## Task 1 - 090921

At a class party, the students present their math teacher with the following problem:

The students inform their teacher that they have secretly divided themselves into three groups such that every student in the class belongs to exactly one group. The students in the first group call themselves the "Trues" because they answer every question truthfully.

The students in the second group call themselves the "Falses" because they answer every question falsely. The students in the third group call themselves the "Inconstant" because each of them answers series of consecutive questions alternately (alternating) true and false; however, it is uncertain whether they will answer the first question of a series true or false.

Each student answers a question posed to them only with yes or no; questions that require other answers are not allowed. The teacher is to determine, by asking questions directed to any student in the class, which only concern the student's affiliation with one of the mentioned groups, whether the student is a "True," a "False," or an "Inconstant."

a) What is the smallest number of questions that suffice for this?

b) Provide a way to determine the affiliation of a student with this smallest number of questions!
a) One question is not enough, as it can only distinguish between two cases ("yes" vs. "no" answer), but there are three groups. That it can be done with two questions is shown in part b).

b) Ask the same question twice: "Are you an 'Unsteady'?"

A "True" will answer "no" both times, an "Untrue" will answer "yes" both times, and an "Unsteady" will answer once with "yes" and once with "no" (in some order).