Truthteller, Liar and Randomizer *

A Recreational Logic Problem

      There are three people who will answer any question you ask with "yes" or "no." One will always tell the truth, one will always lie and the last one will answer either way at random (meaning that it's quite possible for this third person to give the same answer two or even three times in a row... so asking the same person the same question might not be very productive). We'll label these three people as X, Y and Z since we haven't a clue (yet) as to who they are... but, X, Y and Z do know who each of the other two people are (hint!). Your assignment is to determine who is the truthteller (T), the liar (L) and the randomizer (R) by asking one or more of these people only three questions.
[ To be clear: That's only three questions in total  that you are allowed to ask! ].

This may require you to come up with (devise) more than just three questions, since your second and third questions can be based upon the answers you receive from the first and second questions. Therefore, in order to solve this problem, you must show how any group of three answers, to whatever set of questions you ask, allows you to prove the true identity of X, Y and Z.


Please do spend some time thinking about this, if you don't, then what was the point of reading the page? If you really have tried to figure out how to at least start working on the problem, you may click here for more hints on how to solve it... But I will never give you the complete solution (unless you show me some work on your part! — Or am I being a Liar or a Randomizer? Just teasing!).

Although Martin Gardner presented this puzzle in Scientific American some years before it was included in his work, the Sixth Book of Mathematical Games from Scientific American (W.H. Freeman, 1971), chapter 20, section 4, "Truthers, Liars and Randomizers," page 197, he said there, "Logic problems involving truth-tellers and liars are legion, but the following unusual variation — first called to my attention by Howard De Long of West Hartford, Connecticut had not to my knowledge been printed before it appeared in Scientific American." So, if you really want to know when it first came into existence and by whom, you'll have to find out from someone else. Oh, this may be the same Howard DeLong who wrote, A Profile of Mathematical Logic (Dover Books on Mathematics).

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