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Every professor in the math department is a truthteller (always tells the truth) or a liar (always says false statements). Five professors make the following statements:

•Alice: “If I am a liar, then so is Bob”.

•Bob: “If I am a liar, then so is Carol.”

•Carol: “If I am a liar, then so is Dave.”

•Dave: “If I am a liar, then so is Eve.”

•Eve: “If I am a liar, then so is Alice.”

What is the largest number of them that could possibly be liars?

I am really having trouble with this problem and can't seem to get anywhere. I would appreciate some help.

Gerry Myerson
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UT-HJ
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    Hint: you can't have consecutive liars. – Cheerful Parsnip Sep 08 '22 at 19:43
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    This kind of situation comes up all the time in my department... it's annoying... :) – paul garrett Sep 08 '22 at 19:51
  • Hint: Assume that one of them is a liar. Then assume the other, and the other, and so on. – Kamal Saleh Sep 08 '22 at 19:58
  • Write $A$ for "Alice tells the truth" (so $\neg A$ means "Alice is a liar"), and similarly for $B, C, D$ and $E$. The five statements are equivalent to: $A \lor \neg B $, $B \lor\neg C $ , $C \lor\neg D $ , $D \lor\neg E $ and $E \lor\neg A $. Then, for example, if Alice is a liar then $A \lor\neg B $ is false so Bob tells the truth, and then Carol could be also a liar, etc. – Sam Sep 08 '22 at 21:16

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Since we can't have two consecutive liars because then the first liar's statement would actually be true, so the maximum number of possible liars is two. Since Alice and Carol being liars makes it work, the maximum number of liars is two.

mathlander
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