Question

Directions: Professor Mukhopadhay works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities: lecturing, conducting quizzes, evaluating quizzes, and working on consultancy projects. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. Her weekly schedule must satisfy all of the following restrictions:

1. She conducts quizzes on exactly three mornings.
2. If she conducts a quiz on Monday, she does not conduct a quiz on Tuesday.
3. She lectures in the afternoon on exactly two consecutive calendar days.
4. She evaluates quizzes on exactly one morning and three afternoons.
5. She works on a consultancy project on exactly one morning.
6. On Saturday, she neither lectures nor conducts a quiz.

Which of the following statements must be true?

• A. There is one day on which she evaluates quizzes both in the morning and in the afternoon
• B. She works on the consultancy project on one of the days on which she lectures
• C. She works on the consultancy project on one of the days on which she evaluates quizzes
• D. She lectures on one of the days on which she conducts a quiz

Hint:

Notice the whole puzzle really has only two open choices, which three mornings get the quiz and which two afternoons get the lecture, then check each statement against every combination.

Answer 1

The Correct Option is D

Five different claims are offered about the professor's week, and only one of them holds no matter which valid version of the week she actually follows. The quickest way to sort this out is to note that the entire schedule really only has two open choices: which three days get the morning quiz (Monday-Wednesday-Friday, or Tuesday-Wednesday-Friday) and which two consecutive days get the afternoon lecture (Monday-Tuesday, or Tuesday-Wednesday). Every claim can be tested against just these two choices.

1. There is one day on which she evaluates quizzes both in the morning and in the afternoon: The single evaluation morning falls on whichever of Monday, Tuesday or Saturday is not already taken by a quiz or consultancy, and this is a free choice between two days in every version of the week. There is no reason this day must also carry one of the three evaluation afternoons, and it is entirely possible to pick the evaluation morning to be a day, like Tuesday, that never gets an evaluation afternoon. So this is not forced to be true.
2. She works on the consultancy project on one of the days on which she lectures: The lecture afternoons are always Monday-Tuesday or Tuesday-Wednesday, while the single consultancy morning is free to land on whichever non-quiz morning is left over, which can easily be a day outside the lecture pair. This overlap is not guaranteed, so the claim is not always true.
3. She works on the consultancy project on one of the days on which she evaluates quizzes: The consultancy morning and the evaluation days are decided somewhat independently of each other, since the consultancy day is whichever leftover morning is not chosen as the evaluation morning, so there is no rule that forces the consultancy day to also carry an evaluation afternoon. This can fail, so it is not always true.
4. She lectures on one of the days on which she conducts a quiz: The quiz mornings are always three of Monday, Wednesday, Friday or Tuesday, Wednesday, Friday, and the lecture afternoons are always Monday, Tuesday or Tuesday, Wednesday. Every one of the four ways to pair a quiz set with a lecture pair shares at least one common day: Monday, Wednesday, Friday with Monday, Tuesday shares Monday, Tuesday, Wednesday, Friday with Monday, Tuesday shares Tuesday, Monday, Wednesday, Friday with Tuesday, Wednesday shares Wednesday, and Tuesday, Wednesday, Friday with Tuesday, Wednesday shares both Tuesday and Wednesday. Since there is no way to avoid an overlap, this claim holds in every valid week.

Only the fourth claim survives every combination of the two open choices in the puzzle, since a quiz set and a lecture pair can never be chosen without sharing at least one day.

Let's summarize:

• The whole week really has only two open choices: which three days get the morning quiz, and which two consecutive days get the afternoon lecture.
• Testing a must be true claim means checking it holds across every combination of those two choices, not just one convenient week.
• Only she lectures on one of the days on which she conducts a quiz survives all four combinations.

So the correct statement is that she lectures on one of the days on which she conducts a quiz, which is option (EE).