The fact that |z|.|w| = |zw| for complex numbers z and w leads to a "sums of squares identity of type (2,2,2)", namely [x1^2 + x2^2] [y1^2 + y2^2] = [x1y1 - x2y2]^2 + [x1y2 + x2y1]^2. Similarly, quaternion and octonion numbers lead to analogous identities of type (4,4,4) and (8,8,8). The general problem, posed by A. Hurwitz in 1898, is to express [x1^2 +...+ xr^2][y1^2 +...+ ys^2] in the integral polynomial ring Z[x1,...,xr; y1,...,ys] as a sum of squares with as few summands as possible. This problem remains open up to the present day. In fact, the least number of summands needed, denoted by r*s, has not even been asymptotically estimated.
Signed intercalate matrices furnish a combinatorial tool to tackle the Hurwitz problem. I shall present some joint work with J.Zhu, and explain how a recently uncovered identity of type (8,8,14) leads unexpectedly to a new harmonic map between high dimensional spheres.