Suppose vial C tests positive and vials A, E and H test negative. Which patient has the disease?

The correct answer is Patient 6.

Step-by-Step Explanation:

In group/pooled testing, samples from multiple individuals are mixed together into different vials. A positive result indicates that at least one person in that vial has the disease, while a negative result confirms that none of the individuals in that vial have the disease. Let us list the patients assigned to each of the relevant vials based on the puzzle design:
• Vial C contains samples from patients: 5, 6, 7, 8, 13, 14, 15, and 16.
• Vial A contains samples from patients: 9, 10, 11, 12, 13, 14, 15, and 16.
• Vial E contains samples from patients: 3, 4, 7, 8, 11, 12, 15, and 16.
��� Vial H contains samples from patients: 1, 3, 5, 7, 9, 11, 13, and 15.

We are given that Vial C tested positive, while Vials A, E, and H tested negative.

Since Vial C is positive, the diseased patient must be one of the patients represented in Vial C:

$$\text{Diseased Patient Set}=\{5,6,7,8,13,14,15,16\}$$

Since Vials A, E, and H tested negative, any patient who has a sample in any of these three vials cannot be the carrier of the disease. Let us check the list of candidates in our positive set and eliminate them:
• Patient 5: Sample is in Vial H (which is negative) → Eliminated.
• Patient 7: Sample is in Vials E and H (both negative) → Eliminated.
• Patient 8: Sample is in Vial E (which is negative) → Eliminated.
• Patient 13: Sample is in Vials A and H (both negative) → Eliminated.
• Patient 14: Sample is in Vial A (which is negative) → Eliminated.
• Patient 15: Sample is in Vials A, E, and H (all negative) → Eliminated.
• Patient 16: Sample is in Vials A and E (both negative) → Eliminated.

After eliminating all patients whose samples were included in the negative vials, only Patient 6 remains. Since Patient 6's sample was in the positive Vial C and not in any of the negative vials, Patient 6 must be the patient with the disease.