Created
by logician Raymond Smullyan and popularized by his colleague George Boolos, this
riddle has been called the hardest logic puzzle ever.
You and your team have crash-landed on an ancient planet.
The only way off is to appease its three alien overlords, Tee, Eff, and Arr, by
giving them the correct artifacts. Unfortunately, you don't know who is who. From
an inscription, you learn that you may ask three yes or no questions, each
addressed to any one lord.
Tee's answers are always true, Eff's are always false, and
Arr's answer is random each time. But there's a problem. You've deciphered the
language enough to ask any question, but you don't know which of the two words
'ozo' and 'ulu' means yes and which means no. How can you still figure out
which alien is which?
Stop
here if you want to figure it out for yourself!
Answer
in:
3
2
1
But
downright impossible.What good is asking a question if you can neither
understand the answer nor know if it's true?
But
it can be done.
The
key is to carefully formulate our questions so that any answer
yields
useful information. First of all, we can get around to not knowing what 'ozo'
and 'ulu' mean by including the words themselves in the questions, and secondly,
if we load each question with a hypothetical condition, whether an alien is
lying or not won't actually matter.
To
see how that could work, imagine our question is whether two plus two is four.
Instead
of posing it directly, we say, "If I asked you whether two plus two is four,
would you answer 'ozo'?"
If
'ozo' means yes and the overlord is Tee, it truthfully replies,
"ozo."
But
what if we ask Eff? Well, it would answer "ulu," or no to the
embedded question, so it lies and replies 'ozo' instead. And if 'ozo' actually
means no,
then
the answer to our embedded question is 'ulu,'and both Tee and Eff still reply
'ozo,' each for their own reasons.
If
you're confused about why this works,the reason involves logical structure. A
double positive and a double negative both result in a positive.
Now,
we can be sure that asking either Tee or Eff a question put this way
will
yield 'ozo' if the hypothetical question is true and 'ulu' if it's false regardless
of what each word actually means.
Unfortunately,
this doesn't help us with Arr. But don't worry, we can use our first question
to identify one alien lord that definitely isn't Arr. Then we can use the
second to find out whether its Tee or Eff. And once we know that, we can ask it
to identify one of the others.
So
let's begin.
Ask
the alien in the middle,"If I asked you whether the overlord on my left is
Arr, would you answer 'ozo'?"
If
the reply is 'ozo,' there are two possibilities. You could already be talking
to Arr, in which case the answer is meaningless. But otherwise, you're talking
to either Tee or Eff, and as we know, getting 'ozo' from either one means your
hypothetical question was correct, and the left overlord is indeed Arr. Either
way, you can be sure the alien
on
the right is not Arr.
Similarly,
if the answer is 'ulu,' then you know the alien
on the left can't be Arr. Now go to the overlord you've determined isn't
Arr and ask,
"If
I asked 'are you Eff?' would you answer 'ozo'?"
Since
you don't have to worry about the random possibility, either answer will establish its identity. Now that you know whether
its answers are true or false, ask the same alien whether the center overlord
is Arr.The process of elimination will identify the remaining one. The
satisfied overlords help you repair your ship and you prepare for takeoff. Allowed
one final question, you ask Tee if it's a long way to Earth, and he answers
"ozo."
Too
bad you still don't know what
that means.
Source: TEDED
