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 Open By Majority (Posted on 2004-03-03)
A group of five people want to put a set of locks on a chest and distribute keys to the locks amongst themselves in such a way that all the locks on the chest could be opened only when at least three of them were present to open it.

How many locks would be needed, and how many keys?

 See The Solution Submitted by Brian Smith Rating: 4.1429 (7 votes)

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 re(2): key answer +an EXTRA key for GAMER | Comment 7 of 45 |

G'

Neither the proof nor the actual distribution of the keys was requested, but I will obliged you. I felt that something should be left for others. Anyway:

Why is this minimum? If there was a lock for which 2 or fewer people had keys, the 3 or more people left over would be unable to open the box, which is unacceptable; so all locks must share their keys among 3 or more people. Now if there was some 3-person group that was not the set of key owners for a lock, the group consisting of the remaining 2 people would be able to open the box, since it shares at least one person with every other 3-person group, and every group with 4 people or more. Hence every 3-person group must be the set of key owners for some lock, which means that any solution must use at least the locks and keys above (you can use more if you like.)

As to the distribution list - simply follow the solution and
allocate the keys: subsets of a group 1,2,3,4,5 are:
123,124,.....345
all members of the first 3-tuple get EACH the A-key for lock A
all members of the SECOND 3-tuple get EACH the B-key for lock B
....YHE LAST SUBSET the J-kay

Hope you find it satisfactory.

 Posted by Ady TZIDON on 2004-03-03 18:15:00

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