Envy-free division using mapping degree

In this paper we study envy-free division problems. The classical approach to some of such problem reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz theorem) is a boundary condition. This time we try to replace the boundary condition by a certain equivariance condition under all permutations, or a weaker condition of "pseudoequivariance", which has some economic meaning for the problem of partitioning a segment. Such versions of the problem have positive solutions when $n$ is a prime power, and in some cases we provide respective counterexamples for the case when $n$ is not a prime power.