I know the relationships between the ministers in the below problem can be represented by a $K_n$ graph since each one of the n ministers has connections to the n-1 other ministers/ministries they represent. However, I can't seem to figure out where to go next. Here is the problem:
In a certain government there are n cabinet ministers. Each of these has a senior aide. Exactly k of these n aides leak important information to the press. None of the ministers know whether their own aide leaks information or not. However,because of interdepartmental rivalries, every minister has many sources in the other ministries, and thus is aware of all the leaks outside of her ministry. There is an unspoken law in this government that as soon as any minister has irrefutable proof that their aide leaked information, they must re the aide at their first press conference. Every ministry has one press conference per day at exactly 8:00 A.M. For a long time no one is fired, because no minister is aware of the leaks, if any, made by their own aide. One afternoon a newspaper runs a story, and announces that news are leaked by some of the top aides. Exactly k days later, all the guilty aides are fired. What happened?
