We answer the question of when a new point can be added in a continuous way to configurations of n n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of n n points if and only if n ≠ 1 n \neq 1 . On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if n = 2 n = 2 . These results generalize the Brouwer fixed-point theorem, which gives the negative answer when n = 1 n=1 . We also show that when n = 2 n=2 , there is a unique solution to both the ordered and unordered versions of the problem up to homotopy.