Question

E-1, E-2 and E-3 are three engineering students writing their assignments at night. Each of them starts at a different time and completes at a different time. The digit in their name and the order of their starting and completing the assignment is certainly not the same. The last student to start is the first to complete the assignment.
Who is the first student to start writing the assignment?

• A. E-1
• B. E-2
• C. E-3
• D. Cannot be decided

Hint:

Use the fact that no student's start or finish position can match their own name digit, and that the last starter finishes first, to build a valid order.

Answer 1

The Correct Option is C

Instead of building the full derangement first, we can reason directly about which student could possibly start first by testing each candidate.

1. The rule is that no student's starting or completion position can equal their own name digit, and the last student to start must be the first to complete.
2. Test E-1 as first to start: this would mean S(E1) = 1, which equals E-1's own name digit 1, directly breaking the rule. So E-1 cannot be first to start.
3. Test E-2 as first to start: S(E2) = 1 does not equal 2, so this is allowed on its own. The remaining positions 2 and 3 for starting go to E-1 and E-3; since E-3 cannot be at position 3, E-3 takes position 2 and E-1 takes position 3, meaning E-1 starts last. Since E-1 starts last, E-1 must complete first, so C(E1) = 1. But E-1's own digit is 1, so this breaks the rule. So E-2 cannot be first to start.
4. Test E-3 as first to start: S(E3) = 1 does not equal 3, so this is allowed. The remaining positions 2 and 3 go to E-1 and E-2; since E-2 cannot be at position 2, E-2 takes position 3 and E-1 takes position 2, so E-2 starts last. Since E-2 starts last, E-2 must complete first, so C(E2) = 1, which does not equal 2, so this is valid with no contradictions.

Since only the scenario with E-3 starting first avoids all contradictions, E-3 is the first student to start writing the assignment, confirming option (3) as the correct answer.