The Pile of Squares: Two Roads to Ten

A single hand-worked page is recorded: a pile of ten square layers totaling 385 (the square pyramidal number at n = 10); the pyramid-in-a-box working (27 the box, 9 the pyramid) by which the author independently derived why the pile's volume is one third of its box; and a 6-8-10 Pythagorean triangle set beside the pile. The two constructions meet at the number 10 — the pile's height in layers and the triangle's hypotenuse. The note claims priority for the derivation, the page, and the reading of that meeting, and states plainly that the underlying arithmetic is classical. Version 2 adds a naming addendum fixing the names of the construction: the Ruano Squares, the Ruano Equation, and the Ruano Pile.