It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the creation of groups of students, teachers and mathematicians called “Mathematical Circles”. The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport—without necessarily being competitive. This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of the school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be “extracurricular mathematics”.

The paper underscores the effectiveness of leveraging graph theory in urban planning and establishes a solid foundation for implementing sustainable, accessible city models that can adapt to the unique needs of various urban landscapes.

In this paper, we explore a non-conventional proof of the Seven Circles Theorem using several concepts from hyperbolic geometry. We attempt to represent the picture, claimed by the statement, in the Klein model—followed by the Poincar´e’s hyperbolic disk model of hyperbolic space—in order to analyze the claim. We consider an ideal hexagon to have been formed by the points of intersection of each of the six inner circles and the ideal boundary. We then assume that there exists a non-ideal hyperbolic triangle that is formed as a result of intersections between the three main diagonals of the hexagon. We then go on to contradict this claim by proving that the area of the non-ideal triangle is zero.