A very secret company develops a hi-tech machine. There are 20 different blueprints needed to construct it. Every employee has access to exactly 5 different blueprints and every combination of 5 different blueprints is accessible by at least one employee. The CEO wants to distribute employees into departments so that no department could make the machine by itself. That is, there should not be any department where every blueprint is accessible by at least one of its members. What is the smallest number of departments the CEO needs to create?
I've been thinking of Steiner system, but there's no number of employees so I'm stuck. Please guide me to solve this problem. Much thanks!
Suppose there are no more than 5 departments. For every department, choose a blueprint that is not accessible here, 5 blueprints in total. If there are less than 5 departments or some of the chosen blueprints are the same, just add random blueprints to get 5. But, the person that has access to these 5 blueprints cannot working any department. Therefore, it is not possible to have 5 departments or less.
If there are 6 departments, the CEO can send the employees who do not have access to the 1st blueprint to 1st department, the employees who do not have access to the 2nd blueprint to the 2nd department, and so on, the employees who do not have access to the 5th blueprint – to the 5th department. Then, there are left only those employees who have access to blueprints 1-5. They are all sent to the last department.
Answer：6