11 Pirates and the Majority Safe
Design a lock-and-key system so that any 6 or more pirates can open the safe, but no group of 5 or fewer can.
Interactive coalition test
Click pirates to form a group.
Selected: 0
Safe stays closed
Select some pirates to test the construction.
The construction
Make one lock for every possible set of 5 pirates.
For a lock labeled by a 5-person set S, give copies of that key
to exactly the 6 pirates not in S.
462
locks
C(11,5)
252
keys per pirate
C(10,5)
Why it works
≤5
Any forbidden group of at most 5 pirates is contained in some 5-person set
S. None of them has the key to lock L(S),
so they cannot open the safe.
≥6
For every 5-person set
S, a group of 6 or more must contain
at least one pirate outside S. That pirate has the key to
L(S), so every lock can be opened.
One lock visualized
Example lock L{1,2,3,4,5}:
pirates 1–5 get no key, while pirates 6–11 each get a copy.
NO KEY
1
2
3
4
5
🔒
L{1,2,3,4,5}HAS KEY
6
7
8
9
10
11
Core idea:
focus on the maximal forbidden groups, which are the 5-person subsets.
Give each such group its own lock that none of those 5 pirates can open.
Therefore the number of locks is
C(11,5) = 462.