Directions: Chatia, Matia and Toni participated in a race and one of them won…

Concept: In a truth-teller / liar / alternator puzzle, test one person's community as an assumption, derive the truth value of every statement made by everyone under that assumption, and accept it only if every statement stays consistent with the assigned communities and the actual race winner. A Sororian's statements are all true, a Nororian's are all false, and a Cororian's alternate between true and false across successive statements.

Application: Test each possible community for Toni in turn.

1. Assume Toni is Sororian: both of Toni's statements are true, so Matia won the race. Since Matia cannot also be Sororian, testing Matia as Nororian makes “Toni is not a Nororian” false, i.e. Toni is a Nororian — contradicting the assumption that Toni is Sororian. Testing Matia as Cororian (with “Chatia won the race” false, since Matia actually won) forces Chatia into the Nororian slot, but Chatia's claim “Toni always speaks the truth” would then be a true statement sitting among a Nororian's supposedly all-false statements — a contradiction either way, so Toni is not Sororian.

2. Assume Toni is Nororian: both of Toni's statements are false, so Matia did not win the race. Chatia's claim “Toni always speaks the truth” is factually false under this assumption, so Chatia cannot be Sororian (a Sororian's statements must all be true); Chatia must then be the alternator, leaving Matia as Sororian. But a Sororian's statements are all true, so Matia's claim “Toni is not a Nororian” would have to be true — directly contradicting the assumption that Toni is a Nororian. So Toni is not a Nororian either.

3. By elimination, Toni is the alternator. Assigning Matia as the truth-teller and Chatia as the liar now holds together: Matia's true statement gives Chatia as the winner; Chatia's false statements (“Toni is the winner” and “Toni always speaks the truth”) both check out as false, since Chatia herself won and Toni is an alternator rather than an unconditional truth-teller; and of Toni's own two statements, the claim “Matia won the race” is false (Chatia won), so the other statement is true — exactly the one-true-one-false split an alternator's statements must show. Chatia's own first statement — that she would have won the race had Toni not obstructed her — carries the implicit premise that she did not win; since she in fact won the race (per Matia's true statement), that premise fails, so this statement is false too, regardless of whether Toni obstructed her — consistent with Chatia being the all-false liar.

Cross-check: Every statement across all three participants resolves without contradiction only under this assignment (Matia = truth-teller, Chatia = liar, Toni = alternator, with Chatia as the winner), confirming it as the unique solution.

Toni belongs to the Cororian community.