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Logic to the Rescue

Homework Assignment Puzzle Answers

The following problems are adapted from their numbered equivalents in Raymond Smullyan's What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (Prentice-Hall, 1978).  Each correct solution is highlighted.

Click on the number of the puzzle for which you want to see a solution.

1 2 3 4 5
6 7 8 9 10
  1. It is Saturday afternoon. As you walk down the beach, you meet 3 residents: Tom, Dick and Harry. You know that none are Normals. You ask Tom, "Are you a Knight or a Knave?" He answers, but you cannot make out what he said; so you ask Dick, "What did he say?" Dick responds, "Tom said that he is a Knave." At this point, Harry says, "Don't believe Dick, he is lying." What are Dick and Harry? (Smullyan's #26)

Possible solutions 

Dick is a Knight. 
Dick is a Knave.
Harry is a Knight.

Harry is a Knave.

Fact list

Knights always tell the truth. 
Knaves always lie. 
Neither of the men is a Normal.
Dick says, "Tom said that he is a Knave."
Harry says, "Don't believe Dick, he is lying."

Evaluation

If Dick is telling the truth, Tom's statement would be impossible. If he was a Knight it would be a lie. If he was a Knave it would be the truth. Therefore, Dick must be lying. This makes Harry's statement true. Thus Dick has to be a Knave, and Harry a Knight.

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  1. It is Saturday afternoon. As you walk down the beach, you meet 3 residents: Tom, Dick and Harry. You know that none are Normals. You ask Tom, "How many Knights are among you?" He answers, but you cannot make out what he said; so you ask Dick, "What did he say?" Dick responds, "He said that one of us is a Knight." At this point, Harry says, "Don't believe Dick, he is lying." What are Dick and Harry? (Smullyan's #27)

Possible solutions 

Dick is a Knight.
Dick is a Knave. 
Harry is a Knight. 

Harry is a Knave.

Fact list

Knights always tell the truth. 
Knaves always lie. 
Neither of the men is a Normal.
Dick says that Tom said, "...that one of us is a Knight."
Harry says, "Don't believe Dick, he is lying."

Evaluation

If Dick is telling the truth, both he and Tom must be Knights which is impossible. Thus Dick is lying. This makes Harry's statement true.

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  1. You meet two residents of the island; Tom and Henry. Neither of them is a Normal. Tom makes this statement, "At least one of us is a Knave." What is each one? (Smullyan's #28)

Possible solutions 

Henry is a Knight. 
Henry is a Knave.
Tom is a Knight.

Tom is a Knave.

Fact list

Knights always tell the truth.
Knaves always lie.
Neither of the men is a Normal.
Tom says, "At least one of us is a Knave."

Evaluation

If Tom is telling the truth, he cannot be a Knave, so Henry must be. If he is lying neither can be a Knave, so he cannot lie - an impossibility. Thus he must be telling the truth. So Tom is a Knight and Henry a Knave.

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  1. One evening you sit down to dinner with two residents: Susan and Ralph. You know that neither of them is a Normal. Susan says, "Either I am a Knave, or Ralph is a Knight." What are Susan and Ralph? (Smullyan's #29)

Possible solutions 

Ralph is a Knave.
Ralph is a Knight.
Susan is a Knight.

Susan is a Knave.

Fact list

Knights always tell the truth.
Knaves always lie.
Neither of the people is a Normal.
In multipart statements connected with OR, all parts must be false for the statement to be false.
Susan says, "Either I am a Knave, or Ralph is a Knight."

Evaluation

If Susan is lying, the first part of her statement is true. So the entire statement is true. As Knaves cannot tell the truth, it is impossible for her to be lying. Since she cannot be a Knave, Ralph must be a Knight. Since she cannot be a Knave, she must also be a Knight.

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  1. One evening as you are out for a stroll, you walk by a doorway labeled "No Normals allowed." You hear three voices from within. Curious, you listen and hear the following. Voice one: "All of us are Knaves." Voice two: "Exactly one of us is a Knight." What are the three people inside? (Smullyan's #31)

Possible solutions 

Voice 1 is a Knight.
Voice 1 is a Knave.
Voice 2 is a Knight.

Voice 2 is a Knave.
Voice 3 is a Knight.
Voice 3 is a Knave.

Fact list

Knights always tell the truth.
Knaves always lie.
None of the people is a Normal.
Voice one says, "All of us are Knaves."
Voice two says, "Exactly one of us is a Knight."

Evaluation

If Voice 1 is telling the truth, then s/he would have to be lying; an impossibility. Thus Voice 1 must be a lying Knave. We also know that at least one of the others is a Knight. If Voice 2 is lying, s/he would be a Knave as would Voice 3. All would then be Knaves, something we have proven impossible. Therefore, Voice 2 must be a Knight. Voice 3 therefore must be a Knave.

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  1. As you are walking on the island, you meet two of the inhabitants, Tim and Grace. You know that neither of them is a Normal. Tim says, "I am a Knave, but Grace isn't." What are Tim and Grace? (Smullyan's #33)

Possible solutions 

Tim is a Knight.
Tim is a Knave.
Grace is a Knight.
Grace is a Knave.

