Envy-free division in the presence of a dragon

Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing envy-free division problem in presence of an unpredictable (secretive, non-cooperative) player, called dragon. There are two basic scenarios. (1) $$r-1$$ players Once “cake” divided into r parts, dragon makes his choice grabs one pieces. After that, should be able to share remaining pieces fashion. (2) $$r+1$$ who divide cake A ferocious comes swallows players. The need cut advance such way that no matter unlucky player swallowed by dragon, can tiles manner. We emphasize both settings, some may prefer choose degenerate cake. Moreover, construct decision tree, allowing them minimize uncertainty what given each