Few people are willing to tell their boss bad news. The boss will just get angry at the messenger. But bad news must be heard or nothing will change.

A company¿s five division managers¿Arthur, Barbara, Carol, Daniel and Ellen--get together in a room. Three of them think the news really is bad. Two disagree. Still, all five agree that the boss should understand that at least three think the news is bad but that not all five think it's bad. We call these the Three Bad and Some Good conditions. They also agree that the boss should not learn what any single one of them thinks. We call this the Anonymity condition.

They know the boss will interview each of them individually and will be satisfied only with a statement that refers to the person questioned and at most two other people. We call this the Limited Reference condition. That is, the boss won't accept generalities such as "At least three out of all five people think the news is bad." Instead he wants statements such as, "At least two of the three of us -- Ellen, Daniel and me -- think the news is bad" or "Of Arthur and me, at least one of us thinks the news is not bad.¿ The boss is happy to hear more than one statement from a person as long as the statements stay within this Reference Limit of three.

Warm-up: What can the five managers say to prove Three Bad and Some Good while still satisfying the Anonymity and Limited Reference conditions?

Solution to Warm-up: Let's call the division heads A, B, C, D, E for short. As it happens, A, C and D think the news is bad. Suppose that A and B both say, "At least one of A and B thinks the news is 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 A, B, C thinks the news is not bad." The first two statements ensure that at least three think the news is bad (Three Bad). The last ensures that at least one thinks the news is not bad (Some Good). In these circumstances, either A or B could think the news is bad and any of C, D, or E could think the news is not bad (Anonymity).

Problems
1. Under the same rules from the boss and with the same goal, could the managers prove that EXACTLY three of the five people think the news is bad? If so, how? If not, why not?

2. Again under the same rules and with the same goal, could they prove that EXACTLY four people think the news is bad? If so, how? If not, why not?

3. Would your answer to Problem 2 change if you had to prove only that AT LEAST four of the five people think the news is bad?