Derivation and Applications of a Formula for Balancing Numbers Using Range Endpoints

This paper presents a mathematical derivation of a formula to determine the balancing number x within a given range [A, L]. A balancing number is a unique integer in the range that equally partitions the sum of integers on both sides. The method relies solely on the starting A and ending L numbers, eliminating the need for a balancer. This efficient approach is validated with examples and visualizations, showcasing its accuracy and potential applications in resource allocation and optimization. Keywords: Balancing Numbers, Number Theory, Optimization, Math, Data Visualization, Diophantine Equations