lying in the hospital - answers

if you use the tree-death (printable) version, you could actually hold this in your very own hands...

Problem 1

From the first statement, the woman could be anything: a truth-teller who wants to become a liar, an alternator who wants to become a liar (first statement to us, so it must be true), or a liar who wants to become something other than a liar.

But the second statement is a bit more limiting: the woman cannot have become a liar, because if she has, she would be telling the truth about becoming a liar, which would be impossible for a liar to do. This said, she also cannot be a truth-teller, because she would be admitting to being a liar, which would be a lie. So, she must have emerged from the operation as an alternator.

This being the case, she must have gone into the operation as a truth-teller and had a failed liar operation (she can’t have had an alternator operation and stayed an alternator).

Problem 2

Clearly, the old man isn’t a truth-teller, because this would mean he had no surgery and then lied to us. He also can’t be an alternator in the first instance, because this would mean he had no surgery and then told us the truth on his second statement, which is impossible. So, he must have started as a liar, which means he did have surgery. The surgery must have been unsuccessful (otherwise, he would have admitted to being a truth-teller), which makes him an alternator in the second encounter, and his first statement to us as an alternator is true.

Problem 3

Obviously, one of the twins is a liar now, for if their first statements to us contradict one another. Let’s say Twin B is the liar. This means that Twin A is either a truth-teller or an alternator. It also means Twin B had the surgery, which was successful. This means that Twin B started as a truth-teller, which means that Twin A is a liar (based on the mother’s initial statement to us), but this doesn’t work.

Clearly Twin A is a liar now. This means that Twin B had the surgery, and it was unsuccessful. Twin B clearly started as a truth-teller, had unsuccessful surgery, and emerged an alternator (his first statement to us is true, and his second one is false).