Every member of the TL Club is either a truther, who always tells the truth when

Question

Every member of the TL Club is either a truther, who always tells the truth when asked a question, or a liar, who always answers with a lie. When I visited the club for the first time, I found its members, all men, seated around a large circular table, having lunch. There was no way to distinguish truthers from liars by their appearance, and so I asked each man in turn which he was. This proved unenlightening. Each man naturally assured me he was a truther. I tried again, this time asking each man whether his neighbor on the left was a truther or a liar. To my surprise each told me the man on his left was a liar.

Later in the day, back home and typing up my notes on the luncheon, I discovered I had forgotten to record the number of men at the table. I telephoned the club's president. He told me the number was 37. After hanging up I realized that I could not be sure of this figure because I did not know whether the president was a truther or a liar. I then telephoned the club's secretary. "No, no," the secretary said. "Our president, unfortunately, is an unmitigated liar. There were actually 40 men at the table." Which man, if either, should I believe? Suddenly I saw a simple way to resolve the matter.

How many men were there and how many were truthers; how many liars?

Explanation / Answer

There are 40 men in total including 20 truthers and 20 liars.

Now we can arrive at this conclusion by using the two conclusions given in the first paragraph.

1. Every man assured he was a truther which is obvious considering that a . a truther will say that he is a truther since he speaks only truth and

b. a liar will say that he is a truther since he lies .

Now looking at the second conclusion , "every man told that the man on his left is a liar" ,

we can say that there are even number of truthers and liars because

a. Say we asked a truther first whether the guy on his left is a liar , since he speaks truth and since it is said that he said that the man on his left is a liar , it should be true. So a liar sits to the left of a truther.

b. Now lets ask the liar whether the guy on his left is a liar. Since he speaks lies and since it is said that he said the guy on his left is a liar, we can say that the guy on his left must be a truther. So ,a truther sits to the left of a liar.

c. Now we ask the truther again and we can safely say by our first point that the guy on his left will be a liar. Since all guys said the same thing , the seating arrangement is repeated i.e truther ,liar , truther , liar in a circle.

We can say that it is the only possible combination . Say we have two truth guys beside each other. So when we ask the first guy whether the guy on his left is a truther or liar ,since he speaks truth and the guy beside him is a truther ,he would have said that he was a truther , which is a contradiction to what is given( all said that the guys on their left are liars.). So it is not possible to have two truthers beside each other.

Similarly let us assume that there are two liars beside each other. So when we ask the first guy whether the guy on his left is a truther or liar, since he is a liar and the guy beside him is a liar , he would have said that the guy on his left is a truther which is again a contradiction to what is given ( all said that the guys on their left are liars.). So it is not possible to have two liars beside each other too.

Hence the only possible combination is a truther and a liar beside each other for the whole circle. Now if there are odd number of people say 5 for example ,the arrangement would be 1 truther , 2 liar ,3 truther ,4 liar , 5 truther (who would sit beside a 1 truther since they sit in a circle ) .Since it is not possible to have two truthers beside each other , we can say that there cannot be odd number of people in the arrangement.

Hence we can safely say that there can only be even number of people . Since the secretary said that there are 40 which is even ,we can say that she is a truther . Hence the secretary is a truther and the president is a liar (since he said an odd number when there are actually an even number of people) and there are 40 people in total.

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