## Problem Solving

### Techniques

My oldest plays the piano! Thieves! The Jordan Curve Theorem
Handshakes Metapuzzles Borsuk-Ulam and Fixed Point
A geometric fallacy What day of the week? The Cantor Set
Next number? Exact expression A space-filling arc in the plane
Remainders Chess puzzles The Rigidity of the sphere
Probability and Divisibility Points on a plane A non-measurable set
Triangle-square polygons Sequence of triangles Compact Surfaces
Fibonacci Alleys
Arithmetic Expressions Three circles
Unusual dice El Pais challenge
Boundary Operators Powers of two
The Ultimate Metapuzzle Trigonometry
Stones, balances, matrices

Ever since I started reading Martin Gardner in high school, I have had a natural fascination for puzzles and riddles.  I was encouraged to follow Gardner’s column in Scientific American by one of the old Physics instructors in school, and I borrowed from him a copy of Mathematical Circus.  I devoured that book in a few nights, stealing hours from my sleep until exhausted.  Every now and then I would start playing around with a particular riddle after dinner, only to get startled by the sound of the alarm clock in the morning, realizing that, once more, the sleeping I skipped would have to be done in Literature or History class.

Later on, the Philosophy instructor introduced us to the beauty of logic and logic riddles.  He recommended us to follow one author in particular: Raymond Smullyan (read, for example, What is the name of this book?).

It is hard to explain the rewarding sensation of accomplishment after figuring out not only a solution to the puzzle at hand, but also a generalization of the techniques employed, and even better, more puzzles of my own.

It was then no surprise to anybody that I chose as a trade one that would allow me to face these puzzles for a living: I was equally interested in engineering, physics, architecture, but nothing felt so much “at home” as those challenges that Mathematicians encountered daily.  I grew in awe with Euler, Gauss, Newton, Fermat—to name a few of the most influential and inspiring mathematicians of all times.

Learning Mathematics to me was a matter of acquiring as many techniques as possible,  and using them to solve as many problems I could find. Memory played always a second role: if intuition tells you that certain technique can be useful, go ahead and re-read the sources where you learned a particular subject, and work through the statements, examples and remarks until you find your way.  Eventually, you will not need to search your memory: the solution pops naturally in your mind.

In these pages, I plan to revisit some classical puzzles and riddles.  Also, I would like to introduce the reader to some of the most well-known techniques and serious mathematical problems that I used to mature as a scientist.  I do hope you find them as stimulating and fascinating as I did the first time I faced them.  In the case of the riddles, make sure to work them out before reading the solution, although just in case, I will always leave the “punch-line” for you to finish them.