Suppose vial A tests positive and vials D and G test negative. Which of the following vials should we test next to identify the patient with the disease?

The correct answer is Vial E.

To understand why this is the correct next step, let us analyze the distribution of blood samples for each vial and perform a step-by-step elimination process. The 16 patients are numbered 1 to 16, and their blood samples are distributed into 8 vials (A to H) as follows:
• Vial A: {9, 10, 11, 12, 13, 14, 15, 16}
• Vial B: {1, 2, 3, 4, 5, 6, 7, 8}
• Vial C: {5, 6, 7, 8, 13, 14, 15, 16}
• Vial D: {1, 2, 3, 4, 9, 10, 11, 12}
• Vial E: {3, 4, 7, 8, 11, 12, 15, 16}
• Vial F: {1, 2, 5, 6, 9, 10, 13, 14}
• Vial G: {2, 4, 6, 8, 10, 12, 14, 16}
• Vial H: {1, 3, 5, 7, 9, 11, 13, 15}

Step 1: Analyze the positive test result
We are given that Vial A tests positive. This means the patient with the disease must be one of the patients whose blood sample is in Vial A:
Disease carrier belongs to: {9, 10, 11, 12, 13, 14, 15, 16}

Step 2: Eliminate candidates using negative test results
We are given that Vials D and G test negative. This means that none of the patients whose blood samples are in Vial D or Vial G can have the disease.
• Patients in Vial D: {1, 2, 3, 4, 9, 10, 11, 12}
• Patients in Vial G: {2, 4, 6, 8, 10, 12, 14, 16}

By eliminating these patients from the candidate set of Vial A:
• Since Vial D is negative, we eliminate patients 9, 10, 11, and 12. This leaves {13, 14, 15, 16} as the potential candidates.
• Since Vial G is negative, we eliminate patients 14 and 16 from this remaining set.

This leaves us with exactly two potential candidates who could have the disease: Patient 13 and Patient 15.

Step 3: Determine the next test
To identify which of these two patients has the disease, we need to test a vial that contains exactly one of them (either Patient 13 or Patient 15, but not both). Let us check the contents of the given options:
• Vial E contains Patient 15, but does not contain Patient 13.
• Vial H contains both Patient 13 and Patient 15.
• Vial C contains both Patient 13 and Patient 15.
• Vial B contains neither Patient 13 nor Patient 15.

Since Vial E contains Patient 15 but not Patient 13, testing it will uniquely identify the diseased patient:
• If Vial E tests positive, the diseased patient is Patient 15.
• If Vial E tests negative, the diseased patient is Patient 13.

Therefore, we should test Vial E next.