# 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).

**The
Starman's Math Index**