Arithmetic Billiards

Arithmetic billiards is a concept in recreational mathematics that utilizes a geometrical approach to determine the least common multiple (LCM) and greatest common divisor (GCD) of two natural numbers. This is achieved by analyzing a path within a rectangle defined by the two numbers, where the path reflects off the sides at a 45° angle. The path's characteristics depend on the relationship between the two numbers: if one divides the other, the path is a simple zigzag; otherwise, it exhibits self-intersections. The number of unit squares crossed by the path corresponds to the LCM, while the GCD can be determined from the first segment of the path. The path's symmetry and distribution of contact points along the rectangle's perimeter are also notable features. Additionally, periodic paths can be established if the starting point is any integer coordinate within the rectangle, provided the sides are not coprime. The study of arithmetic billiards has been popularized by mathematicians such as Hugo Steinhaus and Martin Gardner, and it serves as a source of mathematical puzzles and educational tools.

- Arithmetic billiards help determine LCM and GCD using geometric paths.

- The path reflects at 45° angles within a rectangle defined by two numbers.

- The path's characteristics vary based on whether one number divides the other.

- The concept has been popularized in recreational mathematics and educational contexts.

- Periodic paths can exist unless the rectangle sides are coprime.