Dissection of a n x n Square into n Congruent Squares where n is an integer, by Lagrange Four-Square and Recursive Closest-Lin Dissection Methods

We present efficient unique dissections for n=2,3,5 (Fig.1-3) and a general constructive method for any integer n. By Lagrange four-square theorem n=a^2+b^2+c^2+d^2, we grid a^2 squares of side sqrt(n) at corner A (Closest square) leaving an L-shaped remainder L1 of area (b^2+c^2+d^2)n. By Wallace-Bolyai-Gerwien theorem L1 is equidecomposable to an intermediate square M1 of side sqrt(n k1) where k1=b^2+c^2+d^2. Recursion n->k<n has depth O(log n) and terminates at Terminator Set T={2,3,5} U {k^2}. The set T is extensible: whenever a most efficient dissection for b>v is proved, T extends to T U {b}. The method proves a sum-of-four-squares system is far more efficient than a prime-product system. L->M dissections are given in Appendix A-C (Fig.11-14) as starting points for further work. DOI: 10.5281/zenodo.23998309 Author: Xu Brett Lin, Independent Researcher, Sydney, NSW, Australia Keywords: geometric dissection, Bolyai-Gerwien, Lagrange four-square, Closest-Lin method, Terminator Set