Logical Brain Teasers
Logic Brain Teasers
Do you like brainteasers? If you do, then you will probably enjoy the logic puzzles invented by the philosopher and logician Raymond Smullyan. The key to solving logic puzzles such as these is to develop a grid or table that systematically shows all the logical possibilities, and then you find the solution by eliminating, on logical grounds, all the possibilities except one, which will inevitably be the only one that “works” or solves the puzzle. Let us solve some of Professor Smullyan’s classic problems together.
For more on Raymond see:
The Magical Island of Knights and Knaves
Each of the following problems is adapted from Raymond Smullyan, What is the Name of this Book? (New York: Touchstone, 1986).
Tall Guy and Short Guy Puzzle
Imagine you have been transported to the mysterious “Island of Knights and Knaves” where every inhabitant is either a knight or a knave. Every knight always tells the truth and every knave always lies. In this problem, you have are given the following information: “The tall one is a knave and/or the short one is a knight.”
First of all, there are only four possibilities:
- Both are knights.
- Both are knaves.
- Tall one is a knight and short one is a knave.
- Tall one is a knave and short one is a knight.
Solution. Those are all the possible combinations, and one of them must be the solution.
To proceed we will formulate a hypothesis, and then we will test it. If the hypothesis fails, then we will try another, and another, until we find a solution. Let us hypothesize that the tall one (hereafter “Tall”) is a knave. This is just a guess. Now, what are the logical consequences of this hypothesis? Well, if Tall is a knave, then Tall must be lying when he says “I am a knave and/or the short one is a knight.” (Tall is lying because knaves always lie and we are assuming Tall is a knave.) It logically follows that the entire statement ("I am ...etc.”...) is false (since it is a lie). That is, it is false as a whole. Now, an “and/or” statement asserts that one side or the other side or both sides of the statement are true. The left side says in effect that Tall is a knave, while the right side says that “the short one is a knight.”
As you know from middle school Language Arts class (!), this kind of statement is false as a whole only in one case: if both of its sides (called “disjuncts”) are false, i.e., if and only if both sides of the “and/or” are false. So, if the whole thing is false then its left side is false as well as its right side. In this case it is false that Tall is a knave, which is to say that Tall is not a knave. But we were assuming Tall is a knave. So, if we begin by supposing that the tall one is a knave, as we did at the beginning, then it follows that it is false that the tall one is a knave. A big fat contradiction. Since a contradiction is a paradigm case of the impossible, it follows that it is impossible that Tall is a knave. So, now we know that Tall must be a knight.
We now have one piece of the puzzle. Actually, we have half the puzzle. Let’s go for the rest. If Tall is a knight (which has now been proven) then Tall’s statement is true. That is, the overall and/or statement “I am a knave and/or the short one is a knight” is true. How can this be? We already know that the left side of the statement is false. Answer: If the right side is true, for an and/or statement with only one side true is still true! Thus the right disjunct must be true, which means the short one must be a knight.
The Strange Case of the Silent One in Red
“Later that day, you come across three strangers, one dressed in black, one in white, and one in red. The one in red remains silent, but the other two speak:
Black: All of us are knaves
White: Exactly one of us is a knight.
What are these individuals? Knights? Knaves? Which is which?”
Solution. The problem tells us that Black says “All of us are knaves.” This statement (“All of us are knaves”) is either true or it is false. It cannot possibly be true, for if it were true, then they would all be knaves, but then Black would be a knave speaking a true statement! This cannot be right because a knave can never tell the truth. So, Black’s statement cannot be true, which means two things: First, it is not true that they are ALL knaves; and second, the statement is false which means that Black must be a knave. So, Black must be lying when he says they are all knaves; they are therefore not all knaves and therefore at least one must be a knight. So, White or Red must be a knight, or both are.
White says: “Exactly one of us is a knight.” Suppose White is a knave. If White is also a knave, along with Black, then Red is the only knight (remember: we know there is at least one knight). But this would mean that White, a knave, is telling the truth when he says there is exactly one knight. But a knave couldn't tell the truth, so, White could not be a knave (because if he is a knave, then he is a knave telling the truth, which is impossible). So, White must be a knight. Therefore White must be telling the truth when he says there is only one knight. Therefore, there must be only one knight in the group and White must be the only knight. Therefore, Red must be a knave.
Three Weird Looking Dudes
Three individuals approach you. The first says, “The second one is a knave.” The second says, “The first and third are of the same type.” What kind of being is the third?
Solution. Consider the possibilities:
- If the first is a knight, then the first speaks the truth. Then the second is indeed a knave and is therefore lying and the third is therefore not of the same type as the first. The third is in this case a knave (since the first is a knight and they are not of the same type).
- If the first is a knave, then he lies when he says the second is a knave. So the second is a knight and speaks the truth. In this case, the third is of the same type as the first, which means the third is a knave.
- The first is either a knight or a knave. There are no other alternatives.
- Thus, either way, the third individual is a knave. So, the third must be a knave.
The Case of the Mysterious Circler
That evening, you come across an individual who slowly circles around you and quietly says, "Either I am a knave or 2 + 3 = 5.” What is he? Knight or knave?
Solution. The statement says, “Either I am a knave or 2 + 3 =5.” First, notice that this statement is a disjunction, with the left disjunct being “I am a knave” and the right disjunct being “2+3=5.” Next, we already know that the right disjunct is true, since it is true that 2 + 3=5. But if even one disjunct of a disjunction is true, then the disjunction as a whole is true. So, this statement is therefore true. The only person on the island who could speak a true statement is a knight. The speaker therefore must be a knight (because the statement as a whole is true).
"Either I am a knight or I am not a knight.”
You stop for gas and the attendant says, “Either I am a knight or I am not a knight.” What is he?
Solution. The sentence "Either I am a knight or I am not a knight." is equivalent to “Either I am a knight or I am a knave” since every inhabitant of the island is a knight or a knave. Therefore, what the cashier says must be true, since every inhabitant is either knight or knave. But only a knight could speak a true sentence. So, the attendant must be a knight.
The Case of the Mysterious Guard
You approach an old fort and the guard in front says: “I am a knight and I am not a knight.” What is he?
Solution. The man’s statement is a self-contradiction. It is a contradiction to say, “I am a knight and I am not a knight.” Since contradictions are always false, his statement is false. Therefore, he must be a knave, since only knaves can speak a false sentence.
Could Anyone Here Say That?
Is the following event possible? An inhabitant of the island speaks the words “I am a knave.”
Solution. Not possible because a knight could never make that statement (since he can only speak the truth and that statement, “I am a knave,” spoken by a knight, would be a falsehood). But a knave could never make that statement either, for if a knave were to say “I am a knave” that would be to speak the truth and knaves only lie. Therefore it would not be possible for an inhabitant of the island to make that statement.
Two Strange Looking Guys
Two strange looking individuals cautiously approach you. The first says, “Neither of us is a knight.” The second is silent. What are they?
Solution. First, the speaker cannot be a knight because he says that neither of them are knights, and a knight couldn’t say this without lying and knights can’t lie. (If a knight said, “Neither of us is a knight,”he would be lying.) So, the speaker must be a knave. So, the speaker is lying when he says neither is a knight. So, at least one must be a knight, and that can only be the silent one. Thus, the one speaking must be a knave and the silent one must be a knight.
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