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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.

Question 11.

Which two people tapped an equal number of times in total?

A
B
C
D
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Text Explanation
We are given that Alia tapped 6 times, Dilshan tapped 11 times, and Ehsan tapped 9 times. We also know that the total number of taps is 40. So, the sum of taps by Badal and Clive can be calculated as,
6 + Badal + Clive + 11 + 9 = 40
Badal + Clive = 14
We are given that taps by Clive are more than taps by Badal.
So, the possible pairs for taps by Clive and Badal such that the sum is 14 are (8, 6), (9, 5), (10, 4)...(14, 0).
We are given that no one gave more than two 'Yes' answers. We know that yes means one tap. So, the maximum number of 1 taps out of the 4 questions answered can be 2. The minimum possible number of taps for any person for the four questions would be 1 + 1 + 2 + 2 = 6. It is not possible for any person to have fewer than 6 taps as the maximum number of yes is 2.
So, out of all the possibilities, the only possible case for Badal is 6, and Clive is 8, as in all the other cases, the number by Badal was less than 6.
We are given that Alia answered Yes to Clive and Dilshan's questions. We know that the total number of taps by Alia is 6, and she tapped once for both those questions. The number of taps by Alia to the questions by Badal and Ehsan must be greater than 1, as we know that a person must have a maximum of two single taps. The sum of taps by Alia to Badal and Ehsan can be calculated as,
Badal + 1 + 1 + Ehsan = 6
Badal + Ehsan = 4
We know that both the values are greater than 1 and their sum is 4, so the only possibility is for both of them to be equal to 2.
So, Alia tapped twice for both Badal and Ehsan.
We are given that Dilshan answered no to Badal's question which means he tapped twice. The total number of taps by Dilshan is 11. So, the sum of taps from Alia, Clive and Ehsan's questions can be calculated as,
Alia + 2 + Clive + Ehsan = 11
Alia + Clive + Ehsan = 9
We know that the maximum possibility for each of the values is 3, and for the sum to be 9, all the values must be equal to 3.
We are given that taps received for the questions by Badal, Dilshan and Ehsan are equal, so let us assume the value to be a.
Putting all the calculated values in the table, we get,
image Alia and Badal are the only people with an equal number of taps.
Hence, the correct answer is option D.
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