Muddy Head Puzzle

April 11, 2023

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.

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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