Mathematical Reasoning Assignment: Logic Puzzle of Knights and Knaves

This assignment presents a detailed solution to the Knights and Knaves logic puzzle, a classic problem in mathematical reasoning. The solution meticulously analyzes the statements of seven individuals—Joe, Sue, Sally, Bozo, Dave, Zed, and Alice—who are either knights (always truthful) or knaves (always liars). The analysis begins by identifying Alice as a knave based on her statement, and then proceeds to deduce the nature of each individual's truthfulness through a series of logical steps. Joe is identified as a knight, and the solution continues to unravel the relationships between the statements, such as Sue's claim about Bozo and Joe, Sally's claim, and Dave's statement about Alice and Bozo. Through careful consideration of each statement and its implications, the solution determines the identities of each person as either a knight or a knave, providing a clear and concise resolution to the puzzle. The solution demonstrates the application of logical deduction and critical thinking to solve the problem effectively.

Running head: MATHEMATICAL REASONING
Mathematical reasoning
Student
Institutional affiliation

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MATHEMATICAL REASONING
A very special island is dominated by only knights and knaves. Knights always tell the
truth, and knaves always lie.
You meet seven inhabitants: Joe, Sue, Sally, Bozo, Dave, Zed and Alice. Joe says that Dave
could claim that Alice is a Knave. Sue says that Bozo and Joe are both knights or both
knaves. Sally claims, “At least one of the following is true: that I am a knight or that Sue is
a Knave.” Bozo claims, “Sally is a knave.” Dave claims that Alice is a knight or Bozo is a
knight. Zed claims, “Of I and Sally, exactly one is a knight.” Alice says, “Dave could say
that Zed is a Knave”
The task is to determine which of the seven are knights and which are knaves.
Solution
Alice:
Alice says, “Dave could say that Zed is a Knave”
But with regards to Dave’s claim i.e. Dave claims that Alice is a knight or Bozo is a knight, it
clearly implies that Alice is lying since her claim does not concur with Dave’s
Implication: Alice is a knave (knaves always lie)
Joe:
Joe says that Dave could claim that Alice is a Knave.
But with regards to Dave’s claim i.e. Dave claims that Alice is a knight or Bozo is a knight
Taking close analysis of Dave’s claim, it implies that if Alice is a Knight, then Bozo is a knight
or, if Alice is a knave then Bozo is a knight
So Joe is telling the truth since his argument act in accordance with Dave’s claim.
Implication: Joe is a knight (knights always tell the truth)

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MATHEMATICAL REASONING
Sue:
Sue says that Bozo and Joe are both knights or both knaves.
Since Joe is a knight, and if Sue is telling the truth (a knight), it would imply Bozo is a knight
(since Sue claims that Bozo and Joe are both knights or both knaves)
But with regards to Bozo’s Claim, Bozo claims, “Sally is a knave.”
This Bozo’s statement would only implying that;
If Bozo is a knight, then Sally is a knave since knights always tells the truth, and If Bozo is a
knave, then Sally is not a knave (implying Sally is a knight) since knaves always lie.
The only implication of the above statement would mean that if Bozo is a knight, then Sally is a
knave, or if Bozo is a knave, then Sally is a knight
Reviewing Sue’s statement above, if Sue is telling the truth (a knight), it would imply Bozo
is a knight. Superimposing this with Bozo’s statement implying if he is a knight, then Sally is a
knave
But with regards to Sally’s claim, Sally claims, “At least one of the following is true: that I
am a knight or that Sue is a Knave.” Implying if Sally is a knave, then none of the above
statements is true which further implies that sally is a knave and sue is a knight
Since the above statement implies Sally is a knave and according to Zed’s statement, it would
imply that Zed is a knight i.e. Zed claims, “Of I and Sally, exactly one is a knight.” This
Zed’s claim would only mean:
If Zed is a knight, it would imply Zed is a knight and Sally is a Knave (exactly one) or Sally is
a knight and Zed is a Knave, and
If Zed is a knave, it would imply both Zed and Sally are knight, or both Zed and Sally are knaves
The highlighted statement complies with our argument.

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MATHEMATICAL REASONING
Dave on the other hand claims that Alice is a knight or Bozo is a knight, but have analyzed
that Alice is a knave and Bozo is a knight implying that Dave’s claim is true so Dave is a knight
From the logical analysis, we have found out that
Alice is a knave
Joe is a knight
Sally is a knave
Sue is a knight
Bozo is a knight
Zed is a knight
Dave is a knight
NB: A divergent logical analysis would arise if we consider Sue to be lying, which in turn will
be ambiguous to analysis since the words “both” and “or” are utilized in her statement.