Publications

📚About the book:

This book represents my personal journey of discovery, getting stuck, asking fundamental “why” questions, and trying to build understanding in mathematics, physics, and computation from first principles. Much of what appears here grew out of ponderings and explorations that brought me immense joy, or as Richard Feynman put it, “the pleasure of finding things out.”

🌟Scopes and Themes:

The book spans across three parts: ‘Mathematical Curiosities’, ‘Ponderings on the Physical Sciences’, and ‘Computational Methods’, covering the most intriguing parts of my journey, ranging from finding out asymptotic lengths of factorials, visualising how expanding cubes can solve cubic equations, deciphering how wheels move despite having stationary contact points, experimental approaches to find out the power law governing pulls of magnets, programming an engine to play combinatorial games like Nim, and many more.

✍️ Acknowledgements: 

I am especially grateful to Dr. Thomas Britz, Editor-in-Chief of the Parabola Journal, and Dr. James Tanton, Mathematician-at-Large at the Mathematical Association of America, for their invaluable reviews of the manuscript and insightful suggestions. I also extend a heartfelt thank you to Dr. Britz for making time out of his busy schedule to write the Foreword of the book.

🌱 Educational Commitment

In keeping with the educational spirit of the work, all author royalties from this book are donated to support the charitable and philanthropic initiatives of taaroop.org.

• Md Faiyaz Siddiquee Taaroop

The geometric proof employs the classic construction of the three squares on the three sides, but (at least to me) much more intuitive than Euclid's subsequent construction. I think that the equivalence between squares ACHI and ACKL (see the paper!) is quite nice.

The mechanical proof occurred to me as I watched my trusty night lamp (turned into a rod for the purposes of the paper) slide down my bedside desk (and proceed to break both its glass covering and my heart). Essentially, a sliding hypotenuse (the lamp) can cover all possible angles, from being completely vertical to being horizontal, thus "encoding" right-angled triangles of all shapes and sizes (as long as you just scale the diagram up or down). Encapsulating this changing phenomenon using a differential equation and appreciating that the length of the rod itself was invariant led to this proof, which is very close to my heart.

• Md Faiyaz Siddiquee Taaroop

The topic of my paper was inspired by a problem I encountered at the Bangladesh National Mathematical Olympiad a couple of years ago. The problem asked for the value of T(P) for a particular P. I could only solve it partially, building up to the casework along with the initial equation and inequalities. On a more sustained effort back home, I managed to derive the general algorithm to find T(P) for any P, which was the motivation behind working out the much simpler general formula, which I later found was first presented in 1985 by one of my favorite Number Theory authors - George E. Andrews - via the method of partitions.

• Md Faiyaz Siddiquee Taaroop

A quite elaborate (almost 2000 words) journey into how I often speed-learn math to apply to my own "quirky" problems such as counting subsets with an added constraint of prohibiting neighboring elements. The particular choice of explaining generating functions was due to its almost magical nature when I first learnt it, and the fact that I tried coming up with a novelty in how to motivate the concept, inspired from an unexpected source (The Rest is Science podcast). Hope you enjoy!

Relevant links to the contest's essay exhibition page on Tom Rocks Maths's website:

• https://tomrocksmaths.com/2026/07/22/trm-essay-competition-2026-group-xiv/ (9th entry in the batch)

• https://tomrocksmaths.com/wp-content/uploads/2026/07/md-faiyaz-siddiquee_trm_essay-md-faiyaz-siddiquee-taaroop.pdf