A Square in Several Forms: Weighted Squares and Geometric Rearrangement

For positive integers x and y, put N = 2xy + x + y. The odd number 2N + 1 has factors 2x + 1 and 2y + 1. An explicit dissection of the N-square forms a rectangle of area 2xy(N + 1) and leaves 2y + 1 squares of side x together with 2x + 1 squares of side y. The factors thus appear as counts of square pieces. The remaining weighted area has two continuations. One reconstructs the sum-square and leads to an oblong with one unit left over. The other retains the difference-square and recovers a neighboring-square Match Factoring area. A reversible frame dissection connects the two routes. Worked examples show how a change of factor pair moves an internal boundary and how equal factors make the difference-square disappear. Square-valued weighted areas lead to classical Pell-type and Fibonacci-Lucas families and to divisors of N + 1. The purpose is to make relationships visible through coordinated changes of form, in the spirit of WidiVision. The presentation brings established identities into one geometric setting and offers concrete questions for exploration. Developed through Human-AI collaboration with Bill Widi as curator. Bill Widi supplied the starting reformulation, directed and reviewed the investigation, and guided its interpretation and presentation. ChatGPT, an AI system developed by OpenAI, assisted with mathematical analysis, computational verification, illustrations, drafting, and editing. Bill Widi retains responsibility for the claims and publication.