September 2004 Puzzle Solutions

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1. Suppose again that A, D, and E think the news is bad. A says, "At least one of A and B thinks the news is bad." B says, "At least one of A and B thinks the news is not bad." D and E each say, "At least two of C, D, and E think the news is bad." C says, "At least one of C, D, and E thinks the news is not bad."

These answers preserve Anonymity because A and B are treated symmetrically, as are C, D and E. They satisfy Three Bad by describing the two groups disjointedly and establishing that at least one of A and B and only two of C, D and E think the news is bad. In the same way, they show that at least two managers think the news is ¿not bad.¿

2. To prove that exactly four people believe the news is bad, it is necessary for someone to say that one person in some group doesn't believe the news is bad. Because the group cannot contain more than three people by the Limited Reference condition, the remaining two people in the other group must both think the news is bad. This result violates our Anonymity condition. So, it¿s not possible.

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3. To prove that at least four people believe the news is bad while keeping all the conditions, have some people make more than one statement. For every subset of three people, someone must say that two out of three of them think the news is bad. If only three or fewer people altogether thought the news was bad, then this statement could not be made about some trio.

Tomas Rokicki, a reader and omniheuristic confidant, suggested a nice generalization of these examples: for which numbers of managers M, number in that group who think the news is bad B with Reference Limit L, is it possible to prove to the boss that at least (or as an alternative, exactly) B people think the news is bad while preserving the Anonymity and the Some Good conditions?

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