Logical reasoning — Truth tellers and sometimes-liars (choose the statement that must be true) Facts: Mary said, 'Ann and I both have cats.' Ann said, 'I don't have a cat.' Mary always tells the truth, but Ann sometimes lies. Which statements must also be facts? I: Ann has a cat. II: Mary has a cat. III: Ann is lying.

Introduction / Context:
Puzzles involving truth-tellers and liars require that we honor each speaker's stated reliability. Here, Mary always tells the truth while Ann lies only sometimes. We must determine what is compelled by Mary's statement and whether Ann's claim can be true.

Given Data / Assumptions:

• Mary's statement must be true because she always tells the truth.
• Therefore, the content “Ann and I both have cats” is factual.
• Ann's reliability varies; this particular statement can be checked against the truth we derive from Mary.

Concept / Approach:
First accept Mary's assertion as fact, then compare Ann's denial to that fact. If Ann's denial contradicts a known truth, Ann must be lying in this instance. From there, evaluate each candidate statement.

Step-by-Step Solution:

Verification / Alternative check:
There is no model consistent with “Mary always tells the truth” where Mary's content is false. Hence the contradiction with Ann's denial is unavoidable, forcing III to be true as well.

Why Other Options Are Wrong:

• I only / II only / I and II only / II and III only: Each omits one or more statements that are also compelled by Mary's guaranteed truthfulness.

Common Pitfalls:
Forgetting to prioritize the “always truthful” speaker and treating both statements as equally uncertain. Here Mary's statement sets the ground truth.

Final Answer:
All the statements are facts.