Problem Solving
Puzzles 
Techniques 

My oldest plays the piano!  Thieves!  The Jordan Curve Theorem 
Handshakes  Metapuzzles  BorsukUlam and Fixed Point 
A geometric fallacy  What day of the week?  The Cantor Set 
Next number?  Exact expression  A spacefilling arc in the plane 
Remainders  Chess puzzles  The Rigidity of the sphere 
Probability and Divisibility  Points on a plane  A nonmeasurable set 
Trianglesquare polygons  Sequence of triangles  Compact Surfaces 
Fibonacci  Alleys  
Arithmetic Expressions  Three circles  
Pick a point  Bertrand Paradox  
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 reread 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 wellknown 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 “punchline” for you to finish them.