Muddy Children Puzzle Variation-Personal Research

Research Paper: https://drive.google.com/file/d/1GmMTRTNZOWpo6D6amaRuD-UtjAPklxlR

3 children A, B and C have been playing in the mud and some of them got mud on their face. They sit in a circle so that they could see if the other children have mud on their face, but they do not know if they have mud on their face since they can not see their own face and apparently are not able feel if they have mud on their face. Their father comes and tells them that at least 1 of them has mud on their face. Then the father repeatedly asks the children who know that they have mud on their face to step forward. The children have to step forward simultaneously and they can not communicate with each other. What will happen?

At first, it may seem like it is impossible for the children to figure out if they got mud on their face. After all, how does the father repeatedly asking the question help the child figure out if they have mud on their face?

We could start by looking at the case when only 1 child, let's say A has mud on their face. If we go into the perspectives of each of the children, A will see no children with mud on their face, B and C will see that A has mud on their face. B and C will not step forward since they do not know that there is only 1 child with a muddy face and from their perspective, 1 or 2 children could have a muddy face. However, A will step forward knowing that they have a muddy face because there must be at least 1 child with a muddy face and A could not see any other children with a muddy face.

We could now look at the case that 2 children, A and B have mud on their face. We will again go into the perspective of each of the children. A could see that B has a muddy face, B could see that A has a muddy face and C could see that both A and B have a muddy face. The first time the father asks the question, none of the children could figure out whether they have a muddy face or not. A and B are not sure if there are 1 or 2 children with a muddy face and C is not sure if there are 2 or 3 children with a muddy face. The second time the father asks the question, A and B will step forward since if none of the children stepping forward the first time clarifies that there are at least 2 children with a muddy face. As mentioned above, if there was only 1 child with a muddy face, that child would have stepped forward the first time the question was asked. A and B could only see 1 other child with a muddy face so they would know that they have a muddy face. C would not step forward since there being at least 2 children with a muddy face does not clarify if 2 or 3 children have a muddy face.

We will now look at the case when all of the 3 children, A, B and C have mud on their face. All of the children could see that the other 2 children have a muddy face which indicates that 2 or 3 children have a muddy face. The first time the father asks the question, none of the children steps forward as they are still not sure if 2 or 3 children have a muddy face. The second time the father asks the question, none of the children will step forward as they are still not sure if there are 2 or 3 children with a muddy face which they need to know if they have a muddy face. However, the third time the father asks the question, all of the three children will step forward. This is from the fact that if there are 2 children with a muddy face, they would have stepped forward the second time the question was asked. As a result, the children would know that there must be at least 3 children with a muddy face and would step forward.

If you know about induction, you could try to prove the general case of the puzzle. Hint: Instead of focusing on the total number of children, focus on the number of children which are muddy.

In the research, I looked into a variation where the father says that there are at least x children with a muddy face, where x is a positive integer including, but not necessarily 1.

These puzzles fall into the category of puzzles called induction puzzles. Mathematics is used in induction puzzle as it sometimes looks into combinatorics and induction is sometimes used to prove it. Some induction puzzle also require outside the box thinking. One example of this is the Homes of Sneetchville puzzle. Where there are star bellied and plain bellied sneetches. At Sneetchville, each house is required to have at most 1 star bellied sneetch and 1 plain bellied sneetch. No sneetch is able to see its own belly and they are not able to talk about whether a sneetch is star bellied or plain bellied. Each sneetch can not skip a home unless it is completely sure that it can not move in. How will the sneetches choose their home?

Induction puzzles are also a topic which fields such as computer science, game theory, economics, philosophy and psychology looks into.

I was inspired to research on induction puzzle after seeing a Numberphile video (https://www.youtube.com/watch?v=laAtv310pyk) on the Hat Problem. The results seemed interesting as it is unexpected at first, but starts to make sense. I found that the Hat Problem could fall into the category of induction puzzle and I started to research into other induction puzzles.