New Ways to Balance Numbers

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from 1 to b – 1 is equal to the sum of the numbers from b + 1 to b + r for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.