The given problem requires finding the minimum number of wrong answers a student could have made to score 23 marks on a test consisting of 90 questions. Each correct answer is awarded 1 mark, while each wrong answer is penalized with -1/4 mark, and each unattempted question is penalized with -1/8 mark.

To solve this problem, we can follow these key steps:

1. Identify the scoring system:
- Correct answer: +1 mark
- Wrong answer: -1/4 mark
- Unattempted question: -1/8 mark

2. Determine the equation:
Let x be the number of correct answers, y be the number of wrong answers, and z be the number of unattempted questions. The equation for the total score can be written as:
x - (1/4)y - (1/8)z = 23

3. Determine the constraints:
- The total number of questions is 90: x + y + z = 90
- The number of correct answers cannot be negative: x ≥ 0
- The number of wrong answers cannot be negative: y ≥ 0
- The number of unattempted questions cannot be negative: z ≥ 0

4. Convert the equation to eliminate fractions:
Multiply the equation by 8 to eliminate the fractions:
8x - 2y - z = 184

5. Determine the feasible values for x, y, and z:
- Since all variables represent the number of questions, they must be integers.
- The total number of questions is 90: x + y + z = 90

6. Apply the constraints:
- x ≥ 0, y ≥ 0, z ≥ 0

7. Solve the system of equations:
- List all possible integer solutions for x, y, and z that satisfy the constraints.
- Check the validity of each solution by substituting the values into the equation from step 4 and verifying if it equals 23.

By following these steps, the minimum number of wrong answers can be determined.