So what if guard 1 is random but a liar, guard 2 is truthful, and guard 3 is a liar?
Then you have guard 1 saying to go left, guard 2 saying to go right, and guard 3 not having an answer.
However, if you disregard 1 and go the opposite of 2, you die.
Truth Random Liar
You have no way of distinguishing if the last or first person is the truth teller or the liar.
Man this is really hard to explain in text, lol.
Okay, let me try another way to explain this here.
Let's say Guard #1 is the random, Guard #2 is the truth teller, and Guard #3 is a liar.
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When you ask Guard #1 which way Guard #2 will point, since Guard #1 is the random one, he will point in either direction. You don't learn any information yet.
When you ask Guard #2 which way Guard #3 will point, since Guard #2 is the truth teller and Guard #3 is the liar, Guard #3 will point in the opposite direction of the correct path. SO, when you ask Guard #2 which way Guard #3 will point, Guard #2 will point in the opposite direction of the correct path. You still don't have much information yet.
The last Guard is the key to it all. Because after you ask him, the previous information now comes into play.
When you ask Guard #3 which way Guard #1 will point, since Guard #1 is the random, Guard #3 will be unable to point in any direction. So now you know that the direction Guard #1 is pointing is completely irrelevant because you don't know if he's telling the truth or lying, because he's random. So basically just kill Guard #1, lol. Now, Guard #2 will be pointing in the opposite direction of the correct direction REGARDLESS if he's the truth teller or liar. Because you asked him to point in the direction that Guard #3 would point if he were to point in the correct direction.
If Guard #2 was the truth teller and Guard #3 was the liar, and assuming LEFT is the correct path, Guard #2 would point to the RIGHT path because that's the direction that the liar would point.
If Guard #2 was the liar and Guard #3 was the truth teller, and assuming LEFT is the correct path, Guard #2 would point to the RIGHT path because he's a liar and knows Guard #3 would point to the LEFT.