Adding a bit more explanation... 4. Therefore, the linear function giving the values of the sum simplex is constant by #2. Since any point's value is a weighte

Adding a bit more explanation... 4. Therefore, the linear function giving the values of the sum simplex is constant by #2. Since any point's value is a weighted average of vertex values, and all the vertex values are the same, every point's value must also be the same. 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. The numbers are assigned based on the first coordinate: f(x,y,z) = x, and similarly in higher dimensions. "Rotations" are done by switching x with any other coordinate axis (actually a reflection), which maps the simplex to itself and maps integral points to integral points.