mathschallenge.net

Problem

In talking to three children you ask A if they always tell lies. Although A fully understands you she answers in a language that only B understands.

B says that A just denied being a liar.

C says that although she doesn't know what A said, she is a liar and cannot be trusted.

Given that each of the children will always lie or always tell the truth, how many liars are there?

Solution

It should be clear that a compulsive liar cannot admit to being a liar, so A would deny being a liar whether or not they always lied or always told the truth. In which case B told the truth about what A said and we establish that B is always truthful.

If A is a liar then C told the truth, but if A is not a liar then C lied.

Although it is impossible to establish whether or not A is a liar or a truth teller, we can be certain that only one of A or C is a liar and the other must always tell the truth.

Hence there is exactly one liar amongst the three children.

What if instead C said that A always tells the truth and a fourth child, D, says that B is the only liar out of the three of them?