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From the round of questioning we realise we have two options: Everyone answered No and Richard answered Yes, or everyone answered Yes and Richard answered No.
Let's assume Richard answered Yes:
Regardless of whether he is a knight or liar, Richard would have to be be sat between a knight and a liar for this to happen.
So (using the notation where we represent the people in the circle as a line of K's and L's, with Richard's position marked by brackets) we have either:
"...?K(L)L?..." or
"...?K(K)L?..."
Now, for everyone else to have answered No:
Again, regardless of whether they are knights or liars, everyone else would have to be sat between two knights or between two liars.
Applying this to our above options gives us:
"...LKLKLKLKLK(L)LLLLLLLLL..." or
"...KKKKKKKK(K)LKLKLKLKL..."
As we can see, on both these options the left side and right side can never meet up at the other side of the circle as they have differing patterns.
Therefore it is impossible that Richard answered Yes.
So, Richard answered No:
This gives us 4 options:
"...?K(K)K?..." or
"...?L(K)L?..." or
"...?L(L)L?..." or
"...?K(L)K?..."
We now note that Richard must have said No, he is not sitting between two Knights. This eliminates two of our options, leaving us with:
"...?L(K)L?..." or
"...?K(L)K?..."
What can we infer from the fact that everyone else answered Yes to the round of questioning?
Simply that everyone else must be sat between one Knight and one Liar.
So, we get:
"...KKLLKKLL(K)LLKKLLKK..." or
"...LLKKLLKK(L)KKLLKKLL..."
You will notice in the first option, no matter how many people there are, to link together to form a circle there must be one more liars than knights in the circle
(e.g. "L(K)L", or "KLL(K)LLK", or "LKKLL(K)LLKKL", ...)
In the second option the reverse is true: there is one more knights than liars in the circle
(e.g. "K(L)K", or
"LKK(L)KKL", or
"KLLKK(L)KKLLK", ...)
Thomas, knowing if his good friend Richard is a Knight or Liar (and thus being able to see already which of the above options is the correct one) now states that the Knights are outnumbered by the liars:
There are 4 possibilities:
Richard is a Knight, Thomas is a Knight: There would be x knights and x+1 liars in the circle - including Thomas this would change to x+1 of each - there is no majority and Thomas wouldn't lie about this - IMPOSSIBLE
Richard is a Knight, Thomas is a Liar: There would be x knights and x+1 liars in the circle - including Thomas this would change to x+2 liars - the liars are in the majority, but Thomas wouldn't tell the truth - IMPOSSIBLE
Richard is a Liar, Thomas is a Knight: There would be x+1 knights and x liars in the circle - including Thomas this would change to x+2 knights - the knights are in the majority, and Thomas wouldn't lie about this - IMPOSSIBLE
Richard is a Liar, Thomas is a Liar: There would be x+1 knights and x liars in the circle - including Thomas this would change to x+1 of each - there is no majority and Thomas would lie about the fact.
Therefore, both Richard and Thomas are Liars, and the group has an equal split of Knights and Liars (i.e. n/2 of each) |
