Tessellate S.T.E.M.S - Computer Science - School - Set 2 - Problem 2

A teacher sits down two students, Alice and Bob facing each other. The teacher picks a natural number, $n$ , possibly $0$ , and writes $n$ on one of their foreheads and $n+1$ on the forehead of the other.

Each student can see the forehead of the other student, but not their own. They have the following dialogue with the teacher, in three phases.

Phase I

Teacher: Do you know the numbers on your corresponding foreheads?

Alice and Bob (simultaneuosly): No

$$

Phase II

Teacher: Do you now know the numbers on your corresponding foreheads?

Alice and Bob (simultaneuosly): No

$$

Phase III

Teacher: Do you now know the numbers on your corresponding foreheads?

Alice: Yes

Bob (simultaneuosly with Alice): No

What is the lower of the two numbers written on their foreheads, and whose forehead is it written on?

Assume that:

1. The teacher is truthful, and so are the students.
2. The students are both perfectly logical, which means that if there is an inference that can be drawn from the information present in the current context, they will infer it.
3. It is common knowledge that the students are perfectly logical.

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This problem is a part of Tessellate S.T.E.M.S.