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I am now struggling with the blue flames.
here is how i see it :
1- everyone is telling the truth : false, 5 is lying
2- i am telling the truth : true, why not
3- there are 2 or more liars : true, 1 and 5 are lying
4- i am telling the truth : true, why not
5- everyone is telling the truth : false, 1 is lying
6- there are 2 or more liars : true, 1 and 5 are lying
2. (Mountain) There are two or more liars.
3. (Grain) I am telling the truth.
With a lock pattern of 1-2-3-2-1-3
your proposed alternate solution is lie-truth-truth
Taking the prespective of the doggos:
1. truth-truth-truth
2.
For 2 its a bit tricky since it ends up with a cyclical nature we have to think of the states it will be in. The opposite to two is there is 0 or 1 liars. So if two is false we have 0 or 1 liars. If its true we have 2 or more liars.
In your proposed solution it ends up as a contradiction since it only has 1 liar, with 2 being true.
3. X-X-True
This one has the gradient flow away from both solution's telling us that the solution is away from the swapping state.
the vector is lie-(truth/lie)-truth.
Looking at lie-truth-truth results in a False statement. So we try swapping the middle state.
Looking at lie-lie-truth results in an ambiguous, and incorrect statement. There are two lies, so the middle lie statement is a contradiction and gets flipped back to the lie-truth-truth statement (due to the rules of the second statement). This gives us a false cyclical. With the second space being locked like that as we can no longer pick it as a free variable. another bit must be flipped excluding the second and we end up with a gradient.
Blue flames had me going too until I noticed I could only see all three blue flames lit from one position, Dig there
1. Everyone is telling the truth
2. 2 or more are lying
3. I am telling the truth
if 1 was true then that would mean that 2 would be lying forcing 1 to be a lie.
because we know that 1 is lying, we know that 2 has to be telling the truth, because if 2 was lying then his statement if forced to be true, so he has to be telling the truth.
and to complete 2's statement 3 has to be lying.
maybe someone could explain the thought process to me on how another answer can be achieved?
Scenario 1: Everyone is telling the truth.
If everyone is telling the truth, then statement 2 ("2 or more are lying") would be false. This creates a contradiction because if everyone is truthful, then it's impossible for two or more people to be lying. Therefore, Scenario 1 cannot be correct.
Scenario 2: 2 or more are lying.
Let's consider if exactly two people are lying. If statement 1 is a lie, and statement 3 is a lie, then statement 2 ("2 or more are lying") would be true. This scenario is logically consistent.
What if all three statements are lies? If statement 1 is a lie, statement 2 ("2 or more are lying") is true, which contradicts the assumption that it's a lie. So, all three cannot be lying.
Scenario 3: I am telling the truth.
If statement 3 ("I am telling the truth") is true, then at least one person is telling the truth. This doesn't immediately tell us about the truthfulness of statements 1 and 2.
However, if statement 3 is true, could statement 1 also be true? We already determined that Scenario 1 leads to a contradiction.
If statement 3 is true and statement 1 and 2 is a lie, then statement 2 automatically becomes true which is contradicting.