## Wednesday, October 12, 2011

### Shepherds on Hillsides

In every group of shepherds in Hyperborea, at least one is a Sororean, who always speaks truthfully, and at least one is a Nororean, who always speaks falsely.

A visitor approached four shepherds on a hillside and asked each how many of the four were Sororeans. These answers were given:

• Three of us are Sororeans.
• One of us is.
• There are two of us.
• None of us are Sororeans.

The visitor approached four more shepherds on another hillside and asked how many were Nororeans. Their answers follow:

• We are all Nororeans.
• One of us is.
• Three of us are.
• The fourth shepherd declined to speak.

How many of the shepherds on the two hillsides were Sororeans?

This puzzle comes out of The World's Biggest Book of Brainteasers & Logic Puzzles

1. 2? I don't know i'm confused.

2. Here are my thoughts:

On the first hillside, each of the four shepherds gives a different, conflicting answer--so that at most one of them is telling the truth. Given that this is the case, the first and third statements (which boil down to "three of us always tell the truth" and "two of us always tell the truth") must be false. The fourth statement cannot possibly be true (since if it were, it would mean that the fourth speaker himself was both a liar (not Sororean, by his own statement) and a non-liar (since he told the truth), an obvious contradiction). Since at least one member of each group of shepherds is a Sororean (by assumption), and shepherds 1, 3, and 4 have lied, I conclude that shepherd 2 is the only Sororean in the first group.

On the second hillside, the first three shepherds all give dirrect, conflicting answers--so that at most one of the first three is telling the truth. Since at most one of the first three shepherds are telling the truth, at least two of them are lying, and are therefore Nororeans. It is not possible, therefore, for shepherd 2's statement to be truthful. At least one of the shepherds on the second hillside must be a Sororean (again by assumption in the question), so that shepherd 1's claim to the contrary cannot be truthful either. Since the first two shepherds are definitely Nororeans, and it is not possible for all four to be so, there are either two or three Nororeans in total on the second hillside. If there are only two, then shepherd 3's claim is false--which brings the total Nororean count to at least 3 (shepherds 1, 2, and 3), another obvious contradiction; so that there must in fact be exactly three Nororeans (shepherds 1, 2, and 4) on the second hillside. I conclude, therefore, that there are three total Nororeans on the second hillside; and, as a result, that there is one Sororean there.

By my reasoning, then, there appear to be two Sororeans in total, between the two hillsides.

Correct?

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