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