# Truth or Lie Riddle

July 31, 2015

How to solve those frustrating logic puzzles involving 'liars' and 'truth-tellers'

## The Puzzles

Version 1 (From mathsisfun.com)

You are at an unmarked intersection ... one way is the City of Lies and another way is the City of Truth.
Citizens of the City of Lies always lie.
Citizens of the City of Truth always tell the truth.
A citizen of one of those cities (you don't know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

Version 2 (From mathforum.org)

A princess visits an island inhabited by two tribes. Members of one tribe always tell the truth, and members of the other tribe always lie.

The princess comes to a fork in the road. She needs to know which road leads to the castle so as to avoid the fire-breathing dragon and rescue the prince from the wizard holding him captive in the castle. (Although the princess doesn't know it, the south road leads to the castle and the north road leads to the dragon.)

Standing at this fork in the road is a member of each tribe, but the princess can't tell which tribe each belongs to. What question should she ask to find the road to the castle?

## The Trick

Don't ask about facts. If you ask about a fact, the two (liar or truth-teller) will always give opposite answers and you won't know which is which. Instead, ask about the liar or truth-teller. This will get the same answer from both a liar and a truth-teller, thereby giving you accurate information.

For example:

In version 1 above, asking where the citizen is from will get the same answer from a truth-teller and a liar.

The truth-teller would point to the City of Truth, because they're truly from there.

The Liar would also point to the City of Truth, because they're lying.

Thus, it doesn't matter which citizen you're speaking to, you know where the City of Truth is.

In version 2 above, asking the two tribesmen what the other would say will give the pricness a similar result.

If she asks a truth-teller what the liar would say, the truth-teller will say: "They would tell you to go [wrong way]".

If she asks a liar what the truth-teller would say, the liar will say: "They would tell you to go [wrong way]".

Therefore, regardless of who she asks, the princess knows which way is the wrong way.

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