Cube Extension Method: Why Two Cubes Cannot Sum to a Cube but Three Cubes Can

Abstract: We present a simple geometric method called "cube extension" to compare the Diophantine equations x^3 + y^3 = z^3 and w^3 = x^3 + y^3 + z^3. Starting from the identity (n+k)^3 - n^3 = 3n^2 k + 3n k^2 + k^3, we interpret the right‑hand side as the volume of three slabs added to a cube of side n to obtain a cube of side n+k. We prove that this added volume can never be a perfect cube (which gives an elementary proof of the n=3 case of Fermat's Last Theorem), but it can often be expressed as a sum of two cubes, leading to infinitely many integer solutions of w^3 = x^3 + y^3 + z^3. Explicit primitive solutions and parametric families are provided, including a step‑by‑step numerical illustration of growing a cube from side 40 to side 50. The method is constructive, visual, and suitable for a wide audience.