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On page 30 of Raymond Smullyan's book To Mock a Mockingbird, we are told that there are four brothers, indistinguishable in appearance, of which

  • Arthur is an accurate truth-teller,
  • Bernard is an inaccurate truth-teller,
  • Charles is an accurate liar,
  • David is an inaccurate liar,

where "inaccurate" means that one always believes that true proposition are false and vice versa.

In the puzzle 2 at page 30, the premise is

  • Arthur is married and wealthy,
  • Bernard is married and not wealthy,
  • Charles is not married and wealthy,
  • David is not married and not wealthy.

Upon meeting one of the four brothers, what three-word question is enough to find out whether he's married?

The author gives this solution, which makes use of the property of being wealthy:

Are you wealthy?

However, since the property of being wealthy is owned only by those two brothers which are accurate (though one is truth-teller and one is liar), isn't the following question, which doesn't rely on the property of being wealthy, enough?

Are you accurate?

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  • $\begingroup$ Ok, so the answer could be yes, or no? $\endgroup$
    – AntsPiano
    Apr 9, 2021 at 16:58
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    $\begingroup$ Welcome to Puzzling.SE! We generally don't allow puzzles that don't have attribution. If you could edit in a source, that would be much appreciated! $\endgroup$
    – Voldemort's Wrath
    Apr 9, 2021 at 17:02
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    $\begingroup$ It is actually attributed, just not very explicitly. I'll edit it to make this clearer. $\endgroup$
    – Gareth McCaughan ♦
    Apr 9, 2021 at 18:40

2 Answers 2

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So far as I can see, your alternative question will also work. I don't see any obvious reason to prefer it to the one Smullyan suggests, but it seems fine.

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I'll go into a bit more detail why your proposed question works.

First, let's look at accuracy. The most important point is this:

Everyone believes themselves to be accurate.

Explanation of above point:

An accurate person believes themself to be accurate, correctly. They are accurate that they are accurate. On the other hand, and inaccurate person believes themself to be accurate as well! They are inaccurate about themself being accurate - since "I am accurate" is a false statement they believe it to be true.

Therefore, a wonderful thing occurs:

Since everyone believes the answer to be "yes" to your question, truth-tellers will answer "yes" and liars will answer "no". The actual content of the question doesn't matter at this point of the analysis, just that everyone believes the answer to be "yes".

And to finally get to how this comes back to finding out if they're married or not:

Both married brothers (Arthur and Bernard) are truth-tellers and will answer "yes". Both unmarried brothers (Charles and David) are liars and will answer "no". Therefore if you get a "yes" answer the brother you met is married, and if you get a "no" answer they are unmarried.

This is similar to the author's proposed question because:

Everyone believes themself to be wealthy.

And then the rest follows from there in a similar fashion.

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  • $\begingroup$ About asking "Is 2+2=4?". Truth-tellers answer "yes" [...]; how? Bernard believes 2+2 != 4, so he would answer no, David would have the same belief as Bernard, but he would answer yes. $\endgroup$
    – Enlico
    Apr 10, 2021 at 7:26
  • $\begingroup$ I was worried that part would be confusing... would it be best for me to just edit it out? My point was you've bypassed accuracy; since everyone believes that what you say is true they'll respond as plain truth tellers out liars. The "2+2" example was a simpler case of something a regular person would believe was true, and what your question would be similar to if the bit about accuracy was removed entirely. $\endgroup$
    – bobble
    Apr 10, 2021 at 14:15
  • $\begingroup$ I don't understand. Given "2 + 2 = 4" is the truth; Bernard will believe it's false so he'll answer no to the question is 2 + 2 = 4? $\endgroup$
    – Enlico
    Apr 10, 2021 at 15:54
  • $\begingroup$ Okay, I'm removing that section. It was meant to show for it was a clever simplification, but the answer works without it and it's clearly causing more confusion than it wants. Again, that section was if everyone was accurate. $\endgroup$
    – bobble
    Apr 10, 2021 at 17:38

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