▁Using ▁a ▁certain ▁topological ▁lemma ▁( that ▁is ▁similar ▁to ▁the ▁Kn aster – K ur at owski – M az urk iew icz ▁lemma ), ▁it ▁is ▁possible ▁to ▁prove ▁that ▁the ▁intersection ▁of ▁all ▁U i ' s ▁is ▁non - empty . ▁Hence , ▁there ▁is ▁a ▁partition ▁in ▁which ▁every ▁piece ▁is ▁the ▁unique ▁preference ▁of ▁an ▁agent . ▁Since ▁the ▁number ▁of ▁pieces ▁equals ▁the ▁number ▁of ▁agents , ▁we ▁can ▁allocate ▁each ▁piece ▁to ▁the ▁agent ▁that ▁pre fers ▁it ▁and ▁get ▁an ▁env y - free ▁allocation . ▁ ▁Pro ced ures ▁ ▁For ▁three ▁agents , ▁an ▁env y - free ▁division ▁can ▁be ▁found ▁using ▁several ▁different ▁procedures : ▁ ▁The ▁St rom qu ist ▁moving - kn ives ▁procedure ▁uses ▁four ▁simultaneously - mov ing ▁kn ives ▁- ▁one ▁moved ▁by ▁a ▁ref eree ▁and ▁another ▁one ▁for ▁each ▁agent . ▁ ▁Bar ban el – Br ams ▁moving - kn ives ▁procedure ▁uses ▁two ▁simultaneously - mov ing ▁kn ives . ▁ ▁The ▁Robert son – Web b ▁rot ating - kn ife ▁procedure ▁can ▁be ▁used ▁when ▁the ▁c ake ▁is ▁two - dimensional , ▁and ▁uses ▁only ▁a ▁single ▁moving - kn ife . ▁ ▁These ▁are ▁continuous ▁procedures ▁- ▁they ▁rely ▁on ▁people ▁moving ▁kn ives ▁continu ously ▁and ▁simultaneously . ▁They ▁cannot ▁be ▁executed ▁in ▁a ▁finite ▁number ▁of ▁discrete ▁steps . ▁ ▁For ▁n ▁agents , ▁an ▁env y - free ▁division ▁can ▁be ▁found ▁by ▁Sim