Is There Gold Here?

Logic Level 2

Edgar Abercombie visited an island which had the following features:

  • Every inhabitant was a knight who always tells the truth, or a knave who always lies.
  • Every inhabitant was sane and had only correct beliefs, or was mad and had only false beliefs
  • When asked a yes/no question, they will simply show you a red card or a black card. One card signifies yes, and the other no, and this is dependent on the person.

Is it possible to ask a single yes/no question, to determine if there is gold on the island?

This problem is taken from Raymond M. Smullyan's book Logical Labyrinths.
No Yes

Calvin Lin by Calvin Lin

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2 solutions

Saya Suka
Apr 25, 2021

This isn't a level 2 logic. 55th solver, and current percentage is at 77%. I need to learn more from JJ.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
John Jones
Dec 29, 2017

Here's one question that could be asked: From the following list, is there an even number of true statements?

You are a knight

You are sane

You use red to indicate yes

There is gold on the island

If they show you a red card, there's gold, but if they show you a black card, there isn't. I'll prove this using a truth table. The first column in the truth table will indicate whether the inhabitant is a knight, the second whether the inhabitant is sane, the third whether the inhabitant uses red to indicate yes, the fourth will be whether there is gold on the island, the fifth will be whether the correct answer is true, the sixth will be whether the inhabitant believes the answer to be true, the seventh will be the answer the inhabitant gives as a yes or no, and the eighth will be whether the inhabitant shows the red card in order to give the answer. Notice the fourth and eighth columns match in all rows.

Y Y Y Y Y Y Y Y

Y Y Y N N N N N

Y Y N Y N N N Y

Y Y N N Y Y Y N

Y N Y Y N Y Y Y

Y N Y N Y N N N

Y N N Y Y N N Y

Y N N N N Y Y N

N Y Y Y N N Y Y

N Y Y N Y Y N N

N Y N Y Y Y N Y

N Y N N N N Y N

N N Y Y Y N Y Y

N N Y N N Y N N

N N N Y N Y N Y

N N N N Y N Y N

Notes: the first four columns represent all possible combinations of knight, knave, sane, mad, red for yes, black for yes, gold or no gold. The fifth column comes from determining whether the number of Y's in the first four columns in the corresponding row is even. If the inhabitant is sane, the sixth column will be the same as the fifth. Otherwise, it will be opposite. If the inhabitant is a knight, the seventh column will be the same as the sixth. Otherwise, it will be opposite. If the inhabitant uses the red card to answer yes, the 8th column will be the same as the 7th. Otherwise, it will be opposite.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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