Consider you’re one of the students playing the game. Mark your name tag with a ?. If three consecutive students—such as OO?, O?O, or ?OO—have the same name tag, that student may confirm the character on their name tag in the first round. Therefore, the only case where a student cannot be immediately certain of the character on their name tag in the first round is the form OX?OX.

Divide the name tag into a chain of \(1\)-length blocks (OXOXOX..., hereinafter a \(1\)-chain) and a chain of \(2\)-length blocks (OOXXOOXX..., hereinafter a \(2\)-chain). OX?OX can be either OXOOX or OXXOX, depending on your name tag. Note that in both cases, its label is one end of a \(2\)-chain. Furthermore, we can observe that the ends of all \(2\)-chains appear in the form OX?OX.

Therefore, the students who did not raise their hands in the first round are equivalent to both ends of every \(2\)-chain. Since all students at both ends of a \(2\)-chain did not raise their hands, a student who did not raise their hand in the first round can identify at least one other student who also did not raise their hand in the second round. By checking which side the student wearing a name tag with the same letter as their own is sitting on, they can definitively determine the order of the \(2\)-chains and \(1\)-chains. From this, they can be certain which side—left or right—the \(2\)-chain connected to them is on. Therefore, all students who did not raise their hands in the first round will raise their hands in the second round.