Question 30. 30 people are sitting around a circular table. Some of them are wearing a hat. Those who wear a hat always tell the truth, while those who do not wear a hat can either lie or tell the truth. Every person at the table says, at least one of my two neighbors is not wearing a hat. What is the largest number of people who could be wearing a hat? Before we get started, just welcome to the last problem. Hopefully everybody's been having a good time until now. Just as a little short story, I feel like every single year Math Kangaroo has a nice circle table, people sitting around it problem. So be ready for those. Now finally getting to what you guys have been waiting for, the actual problem. The first step to this is we can start by reducing the problem. And first over here we said every person at the table says at least one of my two neighbors is not wearing a hat. So if you have two neighbors, then the smallest possible version of the problem is a situation with a circular table with three people around it. So each person has two distinct neighbors. So we can consider this circular table with only three people. In this scenario, the maximum number of people wearing a hat is two. And this is NHHN, where they're not wearing a hat, and therefore they are lying. Those who do not wear a hat can either lie or tell the truth. And then you have the two hat wearers who must tell the truth. So every person at the table says at least one of my two neighbors is not wearing a hat. And so in this case, this person who's wearing the hat will say that at least one of my two neighbors is not wearing a hat. That is true because this N is over here and this H is over here. This person can also say that and be true because this H is over here and this N is over here. And then of course the person who's not wearing a hat can also say that because they will be lying. And then we can kind of start increasing the problem and considering the problem in larger scenarios. So what it turns out is that if we increment the number of people by three, we can see that the maximum number of people wearing a hat increases by two. Well, why is this? This is because every person who tells the truth, hat wearers, must be sitting next to one hat wearer, as per the statement, and one not hat wearer. So we have N H H. And then since if we're adding more people, we'll add them next to this hatted person. And that has to be a non-hatter. And then of course, if we have a non-hatter over here, we can restart the pattern N H H. And again, remember that this table is circular. So this hat wearer is also sitting next to this first not hat wearer. So we can kind of repeat this unit 10 times to reach 30 people. And by doing that, in each unit, we have two people who are wearing hats. So we get that therefore there are 20 hats, and the answer is D. And with that, we conclude the Math Kangaroo 2022 5-6 video library. Thank you for listening, and I hope you guys had a great time. Thank you.

Math Kangaroo

circular table

hat wearers

truth-tellers

NHH pattern