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Puzzle:

King Octopus has servants with six, seven, or eight legs.

The servants with seven legs always lie, but the servants with either six or eight legs always say the truth.

One day, 4 servants met:

The blue one says: “Altogether we have 28 legs”;

the green one says: “Altogether we have 27 legs”;

the yellow one says: “Altogether we have 26 legs”;

the red one says: “Altogether we have 25 legs”.

What is the colour of the servant that says the truth?

This is a very famous logical puzzle.

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Answer:

green.

Prove:

First: One of them tells truth. Indeed, if we suppose that all of them lie, then all of them have 7 legs, which makes the blue a truth-teller. Contradiction.
Also only one statement can be true. So there is exactly one truth-teller.
Second: The truth-teller has 6 legs. Indeed, a liar has 7 legs, so if the truthteller has 8 legs, than they have 3*7+8=29 legs in total and all lie.
Third: Now we just add 3*7+6=27.

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We could go over all permutation of 4 servants being 6, 7, or 8, and then logically eliminating permutations based on contradictions, but there's a shortcut we can use.

We know that there are 3 liars since 4 different answers are given. Therefore, the servants must be 7+7+7+6 (total of 27) or 7+7+7+8 (total of 29). Only one of these totals is given (27), and it's given by the green servant.

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  • $\begingroup$ we don't know that there are only 3 liars ;) $\endgroup$ – klm123 May 9 '15 at 15:16
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    $\begingroup$ @klm123 Yes, we do. It's not explicitly stated in the problem, but all four octopi made mutually exclusive statements, so at most, only one of them is being honest. $\endgroup$ – Kevin - Reinstate Monica May 9 '15 at 23:44
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    $\begingroup$ I would say that "What is the colour of the servant that says the truth?" (singular colour) means that there is only once servant speaking the thruth, an hence, 3 liars. But it can also be deduced the way you did. $\endgroup$ – Erwin Bolwidt May 10 '15 at 5:56
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    $\begingroup$ @Kevin, exactly - at most one of them, but now at least. This is what I said. For example what if they told numbers 1,2,3,4 instead of 25,26,27,28. All logic in this answer applied similarly, but leads to wrong result. $\endgroup$ – klm123 May 10 '15 at 10:32
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The green one says the truth:

Every servant says a different number, so only one says the truth. That means that three are liars

3x7 = 21 legs (The servants with seven legs always lie) We must add to that the legs of the servant that says the truth (the servants with either six or eight legs always say the truth).

So the correct total could be 21+6 or 21+8. And thus...

the green one says: “Altogether we have 27 legs”;

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My answer is blue, Because we can say that 28=6+6+8+8=12+16=28. Servant with six or eight legs never lie, so 25=6+6+6+7, 26=6+6+7+7, and 27=6+7+7+7. So in every quality are sevens and we can say that all of them are lying except blue one.

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