  In this paper we study envy-free division problems. The classical approach to
such problems, used by David Gale, reduces to considering continuous maps of a
simplex to itself and finding sufficient conditions for this map to hit the
center of the simplex. The mere continuity of the map is not sufficient for
reaching such a conclusion. Classically, one makes additional assumptions on
the behavior of the map on the boundary of the simplex (for example, in the
Knaster--Kuratowski--Mazurkiewicz and the Gale theorem).
  We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and
replace the boundary condition by another assumption, which has the meaning in
economy as the possibility for a player to prefer an empty part in the segment
partition problem. We solve the problem positively when $n$, the number of
players that divide the segment, is a prime power, and we provide
counterexamples for every $n$ which is not a prime power. We also provide
counterexamples relevant to a wider class of fair or envy-free division
problems when $n$ is odd and not a prime power.
  In this arxiv version that appears after the official publication we have
corrected the statement and the proof of Lemma 3.4.
