This problem involves arranging a schedule for seven people (C, D, E, F, S, T, U) over seven days of a week (Monday to Sunday) based on a set of given conditions. We need to determine the day on which E has their exam.

Understanding the Constraints

• There are 7 people: C, D, E, F, S, T, U.
• There are 7 days: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.
• Each person has an exam on a different day.
• Condition 1: No one has an exam before D.
• Condition 2: Exactly three people have exams between D and T.
• Condition 3: Exactly one person has an exam between T and S.
• Condition 4: C's exam is on Wednesday.
• Condition 5: E's exam is on a day after F's exam, and E's exam is on a day before U's exam. This implies the order F → E → U.

Step-by-Step Deduction

Step 1: Placing C

We are given that C has the exam on Wednesday. Let's represent the weekly schedule:

• Monday: ___
• Tuesday: ___
• Wednesday: C
• Thursday: ___
• Friday: ___
• Saturday: ___
• Sunday: ___

Step 2: Placing D

Condition 1 states that no one has an exam before D. Since the week starts on Monday, the earliest possible day for D is Monday. If D were scheduled on any day after Monday, the days before D would be empty, which doesn't contradict the statement directly, but placing D on Monday is the most constrained starting point fitting "no one before D". Let's assume D is on Monday.

• Monday: D
• Tuesday: ___
• Wednesday: C
• Thursday: ___
• Friday: ___
• Saturday: ___
• Sunday: ___

Step 3: Placing T

Condition 2 states that only three people have exams between D and T. Since D is on Monday, we count three days after Monday: Tuesday, Wednesday, and Thursday. The person immediately following these three days is T. Therefore, T's exam must be on Friday.

• Monday: D
• Tuesday: ___ (Person 1 after D)
• Wednesday: C (Person 2 after D)
• Thursday: ___ (Person 3 after D)
• Friday: T
• Saturday: ___
• Sunday: ___

This arrangement satisfies the condition of having three people (those scheduled on Tuesday, Wednesday, and Thursday) between D and T.

Step 4: Placing S

Condition 3 states that only one person has an exam between T and S. T is scheduled on Friday. If there is one person between T and S, the sequence would be T, Person, S. This means S must be scheduled two days after Friday, which is Sunday.

• Monday: D
• Tuesday: ___
• Wednesday: C
• Thursday: ___
• Friday: T
• Saturday: ___ (Person between T and S)
• Sunday: S

This arrangement satisfies the condition of having exactly one person (scheduled on Saturday) between T and S.

Step 5: Placing E, F, and U

The remaining people are E, F, and U. The remaining available days are Tuesday, Thursday, and Saturday. Condition 5 states that E's exam is after F's and before U's (F → E → U). We must place F, E, and U in the available slots (Tuesday, Thursday, Saturday) in this specific order.

• F must be placed on the earliest available day: Tuesday.
• E must be placed on the next available day: Thursday.
• U must be placed on the latest available day: Saturday.

Step 6: Final Schedule Confirmation

Let's compile the complete schedule:

This schedule satisfies all the given conditions:

• D is first (Monday).
• F, C, E are between D (Monday) and T (Friday) - 3 people.
• U is between T (Friday) and S (Sunday) - 1 person.
• C is on Wednesday.
• The order F (Tuesday) → E (Thursday) → U (Saturday) is maintained.

Determining E's Exam Day

Based on the deduced schedule, E has the exam on Thursday.