Too many trolls

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Trolls.png

This is a logic puzzle I made up for UNC's Math Challenge Problem

Puzzle

While walking through a fictional forest, you come upon a large group of trolls. The trolls all look identical, but you know that some of the trolls are knights who always tell the truth, while the rest of the trolls are knaves who always lie. Each troll makes a single statement.

The first troll says, "Hi, I'm Tucker."

The remaining 41 trolls each say, in order, "If the previous troll is a knight, then by the time I'm done speaking you will have heard more lies that truths."

Is the first troll's name really Tucker? And which of the remaining trolls are knight and which are knaves?

Help

Hint
Suppose the first troll is a knave. What does this tell you about the second troll's statement? Conclude that the third troll is impossible.
Answer
Troll 1 is named Tucker. Trolls will then alternate between being knaves and knights.
Solution
Suppose that the first troll is a knave. This makes the second troll's statement true, since the hypothesis of his implication is false. Thus troll 2 is a knight. Troll 3 cannot be a knight for if he were, then it would follow that by the time he was done speaking, you would have heard more lies than truths, but you would have heard two truths and one lie. But also the Troll 3 cannot be a knave, for this would make his hypothesis true and conclusion false, meaning that you had not in fact heard more lies than truths, but you had heard two lies and one truth, a contradiction. Thus it is impossible for the first troll to be a knave. So the first troll really is a knight (and thus named Tucker). This makes the second troll's statement false (as the previous troll really is a knight and you will not have heard more lies than truths). The third troll will then be telling the truth (the second troll is not a knight). Troll 4 will be lying again, and so on. Every odd-numbered troll will be a knight and every even numbered troll will be a knave.

See also