Four scientists combinatorial problem. Four scientists are working on a secret project.  They wish to lock up the documents in a cabinet such that the cabinet can be open if and only if 3 or more scientists are present.
a. What is the smallest number of locks needed?
b. What is the smallest number of keys that each scientist must carry?
c. Design a scheme with the minimum number of locks needed and the minimum number of keys for each scientist that will actually accomplish the desired security.
 A: As to how many locks, for any group of 2 scientists there must be at least one lock they cannot open.
Moreover, for any two different groups of 2, the:”inoperable” lock must be different for each group.
Thus, at least C(4, 2) = 6 locks are needed.
As to how many keys, whenever scientist S is associated with a group of 2 other scientists, 
scientist S must have the key to at least one lock that the other 2 scientists cannot open.   
Moreover, for any two different groups of 3, the “inoperable” key must be different for each group.  
Thus, at least C(3, 2) = 3 keys are needed.  
Now, we must design a scheme with 6 locks and 3 keys for each scientist that will actually 
accomplish the desired security.  For each of the 6 groups of 2, label a lock with the two names on it.  
This means that a lock has 2 names on it of those scientists who cannot open it.  For each group of 
2 scientists bring in a 3rd scientist.  He must have the key(s) to the lock(s) that the other 2 scientists 
do not have.  This will meet the security requirements.  
Consider the following scheme to accomplish the desired security:
Denote scientists: a, b, c, d
Denote locks: L1, L2, L3, L4, L5, L6
Labels for locks:
L6(a, b):  lock 6 cannot be opened by scientists a, b
L5(a, c):  lock 5 cannot be opened by scientists a, c
L4(a, d):  lock 4 cannot be opened by scientists a, d
L3(b, c):  lock 3 cannot be opened by scientists b, c
L2(b, d):  lock 2 cannot be opened by scientists b, d
L1(c, d):  lock 1 cannot be opened by scientists c, d
Labels for scientists:
a(L1, L2, L3):  scientist a has keys for locks 1, 2, 3
b(L1, L4, L5):  scientist b has keys for locks 1, 4, 5
c(L2, L4, L6):  scientist c has keys for locks 2, 4, 6
d(L3, L5, L6):  scientist d has keys for locks 3, 5, 6
A: Hint: Write out the possible configurations of three you can make out of the four scientists. Then assign each of them keys in a way that all the locks can be opened in each of the configurations. (The discussion below might help you decide how many locks there should be.)
I will show that it is not possible if each lock has a distinct key from other locks, and each scientist only has one key: Suppose we have scientists $A,B,C,D$.
Suppose there's one lock. Then at least one of the scientists must have a key which opens it; WLOG $A$. Then $A$ going alone can open it. So one lock is not possible.
Suppose there are two locks, 1 and 2. Then WLOG $A$ has a key to 1 and $B$ has a key to 2. Then $A$ and $B$ together can open it; again requirement not satisfied.
Suppose three locks: 1, 2, and 3. Then WLOG $A$, $B$, $C$ have keys to 1, 2, 3, respectively. We can give $D$ key 1, 2, or 3; WLOG $D$ gets 1. Then if $A$ and $D$ are in the group of three, whether they add $B$ or $C$ as the third, the three locks cannot be unlocked.
Suppose four or more locks. Then no group of three can open the door.
