On partitions into squares of distinct integers whose reciprocals sum to 1

In 1963, Graham proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of such partitions by proving that 8542 is the largest integer with no such partition

Enumeration of Payphone Permutations

The desire for privacy significantly impacts various aspects of social behavior as illustrated by people's tendency to seek out the most secluded spot when multiple options are available. In particular, this can be seen at rows of payphones, where people tend to occupy an available payphone that is most distant from already occupied ones. A similar tendency was observed for urinals in a male restroom and was stated as one of the rules in the Male Restroom Etiquette documentary by Phil Rice. Assuming that there are $n$ payphones in a row and that $n$ people occupy payphones one after another as privately as possible, the resulting assignment of people to payphones defines a permutation, which we will refer to as a \emph{payphone permutation}. It can be easily seen that not every permutation can be obtained this way. In the present study, we consider different variations of payphone permutations and enumerate them