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Return to Treasure Island (Posted on 2012-04-05) Difficulty: 3 of 5

'Arr, ' said Long John Silver, 'Arr, arr,' by way of clearing his throat, before addressing his 9 remaining companions.

'We've sailed the seas for 7 long years, and plundered a pirate hoard, and the time has come to divvy the loot at last. It goes without saying that each of you lubbers wants all the treasure he can lay his hands on.'

'Now, as ye know, the Rules of the Sea require that the Captain shall propose a division, and all then vote, democratical-like; if the proposal be passed, then all take accordingly. If the lots cast be tied, the Captain votes again, to break the tie. But if the division be rejected, the Captain forfeits his share, and the right to propose passes to the next in rank, and so on until a division be agreed at last.'

Now gather round me hearties, and listen to me proposal, according to rank:

Long John Silver, Cook and Captain: 75 shares
Abe the Master: 1 share.
Ben the Mate: 1 share
Caleb the Gunner: 1 share
Dave the Able: 1 share
Eben the Bosun: 1 share
Fergus the Feeble: 2 shares
Gabe the Gimpy: 4 shares
Ham the Hopeless: 6 shares
Jesse the Cabinperson: 8 shares

'I'm sure you'll agree, lads, I've done you all proud.'

Of course, it's endlessly debatable whether Long John's proposal is fair; but should his proposal be carried, or not?

No Solution Yet Submitted by broll    
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Solution No Subject | Comment 11 of 14 |
My solution depends on the share division: is 0 shares allowed or not?  Given the problem's wording, I thought 1 share was the minimum.

*****************************************************
I don't see how all votes can be predicted and I started from the bottom up.

Suppose all propositions are rejected and we end with Ham and Jesse.

Ham will propose:
Ham 99 and Jesse 1.

Votes will be:
Ham: Pro
Jesse: No
Tie vote Ham Pro
Proposition Accepted

Jesse will accept any proposition that is better than 1 share.
-------------------------------------------------------------
Situation with Gabe, Ham and Jesse.
Gabe will propose:
Gabe: 98
Ham: 1 
Jesse: 2

Votes will be:
Gabe: Pro
Ham: No (Since 1 is less than 99)
Jesse: Pro (Since 2 is more than 1)
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Fergus, Gabe, Ham and Jesse.
Fergus will propose:
Fergus: 95
Gabe: 1
Ham: 1 
Jesse: 3 (2 will only be a draw with the GHJ situation)

Votes will be:
Fergus: Pro
Gabe: No
Ham: No 
Jesse: Pro (Since 3 is more than 2)
Tie vote Fergus: Pro
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Eben, Fergus, Gabe, Ham and Jesse.
Eben will propose: 
Eben: 92
Fergus: 1
Gabe: 1
Ham: 2 (1 will only be a draw with the FGHJ situation)
Jesse: 4 (3 will only be a draw with the FGHJ situation)

Votes will be:
Eben: Pro
Fergus: No
Gabe: No
Ham: Pro (Since 2 is more than 1)
Jesse: Pro (Since 4 is more than 3)
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Dave, Eben, Fergus, Gabe, Ham and Jesse.
Dave will propose:
Dave: 89
Eben: 1
Fergus: 1
Gabe: 1
Ham: 3 (2 will only be a draw with the EFGHJ situation)
Jesse: 5 (4 will only be a draw with the EFGHJ situation)

Votes will be:
Dave: Pro
Eben: No
Fergus: No
Gabe: No
Ham: Pro (Since 3 is more than 2)
Jesse: Pro (Since 5 is more than 4)
Tie vote Dave: Pro
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Caleb, Dave, Eben, Fergus, Gabe, Ham and Jesse.
Caleb will propose:
Caleb 85
Dave: 1
Eben: 1
Fergus: 1
Gabe: 2 (1 will only be a draw with the DEFGHJ situation)
Ham: 4 (3 will only be a draw with the DEFGHJ situation)
Jesse: 6 (5 will only be a draw with the DEFGHJ situation)

