A. The number of false statements here is one.
B. The number of false statements here is two.
C. The number of false statements here is three.
D. The number of false statements here is four.
In this tutorial we look at how to solve this kind of ‘self-referential’ logic problem.
Mindware Strategy Tip
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The trick to solving this type of logical problem is to start with A and systematically work through the logical implications of of sentence A, seeing in the process what else is logically contradicted to help you rule out different options. Once you have worked through the implications of sentence A, move on to sentence B, and so on – until you find a sentence that is logically consistent with all its implications.
The general strategy is to look for a sentence (A, B, C or D) that is internally consistent in all its implications.
Let’s Do It!
Starting with sentence A:
If A is true, two of the remaining sentences (B, C and D) must be true while one of them is false.
If B is true, then C and D must be false (since A is assumed to be true). But this contradicts A. If C is true, then B, C and D must be false – but this contradicts A which states that there is only 1 false statement. If D is true, then A cannot be true!
So A cannot be true.
Moving onto sentence B:
If B is true, two of A,C and D must be false and one true. If A is true this contradicts B. If A is false, this is consistent with B. Then either C is true and D is false, or C is false and D is true. If C is true then this contradicts B. If D is true then this contradicts B.
So B cannot be true.
Moving onto sentence C:
If C is true, sentences A, B and D must all be false. If A is false, this is consistent with C. If B is false, this is consistent with C. If D is false, this is consistent with C. Easy!
So if there’s only one answer, this is it.
Checking sentence D:
If D is true, then D is false, which is a contradiction!
So D cannot be true.
So, by a process of elimination, sentence C is our answer!
Diamond, Coin & Rock Problem
In this problem, you may think you could say to the rich man:
“You will give me the diamond”
But the truth of this sentence depends on the choice of the rich man. If you chooses not to give you the diamond, it will be false, and you will get nothing.
The correct answer is:
“You won’t give me the rock or the coin.”
If this is a false sentence, then the wise man will give you either the rock or the coin, but this contradicts the requirement that if you say something false, the rich man won’t give you any of the three. So the sentence can’t be false.
If this is a true sentence, then the wise man must give you either the diamond, rock or coin. Since it can’t be the rock or coin, he will give you the diamond.
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