Problem 7.5. On an island, there live knights who always tell the truth, and liars who always lie. One day, 35 inhabitants of the island sat down at 7 tables, with 5 people at each table. Each of these 35 inhabitants was asked: "Are there more than three tables where at least 3 knights are sitting?"

(a) (1 point) What is the maximum number of inhabitants who could have answered "Yes"?

(b) (3 points) What is the maximum number of inhabitants who could have answered "No"?
Answer:

(a) (1 point) 35.
(b) (3 points) 23.

Solution. (a) It is clear that all 35 people could have answered "Yes" if they were knights. More than 35 could not be.

(b) Consider two cases.

First case. The statement "There are more than three tables where at least 3 knights are sitting" is true.

Then there are at least 4 tables where at least 3 knights are sitting, meaning there are at least 12 knights in total. Therefore, there are no more than $35-12=23$ liars, so there were no more than 23 "no" answers.

It is not difficult to construct an example in this case:

- At the first four tables, there are 3 knights and 2 liars each;
- At the remaining three tables, there are 5 liars each.

Second case. The statement "There are more than three tables where at least 3 knights are sitting" is false.

Then there are no more than 3 tables where at least 3 knights are sitting. Thus, there are at least 4 tables where no more than 2 knights are sitting. Therefore, there are at least 4 tables where at least 3 liars are sitting, meaning there are at least 12 liars.

Therefore, there were no more than $35-12=23$ knights, so there were no more than 23 "no" answers.

It is not difficult to construct an example in this case:

- At the first three tables, there are 5 knights each;
- At the remaining four tables, there are 2 knights and 3 liars each.