Fact list

Knights always tell the truth.
Knaves always lie.
Neither of the people is a Normal.
Tim says, "I am a Knave, but Grace isn't."
In multipart statements connected explicitly or implicitly with AND, all parts must be true to make the statement true.

Evaluation

Tim must be a Knave. For him to be a Knight, both parts of the statement must be true. It is not possible that the first part could be true. Since this makes the first part true, the 2nd part has to be false; so Grace must be a Knave.

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  1. As a contestant on the island's only TV quiz show, you face 3 residents: Tommy, Babette and Cathy. You are told that 1 is a Knight, 1 a Knave, and 1 a Normal. Tommy says, "I am a Normal." Babette says, "That is true." Cathy says, "I am not a Normal." Your job is to prove with certainty what each is. (Smullyan's #39)

Possible solutions 

Tommy is a Knight.
Tommy is a Knave.
Tommy is a Normal.
Babette is a Knight.
Babette is a Knave.
Babette is a Normal.
Cathy is a Knight.

Cathy is a Knave.
Cathy is a Normal.

Fact list

Knights always tell the truth.
Knaves always lie.
Normals sometimes lie, and sometimes tell the truth.
1 contestant is a Knight, 1 a Normal, and 1 a Knave.
Tommy says, "I am a Normal."
Babette says, "That is true."
Cathy says, "I am not a Normal."

Evaluation

If Tommy is lying or telling the truth, he cannot be a Knight. Therefore he is either a truth-telling Normal, or a liar. If he is lying, he must be a Knave. For if he were a lying Normal, he would be telling the truth - an impossibility. If Babette is telling the truth, she is the Knight, and Tommy the Normal. If she is lying, she is a lying Normal, and Tommy is the Knave. If Cathy is lying, she is a Normal. If she is telling the truth, she must be a Knight. As we have shown 1 of the first 2 must be a Normal, Cathy has to be the Knight. Therefore, Babette is the Normal and Tommy the Knave.

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  1. You are at the airport ready to leave the island and fly home. Before being allowed on the plane, you must pass one final test. You are introduced to two residents, Carly and Earl. Carly says, "Earl is a Knight." Earl says, "Carly is not a Knight." One is telling the truth, but is not a Knight. Who is it? (Smullyan's #40)

Possible solutions 

Carly is the truth-telling Normal.
Earl is the truth-telling Normal.

Fact list

Knights always tell the truth.
Knaves always lie.
Normals sometimes lie, and sometimes tell the truth.
Either Carly or Earl is a truth-telling Normal.
Carly says, "Earl is a Knight."
Earl says, "Carly is not a Knight."

Evaluation

If Carly's statement is true, then he must be the Normal. If it is false, Earl has to be the Normal. If Earl's statement is false, Carly is a Knight and his statement would then be false - an impossibility. Therefore, Earl's statement is true. Thus Carly has to be the Normal.

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  1. It turns out that on one part of the island there is a rule that Normals can only marry Normals, and Knights and Knaves can only marry each other. One day as you are walking in this part of the island, you meet Mr. and Mrs. Jones. Mr. Jones says, "My wife is not a Normal." Mrs. Jones says, "My husband is not a Normal." What are Mr. and Mrs. Jones? (Smullyan's #44)

Possible solutions 

Mr. & Mrs. Jones are Normals.
Mr. Jones is a Knight and Mrs. Jones a Knave.
Mr. Jones is a Knave and Mrs. Jones a Knight.

Fact list

Knights always tell the truth.
Knaves always lie.
Normals sometimes lie and sometimes tell the truth.
Normals always marry Normals.
Knights always marry Knaves.
Knaves always marry Knights.
Mr. Jones says, "My wife is not a Normal."
Mrs. Jones says, "My husband is not a Normal."

Evaluation

If his statement is true, he must be a Knight and she a Knave. If it is false, she is a Normal, and he a lying Normal. However, if she is a Knave and he a Knight, her statement would be true - an impossibility. Therefore they must both be Normals.

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  1. It turns out that on one part of the island there is a rule that Normals can only marry Normals, and Knights and Knaves can only marry each other. One day as you are walking in this part of the island, you meet Mr. and Mrs. Jones. Mr. Jones says, "My wife is a Normal." Mrs. Jones says, "My husband is a Normal." What are Mr. and Mrs. Jones? (Smullyan's #45)

Possible solutions 

Mr. & Mrs. Jones are Normals.
Mr. Jones is a Knight and Mrs. Jones a Knave.
Mr. Jones is a Knave and Mrs. Jones a Knight.

Fact list

Knights always tell the truth.
Knaves always lie.
Normals sometimes lie and sometimes tell the truth.
Normals always marry Normals.
Knights always marry Knaves.
Knaves always marry Knights.
Mr. Jones says, "My wife is a Normal."
Mrs. Jones says, "My husband is a Normal."

Evaluation

If his statement is true, they are both Normal. If it is false, he is a Knave and she a Knight. Her statement would be false - impossible. Therefore, they are both Normal.

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original web posting: Monday, June 4, 2001
last modified: Monday, November 22, 2004