Alice and Basilio make \(20\) bills. On each bill they will write a seven-digit number, so each bill initially has \(7\) empty cells for digits.
Basilio repeatedly calls out a digit, either \(1\) or \(2\). After each call, Alice writes this digit into any empty cell of any bill and shows the result to Basilio. When all cells are filled, Basilio takes as many bills with different numbers as possible (if several bills have the same number, he takes only one of them), and Alice keeps the rest.
What is the largest number of bills Basilio can guarantee to get, no matter how Alice plays?
Detective Nero Wolf is investigating a crime. There are \(80\) people involved in the case. Among them, one is the criminal and another is the only witness to the crime (but the detective does not know who they are).
Each day, the detective may invite any group of these \(80\) people for questioning. If the group contains the witness but does not contain the criminal, then the witness will reveal who the criminal is. Otherwise, nothing happens.
Can the detective guarantee that he solves the case within \(12\) days?
There are \(16\) cities in the kingdom. We would like to build roads between these cities so that one can get from any city to any other without passing through more than one city on the way. To save cost, we would like to have no more than four roads coming out of each city. Prove that such a system of roads is unfortunately impossible to build.