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

TL;DR

This paper explores methods to find parametric solutions for sets of n squares where removing any one results in a sum that is also a square, extending known solutions beyond n=4.

Contribution

The paper introduces two new methods for deriving parametric solutions for n > 4, providing explicit solutions for n=5 to 8 and discussing extensions for larger n.

Findings

Derived parametric solutions for n=5, 6, 7, 8

Presented two methods for generating solutions

Indicated potential for solutions at larger n

Abstract

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 $n$.