Votes will be:
Caleb: Pro
Dave: No
Eben: No
Fergus: No
Gabe: Pro (Since 2 is more than 1)
Ham: Pro (Since 4 is more than 3)
Jesse: Pro (Since 6 is more than 5)
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Ben, Caleb, Dave, Eben, Fergus, Gabe, Ham and Jesse.
Ben propose:
Ben: 79
Caleb: 1
Dave: 1
Eben: 1
Fergus: 1 
Gabe: 3 (2 will only be a draw with the CDEFGHJ situation)
Ham: 5 (4 will only be a draw with the CDEFGHJ situation)
Jesse: 7 (6 will only be a draw with the CDEFGHJ situation)

Votes will be:
Ben: Pro
Caleb: No
Dave: No
Eben: No
Fergus: No
Gabe: Pro (Since 3 is more than 2)
Ham: Pro (Since 5 is more than 4)
Jesse: Pro (Since 7 is more than 6)
Tie vote Ben: Pro
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with Abe, Ben, Caleb, Dave, Eben, Fergus, Gabe, Ham and Jesse.
Abe will propose:
Abe: 76
Ben: 1
Caleb: 1
Dave: 1
Eben: 1
Fergus: 2 (1 will only be a draw with the BCDEFGHJ situation)
Gabe: 4 (3 will only be a draw with the BCDEFGHJ situation)
Ham: 6 (5 will only be a draw with the BCDEFGHJ situation)
Jesse: 8 (7 will only be a draw with the BCDEFGHJ situation)

Votes will be:
Abe: Pro
Ben: No
Caleb: No
Dave: No
Eben: No
Fergus: Pro (Since 2 is more than 1)
Gabe: Pro (Since 4 is more than 3)
Ham: Pro (Since 6 is more than 5)
Jesse: Pro (Since 8 is more than 7)
Proposition Accepted
This is a stable situation, from here on, nobody will be voted away.  Let's see if we can reach an "higher" stable situation.
-------------------------------------------------------------------------
Situation with LJS, Abe, Ben, Caleb, Dave, Eben, Fergus, Gabe, Ham and Jesse.
LJS will propose: 
LJS: 71
Abe: 1
Ben: 1
Caleb: 1
Dave: 1
Eben: 1
Fergus: 3 (2 will only be a draw with the ABCDEFGHJ situation)
Gabe: 5 (4 will only be a draw with the ABCDEFGHJ situation)
Ham: 7 (6 will only be a draw with the ABCDEFGHJ situation)
Jesse: 9 (8 will only be a draw with the ABCDEFGHJ situation)

Votes will be:
LJS: Pro
Abe: No
Ben: No
Caleb: No
Dave: No
Eben: No
Fergus: Pro (Since 3 is more than 2)
Gabe: Pro (Since 5 is more than 4)
Ham: Pro (Since 7 is more than 6)
Jesse: Pro (Since 9 is more than 8)
Tie vote LJS: Pro
Proposition Accepted
*******************************************************************************
Now, given the difference between my calculation and the problem's suggestion, it is likely that I miscalculated somewhere. Or, I'm too good for this world and LJS didn't give a single share to ABCD and E and took all of those shares himself, but that would end him on 76.
*******************************************************************************
Anyway, we're not there yet and I keep following my above calculations.
At this moment, all pirates knwow that LJS will have 71 shares. So there's 29 left.  Will it be possible for LJS to buy 4 votes at a lesser cost?

YES!
If LJS proposes
LJS: 86
Abe: 2
Ben: 2
Caleb: 2
Dave: 2
Eben: 1
Fergus: 1
Gabe: 1
Ham: 1
Jesse: 1
---------------------------------------------------------------------------------
ABCD may want more shares and will put pressure on LJS, such that MJS lands above 71 with the rest of the shares shred between ABCD.  That will be 3 a person, giving the following
LJS: 75
Abe: 5
Ben: 5
Caleb: 5
Dave: 5
Eben: 1
Fergus: 1
Gabe: 1
Ham: 1
Jesse: 1
But, since in the original calculation, Fergus costed only 3 votes, he can be "bought" with 4 votes, leaving LJS with 76.

  Posted by Hugo on 2012-04-06 17:21:05
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