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# Muddy Head Puzzle

## 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.

**Muddy Head **

In Muddy Head Puzzle, a mother tells her son and daughter, to play without getting dirty. However, while playing, both children get mud on their foreheads. The mother says “At least one of you has a muddy forehead”. She then asks the children to answer “Yes” or “No” to the question: “Do you know whether you have a muddy forehead?” The mother asks this question twice. What will the children answer each time this question is asked, assuming that a child can see whether his/her sibling has a muddy forehead, but cannot see his or her forehead?

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

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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