My Troll Puzzle (Examples)
Breaking down the hardest interview question I’ve ever encountered
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(These are examples to help break down an interview problem. When you’re ready, you can move on to the solution)
For these examples, you can pretend that everyone behaves a certain way. The key is to relax one of the constraints of the riddle, and play out what would happen.
In the first two examples, we pretend everyone has the same reasoning, but that reasoning is less than rational. In the final example, we pretend there are fewer people. Each example will get you closer to the final answer without actually revealing it.
Greedy Example
Suppose everyone is irrationally greedy. Pretend everyone wants all $1000 and wouldn’t approve anything less. If Ansel proposes taking $1000 for himself, he would vote for his own plan, but no one else would, because they want all $1000.
We see that Ansel would be thrown overboard.
Brooklyn would also be thrown overboard.
As would Cathy.
But Diego would have exactly 1/2 of the remaining votes, because only 2 people are left. His greedy plan would succeed if it came down to him and Ethan.
Now you’ve determined that Ansel clearly won’t propose taking $1000 for himself in the real problem, because he would be thrown overboard. He could work through this example in his head, and decide to consider a more equitable approach.
Equitable Example
Suppose everyone would propose to split the money equally, if it were their turn to propose a plan. Ansel suggests everyone gets $200. Does his plan get approved?
The answer depends on whether at least two of Brooklyn, Cathy, Diego, and Ethan would vote for this plan. Of course, if they don’t vote for this plan, Brooklyn will be the next to propose the same plan, but divided between the four remaining friends.
And now it becomes clear — even if everyone decides their plan is to divide the money equally, it’s always in their best interest to vote against the existing plan, since it’s preferable to divide the money among fewer people. Ansel, Brooklyn, and Cathy will be cast overboard before it’s down to Diego and Ethan again. Ethan will vote against Diego because he’d rather split $1000 among just himself, but Diego’s own vote is 1/2 of the remaining people at that point, so his plan would pass. Ethan and Diego will each get $500.
If we weren’t pretending that everyone intends to divide the money equally, would Diego give Ethan anything?
3 Person Example
Finally, suppose the original rules are the same, but only Ansel, Brooklyn, and Cathy are on the boat. They are all perfectly rational. Each person’s first priority is to stay dry, and their next priority is to get as much money as they can. Ansel knows that if Brooklyn and Cathy vote to throw him overboard, Brooklyn will get to choose by herself, because she will have 1/2 of the two remaining votes by herself.
Ansel knows that Brooklyn would only vote for his plan if he gave her all $1000, because that’s what she could get if she succeeds in throwing Ansel overboard.
Cathy, however, will get nothing if Ansel gets thrown overboard. Ansel can get Cathy’s vote (and earn a 2/3 majority) by giving her just $1.
With 3 people, Ansel will get $999, Brooklyn will get $0, and Cathy will get $1.
