## Solution

Here we have the solution for the Muddy Head Puzzle:-

• Let S be the statement that the son has a muddy forehead and let D be the statement that the daughter has a muddy forehead.
• When the mother says that at least one of the two children has a muddy forehead, she is stating that the disjunction S ∨ D is true.
• Both children will answer “No” the first time the question is asked because each sees mud on the other child’s forehead.
• That is, the son knows that D is true but does not know whether S is true, and the daughter knows that S is true, but does not know whether D is true.
• After the son has answered “No” to the first question, the daughter can determine that D must be true. This follows because when the first question is asked, the son knows that S ∨ D is true, but cannot determine whether S is true.
• Using this information, the daughter can conclude that D must be true, for if D were false, the son could have reasoned that because S ∨ D is true, then S must be true, and he would have answered “Yes” to the first question.
• The son can reason similarly to determine that S must be true. It follows that both children answer “Yes” the second time the question is asked.

• Assume that both children are honest and that the children answer each question simultaneously.

### Solution

Here we have the solution for the Muddy Head Puzzle:-

• Let S be the statement that the son has a muddy forehead and let D be the statement that the daughter has a muddy forehead.
• When the mother says that at least one of the two children has a muddy forehead, she is stating that the disjunction S ∨ D is true.
• Both children will answer “No” the first time the question is asked because each sees mud on the other child’s forehead.
• That is, the son knows that D is true but does not know whether S is true, and the daughter knows that S is true, but does not know whether D is true.
• After the son has answered “No” to the first question, the daughter can determine that D must be true. This follows because when the first question is asked, the son knows that S ∨ D is true, but cannot determine whether S is true.
• Using this information, the daughter can conclude that D must be true, for if D were false, the son could have reasoned that because S ∨ D is true, then S must be true, and he would have answered “Yes” to the first question.
• The son can reason similarly to determine that S must be true. It follows that both children answer “Yes” the second time the question is asked.

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