We can find many puzzles on mathematical logic principles for learning and fun purposes. Raymond Smullyan (1999—2017) is an American mathematician who is the author of many books on logical puzzles. Let’s look at a few exciting puzzles by Smullyan.
Puzzle 1
There are two kinds of inhabitants of an island: knaves and knights. The knaves always lie, and the knights always tell the truth. We meet two people on this island, and . The person says, “I am a knave, or is a knight.” And does not say anything. We want to determine whether and are knave or knight.
Solution
There are four possibilities here as each one of and can be knave or knight. We will examine all four cases and discard those leading to a contradiction.
Case 1: If both and are knights.
In this case, the statement of should be true as they are a knight. I am a knave, or is a knight, is a true statement because the disjunction of false and true is true.
Case 2: If both and are knaves.
In this case, must lie. But we observe that gives a true statement that I am a knave, which is a contradiction.
Case 3: If is a knave and is a knight.
In this case, must lie as they are a knave. But says, “I am a knave,” which is true. Therefore, this can not be the case as it leads to a contradiction.
Case 4: If is a knight and is a knave.
In this case, must tell the truth as a knight. But says that I am a knave or that is a knight, which is not true. Therefore, this case also leads to a contradiction.
Hence, we can conclude the only possibility that both and are knights.
Puzzle 2
There are two kinds of inhabitants of an island: knaves and knights. The knaves always lie, and the knights always tell the truth. We meet two people on this island, and . The person says, “We both are knights.” And says that “ is a knave.” We want to determine whether and are knave or knight.
Solution
There are four possibilities here as each one of and can be knave or knight. We will examine all four cases and discard those leading to a contradiction.
Case 1: If both and are knights.
In this case, the statement of should be true as they are a knight. says that we both are knights, which is true. But if is a knight, then the statement that is a knave must be true, which is not the case. Hence, this case leads to a contradiction.
Case 2: If both and are knaves.
In this case, must lie. The statement of that we both are knights is a lie. But the words of that is a knave are not a lie. Hence, this case leads to a contradiction.
Case 3: If is a knave and is a knight.
In this case, must lie as they are a knave. The statement of , “We both are knights,” is a false statement. As is a knight, the statement of must be true. says, “ is a knave, which is a truth. Hence, there is no contradiction.
Case 4: If is a knight and is a knave.
In this case, must tell the truth as a knight. But says that we both are knights, which is not the case. Hence, this case also leads to a contradiction.
Hence, we can conclude the only possibility that is a knave and is a knight.
Puzzle 3
Assume three kinds of inhabitants of an island: knaves, knights, and knatives. The knaves always lie, and the knights always tell the truth. The knatives can speak the truth, or they can lie. Assume we meet three people on this island: , , and . One of them is a knight, one is a knave, and one is a knative. We want to determine who is the knight, who is the knave, and who is the knative using the following information:
- says: “I am the knight.”
- says: “I am the knave.”
- says: “ is the knight.”
Solution
There are six possibilities, as shown in the following table:
Possibility | |||
---|---|---|---|
1 | Knight | Knave | Knative |
2 | Knight | Knative | Knave |
3 | Knave | Knight | Knative |
4 | Knave | Knative | Knight |
5 | Knative | Knight | Knave |
6 | Knative | Knave | Knight |
We’ll look at each case and discard those which lead to a contradiction.
Cases 1 and 6
If is the knave, he can not say I am the knave; that is a contradiction.
Cases 3 and 5
If is the knight, he can not say I am the knave; that is a contradiction.
Case 4
If is the knight, and he says is a knight, then should be a knight. But, can not be the knight because it is contradictory, as shown in cases three and five. Therefore, is not telling the truth, which is contradictory.
Case 2
If is a knight, he says I am the knight, which is consistent. If is knative and says that I am the knave, that means is telling a lie, which is consistent. If is the knave, and he says is the knight, he is telling a lie, which is again consistent.
So, after analyzing each case, we can conclude that is the knight, is knative, and is the knave.
Puzzle 4
There are two kinds of inhabitants of an island: knaves and knights. The knaves always lie, and the knights always tell the truth. Assume we meet two people on this island, and . The person says, “I am a knave if and only if is a knave.” And says, “Our kind is different.” We want to determine whether and are knave or knight.
Solution
There are four possibilities here as each one of and can be knave or knight. We will examine all four cases and discard those leading to a contradiction.
Case 1: If both and are knights.
In this case, the statement of should be true as he is a knight. I am a knave, if and only if is a knave is a true statement because . The statement of should also be true as is a knight. says our kind is different, which is not the case, giving a contradiction.
Case 2: If both and are knaves.
In this case, must lie. But we observe that gives a true statement that I am a knave if and only if is a knave, which is a contradiction.
Case 3: If is a knave and is a knight.
In this case, must lie as he is a knave. As says, “I am a knave if and only if is a knave,” which is false; therefore, it is consistent. The statement given by should be true as is a knight. says our kind is different, which is true. Consequently, this case is consistent and does not lead to a contradiction.
Case 4: If is a knight and is a knave.
In this case, must tell the truth as a knight. But says that I am a knave if and only if is a knave, which is not true. Therefore, this case also leads to a contradiction.
So, we can conclude the only possibility is that is a knave and is a knight.
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