Five bankers are sharing 12 golden ingots.
They decide to proceed that way :
The elder one will suggest an ingots
allotment.
The rest will vote for or against it.
If the majority accepts, the sharing is ratified.
If not, the elder will be dismissed.
So, the sharing would be done between the remaining banker with the same
rules.
Knowing that they are set from left to right in a diminishing
order of their ages, how would be the allotment ?
The eldest should allot ingot like 9, 0, 1, 0, 2 among 5 bankers.
Let's name bankers as Banker 5, Banker 4 ...... Banker 1 according to decreasing order of their ages. ------------------------------------------------------------------------------
CASE 1 :
Suppose there are only 2 bankers left then the youngest will always deny whatever elder offers so that he can take away all 12 ingots on his turn.
If case is reduced to 3 bankers then the eldest knows that the youngest is not going to agree with him in any case. With that, the eldest will be dismissed and case reduced to CASE 1 where youngest can take away all.
So, the eldest here proposes allotment as 11, 1, 0. The Banker 2 has no option than to accept this proposal otherwise he won't get anything if case is reduced to 2 bankers as in CASE 1 above.
With 5 bankers, the eldest i.e. Banker 5 should propose allotment 9,0,1,0,2.
Obviously, any how Banker 4 is going to deny any proposal as he wants the distribution among 4 bankers where he will be getting 9 ingots as in CASE 3 above.
And if Banker 5 is eliminated and the case is reduced to CASE 3 where 4 bankers are left then Banker 3 knows he won't be getting anything. So, better he should be happily agree this proposition where he is getting at least1 ingot.
Finally, offering Banker 5 one more extra ingot than the case where 4 bankers will be left, makes him in favor of this proposition.
Notice that the Banker 5 has to give 3 ingots at least to banker 2 to get his vote as he will be getting 2 ingots in case of 4 bankers as in CASE 3. Whereas, in the same case Banker 3 is not getting anything & would be happily agree if getting 1 ingot at least in this case.
