The taxicab metric is useful here because if you compute the components of a distance in the square metric, you must max() them together, while under the taxicab metric you can add them together. We can add a pair of large sums together to get the large sum of pairs, but we can't max() a pair of large sums together to get the large sum of max()es.

One way to get to the taxicab metric verson of the problem is to take w = x + y and z = x - y. From then on all the distances you calculate in wz-space under the taxicab metric will be precisely twice the distance between the same points in xy-space under the square metric.