[Math Lair] Lie Detection Puzzles

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Lie detection puzzles are a type of logic puzzle. They were invented in the 1930s by British puzzlemaker Hubert Phillips ("Caliban"). Lie detection puzzles usually take the following form: Some crime has been committed, and several people make statements related to the crime. You are then told that a certain number of those statements are lies, and asked to find who committed the crime based on that information. Here's a simple example:

Frederick was killed by one of Agnes, Bernie, Carol, David, or Eugene. All five were brought in for questioning. They made the following statements:
Agnes: Eugene did it.
Bernie: Agnes did it.
Carol: Agnes did it.
David: Eugene did it.
Eugene: Bernie did it.
As it turned out, four people lied and only one told the truth. Who was the killer?

Solution: If exactly one person told the truth, then neither Agnes nor Eugene could be the killer, since that would mean two people told the truth. Nor could it be Carol or David, since that would mean that no-one told the truth. The only way in which exactly one statement is true is if Bernie was the killer. So, it must be Bernie.

Working backwards can be a useful strategy for solving lie detector puzzles. Consider the following puzzle:

A murder was committed. In response, the police rounded up five suspects, Al, Bob, Chuck, Doug, and Ernie and took them in for questioning. The five suspects gave the following statements:
Al: Bob didn't do it.
Bob: Chuck didn't do it.
Chuck: Ernie didn't do it.
Doug: I didn't do it.
Ernie: Either Al or Chuck did it.

It turned out that exactly one of the suspects committed the murder, and it turned out that exactly four of the suspects told the truth and one lied. Who committed the murder?

Solution: We could work backwards to solve this one. Set up a table and, for each possible murderer, set up a table listing whether each statement is true or false:
If the murderer is...
Al's statementTFTTT
Bob's statementTTFTT
Chuck's statementTTTTF
Doug's statementTTTFT
Ernie's statementTFTFF
Total # of true statements53433

The only way there can be four true statements is if Chuck committed the murder. Therefore, he must be the murderer.

For more complex problems, it may help to organize everything you know in a table. Consider the following problem:

Laurel was killed on a road two miles from Trenton at 3:30 a.m., February 14. Within a week, the police had captured five suspects in Philadelphia: Hank, Joey, Red, Shorty and Tony. The five men were questioned, and each made four statements, three of which were true and one false. It is known that one of these men killed Laurel. Who did it?

The statements made by the suspects are as follows:
Hank(a)I did not kill Laurel.
(b)I never owned a revolver in my life.
(c)Red knows me.
(d)I was in Philadelphia the night of February 14.
Joey(a)I did not kill Laurel.
(b)Red has never been in Trenton.
(c)I have never seen Shorty before.
(d)Hank was with me in Philadelphia the night of February 14.
Red(a)I did not kill Laurel.
(b)I have never been in Trenton.
(c)I have never seen Hank before.
(d)Shorty lied when he said I did it.
Shorty(a)I was in Mexico City when Laurel was murdered.
(b)I never killed anyone.
(c)Red is the killer.
(d)Joey and I are friends.
Tony(a)Hank lied when he said he never owned a revolver.
(b)The murder was committed on St. Valentine's Day.
(c)Shorty was in Mexico City at that time.
(d)One of us five is guilty.

Solution: To solve this problem, it may help to arrange the information in a table such as the following:
SuspectStatement (a)Statement (b)Statement (c)Statement (d)

We can start with Tony's statements: We know that the murder was committed on Valentine's Day and that one of them is guilty, so Tony's statements (b) and (d) are true:
SuspectStatement (a)Statement (b)Statement (c)Statement (d)

Now, think about this for a moment. Since each person made three true statements and one lie, then we know the following:

  1. If we know that three statements made by the same person are all true, the fourth one must be false.
  2. If we know that a statement is false, the other three statements made by the same person are all true.
  3. Given any two statements by the same person, at least one (possibly two) are true.

Let's look at Red's statements (a) and (d). If he didn't kill Laurel, both statements would be true; if he did, both statements would be false. However, both statements can't be false (see #3 in the list above). So, he did not kill Laurel and both statements are true:
SuspectStatement (a)Statement (b)Statement (c)Statement (d)

Now, Red's statement (d), which we now know to be true, says that Shorty's statement (c) is false. So, that statement is false. That means that the other three of Shorty's statements are true. Now, since Shorty's statement (a) is true, that means that Tony's statement (c) is true; since Tony's statements (b), (c), and (d) are true, his statement (a) is false. Since statement (a) said that Hank lied when he said he never owned a revolver (b), that statement must be true:
SuspectStatement (a)Statement (b)Statement (c)Statement (d)

Similarly, since Shorty's statement (d) is true, that means that Joey's statement (c) is false and so the rest of his statements (a), (b), and (d) are true. Since Joey's statement(b) implies Red's statement (b), Red's statement (b) must be true and statement (c) is false. So, Hank's statement (c) is true, as is statement (d) (based on Joey's statement). The lie that Hank told must then be statement (a):
SuspectStatement (a)Statement (b)Statement (c)Statement (d)

Statement (a) for Hank was "I did not kill Laurel." Since that statement is a lie, then that means that Hank was the killer.

Finally, here is a puzzle composed by Caliban:

St. Dunderhead's School at Fogwell has a high reputation for hockey—but not so high a reputation for veracity. The First XI played a match at Diddleham recently, after which the girls were allowed to go to a concert. Miss Pry, the mistress in charge, collected the team afterwards; she saw ten girls emerge from the concert-hall and one from the cinema next door. When she asked who had been to the cinema, the members of the team replied as follows:
Joan Juggins: "It was Joan Twigg."
Gertie Gass: "It was I."
Bessie Blunt: "Gertie Gass is a liar."
Sally Sharp: "Gertie Gass is a liar, and so is Joan Juggins."
Mary Smith: "It was Bessie Blunt."
Dorothy Smith: "It was neither Bessie nor I."
Kitty Smith: "It wasn't any of us Smith girls."
Joan Twigg: "It was either Bessie Blunt or Sally Sharp."
Joan Forsyte: "Both of the other Joans are telling lies."
Laura Lamb: "Only one of the Smith girls is telling the truth."
Flora Flummery: "No, two of the Smith girls are telling the truth."

Given that, of these eleven assertions, at least seven are untrue, who went to the cinema?

Solution: One way to solve this problem is to make a table, assume in turn that each girl is guilty, and see how many girls told the truth in that case. Doing so, you will find that eight girls lied if Dorothy Smith went to the cinema, but no more than six lied if anyone else went. Therefore, Dorothy Smith went to the cinema. You can also use guess and check to solve the problem. If you do so, you'll notice after a while that the only way you can get enough liars is if all three Smith girls are lying. If they are all lying, then Dorothy Smith must have gone to the cinema.

Sources used (see bibliography page for titles corresponding to numbers): 7, 61.