Pirates and Pigeons

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Q:Six pirates meet on an island. Prove that at least three of them are all mutual friends or at least three of them are complete strangers?
Discrete Mathematics with Applications

A:This is yet another case for applying the Pegionhole principle. Though the way it can be applied is quite subtle. Lets start by enumerating the many possible connections that are possible. For example, if we name the pirates A,B,C,D,E and F, then the possible connections are AB, AC, AD,...,EF.
A would have 5 connections, B four and so on until E has one. This enumerates out to 15 possible connections. The connections for just one pirate is shown in the picture below.

If there are \(n\) pirates, the total number of connections possible is given by \(\frac{n(n-1)}{2}\), the sum of integers to \(n-1\). Each of these connections falls into two states (i.e. our pegionholes). They are either "friends" or "strangers". As an example, assume 10 connections fall into the "friends" state and the remaining 5 connections fall into "strangers". Here is a key point, if 10 connections fall into friends, the number of pirates required to make 10 connections is 5 as \(\frac{5\times (5-1)}{2}= 10\) from the equation above. Such a scenario would automatically prove the question. The worst case scenario is when there is more or less an even distribution of the connections in both states, i.e. 8 in "friends" and 7 in "strangers" or vice versa. If 4 connections are in the strangers state then the least number of pirates that would create 7 states is 4 as \(\frac{4\times (4-1)}{2}=6\). Note it takes only 4 pirates to explain 6 connections, let alone 7 or 8. So if there are more than 6 connections in any given state, then there are at least 4 pirates that go into creating those connections.
This proves that either

1. At least 3 are strangers OR 
2. At least 3 are friends.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)
This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms
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Introduction to Probability Theory
Overall an excellent book to learn probability, well recommended for undergrads and graduate students

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)
A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.

Introduction to Probability, 2nd Edition
A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)
An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job

Understanding Probability: Chance Rules in Everyday Life
This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)
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Discovering Statistics Using R
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A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Probability and Statistics (4th Edition)This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good