Question

Five students are seated in a circle. Each student has either a red disc or a green disc painted on

her forehead. Each student can see the other four discs, but not her own. If a student is wearing a green disc, any state she makes is true; if she is wearing a red disc, any statement she makes is
false.
Student 1 says: I see 3 green and 1 red discs
Student 2 says: I see 4 red discs
Student 3 says: I see 1 green and 3 red discs
Students 4 say: it is raining outside
Student 5 says: I see 4 green discs
Which students have green discs and which students have red discs?​

Answer 1

Students 3 and 4 have green discs. Students 1, 2, and 5 have red discs.

Explanation:

If Student 1 is telling the truth, Students 2 and 3 must be lying. If that were true, Student 1 would see at least 2 red discs, not 1. So Student 1 is lying, and therefore has a red disc.

Therefore, Student 5 is also lying, and must also have a red disc.

If Student 2 is telling the truth, the other four students must be lying. That means Student 2 would have a green disc, and the other four would have red discs. But if that were true, then Student 3 would be telling the truth. Therefore, Student 2 is lying.

So far, we know Students 1, 2, and 5 are lying. If Student 3 is telling the truth, then she and Student 4 each have a green disc. If Student 3 is lying, then all five students would have red discs, which would mean Student 2 was telling the truth.

Therefore, Students 3 and 4 have green discs. Students 1, 2, and 5 have red discs.