This is delightful! I believe these sums-of-powers formulas can be derived fairly easily with discrete calculus techniques (it's in Knuth's Concrete Mathematics, IIRC), but to see a purely geometric approach succeed is fascinating.<p>To understand why this technique works with sums of arbitrary powers, the key is to show that summing together rotations of simplices gets you a simplex of constant values. The following geometric argument works:<p>1. Think of the numbers in the simplex as a function f(x) of the coordinate vector x for the space the simplex is embedded in. Observe that f is linear.<p>2. A linear function on a simplex is determined uniquely by the values it takes at the vertices. Specifically, the value at any point is a weighted average of vertex point values.<p>3. If we create a "sum simplex" by summing together all possible rotations of the simplex, the result will also be a linear function on a simplex, and the values at all the vertices will be the same by symmetry.<p>4. Therefore, the linear function giving the values of the sum simplex is constant.<p>A rigorous proof could be developed by considering the standard simplex generated by the coordinate unit vectors, but scaled by a factor of n. For example, the equilateral triangle is the 2D simplex generated by connecting the vertices at (n,0,0), (0,n,0), (0,0,n). On this simplex, the integral points (x,y,z), where the coordinates x,y,z are integers, are the points assigned numbers in the blog post. "Rotations" are done by switching coordinate axes (actually a reflection), which preserves the integral points.