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Dr. Joe Smythe had been fatally wounded with a single bullet in the
heart. The three suspects were all former girlfriends of the doctor. Each
woman made three statements. After an arrest was made, the detective
noted that Carlene and Paula each made the same number of true
statements (anywhere from zero to three true statements).

Who was arrested for this crime?

Lucinda: 1. Paula has shot a gun. 2. Carlene was in the office this
morning fighting with Joe. 3. I loved Joe.

Carlene: 1. Paula’s statements are all true. 2. Lucinda’s statements are
all true. 3. I haven’t seen Joe for over a month.

Paula: 1. Everything Carlene said is a lie. 2. I’ve never shot a gun in my
life. 3. Lucinda’s statements are all false.

Scroll down for the answer

Lucinda was the one arrested for the murder of Dr. Joe Smythe.

Compare Carlene and Paula’s first statements. If Carlene’s statement
that "All of Paula’s statements are true" was true, this would make
Paula’s statement that "Everything Carlene says is a lie" false. Therefore
Carlene’s first statement cannot be true.

It can also be determined that Paula’s first statement "Everything
Carlene said is a lie" is false, because if it was true, then Paula would
have at least one true statement, but all of Carlene’s statements would
be false. This cannot be since they both gave the same number of truthful
statements. Now it can be determined that Carlene (and therefore
Paula) made at least one
truthful statement.

If Carlene’s second statement "Lucinda’s statements are all true" was
true, then Lucinda’s first statement "Paula has shot a gun" was true.
However, that makes Paula’s statement "I never
shot a gun in my life" false and then Paula’s third statement would also
be false. Paula and Carlene would not have made the same number of
truthful statements. Carlene’s second statement must be false and her
only true statement was "I haven’t seen Joe for over a month". Since she
hasn’t seen Joe for over a month, she can’t be the one who shot him.
Lucinda’s second statement is false.

If Carlene made only one true statement, Paula also only made one true
statement. If Paula’s third statement "Lucinda’s statements are all false"
was the one true statement, Lucinda’s first
statement (about Paula using a gun) would be fals. However that would
make two of Paula’s statements true and only one of Carlene’s
statements was true. This leaves Paula’s second
statement "I’ve never shot a gun in my life" as her only true statement. If
she hasn’t shot a gun, then she isn’t the one who shot Joe.