Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain).
The captain always proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go "Aye", the loot is divided as proposed, as no pirate would be willing to take on the captain without superior force on their side.
If the captain fails to obtain support of at least half his crew (which includes himself), he faces a mutiny, and all pirates will turn against him and make him walk the plank. The pirates start over again with the next senior pirate as captain.
What is the maximum number of coins the captain can keep without risking his life?
Hmmm...Similar question was posted long before :-
http://www.jatland.com/forums/showthread.php?p=76287#post76287
[To summarize, the captain can have 98 gold coins]
Pasting the solution I gave :-
Lets number the persons from N1 to N5----N1 being the juniormost and N5 being the seniormost....
In the riddle,its given that these people are very very greedy and logical too......
So what the N5 can do.....Here is my answer.........
Lets apply induction.
Case1: only one person.
If there is only one person,then he will have all the share.
Case2: two persons
let it be N1 and N2.In this case N2 would want to take all the coins,and he has 50% votes (his own vote) for
doing this. This way N1 won't be getting anything.
Case3: three persons
let it be N1,N2, and N3.So in this case N3 needs support of one more member.N2 wont support him,as he knows that if he is able to get rid of N3,then only two persons will be left and he will be at gain(see case2).So N3 would lure N1 for 1 gold coin.since each of them is extremely greedy and logical,so N1 would think that 1 coin is better than nothing,so he would accept his offer.In this way,N3 will have 99 coins,N2 will have none and N1 will have 1.
Case4: four persons
Let it be N4,N3,N2,and N1. then N4 needs support of at least one member. But he wont get it from N3 (from case3) and recursively from N1. So the only person left is N2. Again, N4 will have 99coins,and N2 will have 1.
Case5: five persons
N5 needs at least 2 more people on his side,to get his plan proposed......If we think logically,like them, we know that N4 would want to kill N5,since he is the next seniormost.......So he will lure N3 with one coin.N3 knows that if N4 comes,he won't get anything,so he accepts that deal...Next comes N2...Now he also would want to kill N5,since he would get advantage then...So the one that will be left is N1.The N5 person can share 1 more coin with N1.
So the solution scheme can be:
Person# No. of coins
N5 98
N4 0
N3 1
N2 0
N1 1
Similarly, there can be more solutions,(if N5 is less greedy),then
Person# No. of coins OR
N5 97 97
N4 0 0
N3 2 1
N2 0 0
N1 1 2