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Brain Teasers - Problem Simplification

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Exploring problem simplification brain teasers
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The Screwy Pirates Brain Teaser

The screwy pirates teaser goes like this:


Five pirates plunder a chest filled with 100 gold pieces. Being democratic pirates, they agree on the following method of deciding the distribution:


The most senior pirate will propose a method of distribution, which is then voted on. His proposal succeeds if he gets at least 50% of the votes, and fails otherwise. If his proposal fails, he is made to walk the plank and the process repeats until a proposal is agreed to. You can make the following assumptions about the pirates:


1. They care about their lives first and getting as much gold as possible second

2. Given a choice of equal outcomes, they will choose to reduce the number of pirates by voting against the senior.


How will the gold be distributed?


The problem may seem a little daunting at first (it certainly was for me), but the solution is simple once you have broken the problem down by reducing the number of pirates the problem includes. Lets start with two pirates as having one pirate is irrelevant. We will define our pirates as pirate 5-1 in order of the seniority.


Two pirates: Pirate 2 will claim all the gold as he will get 50% of the gold regardless.

Three pirates: Knowing pirate 1 will vote against him if he offers him nothing, but also knowing that pirate 1 will get nothing if his proposal is not accepted, pirate 3 only needs to offer pirate 1 a single gold.

Four pirates: Pirate 4 knows that pirate 2 will get nothing if his proposal is voted against, as such, he only needs to offer pirate 2 a single coin to secure 50% of the vote.

Five pirates: Pirate 5 knows that pirate 3 and 1 will get nothing if his plan is voted down, therefore he only needs to offer pirate 3 and 1 one gold each to secure enough votes.


So we can see that, after the fifth case, a clear pattern has emerged. For any 2n + 1 pirates; n < 99, the most senior pirate will only need to offer pirates 1, 3, ..., 2n - 1, a single coin. For any 2n pirates; 1 < n < 99, the most senior pirate will only need to offer pirates 2, 4, ..., 2n - 2, a single coin.

The Tiger and Sheep Brain Teaser

The Tiger and Sheep teaser goes like this:

100 tigers and a single sheep are placed on an otherwise deserted island. The island has only grass and the 101 animals on it. Tigers can eat grass, but they would much prefer to eat sheep. However, when a tiger eats the sheep, it turns into a sheep itself. You can assume that all the tigers are rational and want to survive.

Will the sheep be eaten?

Again, to solve this the best method is to reduce the problem. If we have a single tiger with the sheep, the tiger is going to eat the sheep as it does not matter if it becomes a sheep after. If we have two tigers, they will know that if one of them eats the sheet, the remaining tiger will eat them in return. As such, neither of the tigers will eat the sheep. If there are three tigers, one of them will eat the sheep as the two remaining tigers afterwards will not want to. If there are four tigers, they will know that if they eat the sheep, they in turn will be eaten by one of the remaining three. So none of the four will eat the sheep. 

From this a very simple pattern can be observed. If n = the number of tigers, where n is odd the sheep will be eaten and where n is even the sheep will not be eaten. Since in this question n = 100, the sheep will not be eaten.