Muddy Heads | Puzzles for Interview - Interview Preparation PDF Download

Introduction

A mother asks her two children, a boy and a girl, to play without getting dirty. However, while playing, both children get mud on their foreheads. The mother tells them that at least one of them has a muddy forehead and asks them a question twice.

Part 1: The Question

The mother 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.

Part 2: The Solution

The solution involves understanding the logic behind the children's answers. Let s be the statement that the son has a muddy forehead, and 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.

Part 3: First Question

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.

Part 4: Second Question

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 in a similar way to determine that s must be true. It follows that both children answer “Yes” the second time the question is asked.

Conclusion

In conclusion, the muddy forehead puzzle involves logical reasoning and understanding the disjunction statement. The children's answers depend on their knowledge of their own and their sibling's muddy forehead.