Solve tough gaming problem with highest probability of success (may be unsolveable, unsure)
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Read completely! Friend gave me puzzle that I have not been able to figure out. I extended the challenge, but I stepped way out of my ability to get solution...Ive thought of different approaches...too many variables...help!
In land ruled by a strict king. A lucky man discovers something of great value that he is allowed to keep, if he can avoid the king taking it away from him. He will need to sell it all eventually to make a living.
Conditions: The object can be divided into as many parts up to 10 if desired.
All or parts of the item can be divided in up to 10x the number of other people involved in the scheme. For example...if 5 people involved, there could be the item in 1 of 50, or 10 parts in 20 containers, etc. The containers will be impossible completely impossible to distinguish except for numbers 1-50 on container.
He can use up to 5 other people to help him with hiding away the item (whether item divided up or not) and bury them at known locations. The less people involved the better. For obvious reasons, there is limited certainty they will not give up the others or information pieces if they are under duress. King is ruthless and will punish anyone more the more the conspirator knows...king will always determine everything the conspirator knows.
Multiple types of information/material that can be exchanged. Conspirators can exchange any information and containers they have up to 5 times as described below:
My understanding...man wants to maximize the randomness of the information and locations and knowledge of who one another is, so that any captured people will know limited amount, and hence not be able to give the king the locations of all the containers. (Obviously, it might be problematic if the item is not divided up, and the king happens to find it in one of the containers).
Available information chunks to trade between the conspirators was:
-list information about location of buried containers
-information that links container number to buried location list
-information as to how many times the item was broken up
-information that links pieces of broken up items (whether single or divided up) to container number
-information as to who is involved (each person does not necessarily need to know who is involved), although the fewer the better since trust diminishes the more people involve
-information about containers that house the item pieces can be changed with a record of changes during each step
-any of the information chunkc can be divided up into equally divisible pieces to the number of conspirators involved
-the number of layers of people not known by one another can be two (man knows friend who knows an unknown friend)
I tried to draw diagrams with exchanges by steps, for example...step 1...
man divides into 10 in 20 containers, numbers them, and splits them randomly between 2 people while giving another person the locations and numbers.
step 2...the item container/number reference is changed (a new record of the changes adds another piece of information to the puzzle like mentioned above). these 4 conspirators exchange containers, partial information about locations, etc. I got lost...
At the end of at least 1 exchanges...
-the king should be able to capture up to conspirators minus 1 and not be able to find all the containers.
-no one should know everyone involved
-no one should know enough to find over half the containers
-any other condition you think necessary to make each conspirator least liable to have minimum information or capability to track all the pieces of information and their association
-discovery man MUST be possible to be able to retrieve RNDUP(=0.75*# containers) with the loss of up to conspirators-2
I want at least 3 journal articles referenced from game theory or comp science or topolog geom theory (or relevant ) and diagrams per step to back up your clear explained solution. Stat probabilities of liability at end would be nice too. If not solvable, prove why.