An arbitrary number of squares whose sum, on excluding any one of them, is also a square

This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n>4$, only a finite number of numerical solutions, found by computer trials, are known. In fact, efforts to find parametric solutions for $n>4$ have so far been futile. In this paper we describe two methods of obtaining parametric solutions of the problem, and we apply these methods to get several parametric solutions when $n=5,6,7$ or $8$. We also indicate how parametric solutions may be obtained for larger values of $n$.