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Induction puzzles

These puzzles are examples of applied logic which use the induction principle to determine the solution to a complex problem by solving the simpler problems in contains.

Examples of Induction Puzzles – The Muddy Children Problem

The Muddy Children puzzle presents a situation in which a group of nice little children are informed that at least one of them has a dirty face. No fingers are pointed so the children have no way of knowing if it’s about them or not. The problem assumes all the children have the same reasoning capacities so they will be able to make the same deductions. You’ve probably noticed they do not receive information as to the specific number of children with a muddy face – at least one might mean just one, two, three or indeed all of them. Since they cannot look in a mirror how do they find out if the problem concerns them or not?


How to Solve an Induction Puzzle

The children are standing in a circle so they can see each other and they are told to take a step forward when they hear a stroke if they believe they are dirty.

To understand how the induction principle works it’s easier to start with just two children Jane and John. They know one of them must have mud on her face. If Jane is the dirty one she will see that John is squeaky clean and will step forward rightly concluding that she is at fault. If both kids have mud on their faces, neither will move forward at the first stroke, each assuming that it’s the other who should do so. When none of them moves, they will both realize the other has seen the mud on their face and will understand they are also dirty.As a result, at the second stroke, both kids will move forward. Puzzles involving a larger number of participants can be solved using the same logical step.


Versions of the Muddy Children Induction Puzzle

A well-known version of this puzzle is the King’s Wise Man Hat, in which the rules gathers three of the most brilliant minds in his kingdom and announces that the one who will solve a riddle will be his new adviser. The king places a hat on each man’s head and tells them that at least one of them wears a blue hat, while the rest have white hats. There are other variants with other characters and up to ten participants, but the way to solve them is basically the same.