sum for A and B would be 5, which is impossible using only 1s and 3s. Therefore, Ehsaan's response to Clive must be 1, forcing his responses to Alia and Badal to both be 3. Ehsaan's responses given are: to A is 3, to B is 3, to C is 1, to D is 2.
Finally, let's determine Clive's responses, specifically to Ehsaan. The question asks for Clive's response to Ehsaan's question. Ehsaan's question received responses 1, 2, 2, 3. We know Alia answered 2, and Dilshan answered 3. The remaining responses for Badal and Clive to give Ehsaan are 1 and 2.
Applying the mismatch rule from clue 5 to Ehsaan: no one's answer to Ehsaan can match Ehsaan's answer to them. Ehsaan answered Clive with a 1. Therefore, Clive cannot answer Ehsaan with a 1. Clive must answer Ehsaan with a 2 (which means "No"), leaving Badal to answer Ehsaan with a 1.
Therefore, Clive responded with 2 taps to Ehsaan's question, meaning his response was "No".
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