m ons ' ▁c ake - cut ting ▁protocol . ▁The ▁protocol ▁uses ▁a ▁simple x ▁of ▁partitions ▁similar ▁to ▁the ▁one ▁used ▁in ▁St rom qu ist ' s ▁existence ▁proof . ▁It ▁generates ▁a ▁sequence ▁of ▁partitions ▁which ▁converges ▁to ▁an ▁env y - free ▁partition . ▁The ▁convergence ▁might ▁take ▁infinitely ▁many ▁steps . ▁ ▁It ▁is ▁not ▁a ▁coinc idence ▁that ▁all ▁these ▁algorithms ▁may ▁require ▁infinitely ▁many ▁queries . ▁As ▁we ▁show ▁in ▁the ▁following ▁sub section , ▁it ▁may ▁be ▁impossible ▁to ▁find ▁an ▁env y - free ▁c ake - cut ting ▁with ▁connected ▁pieces ▁with ▁a ▁finite ▁number ▁of ▁queries . ▁ ▁Hard ness ▁result ▁ ▁An ▁env y - free ▁division ▁with ▁connected ▁pieces ▁for ▁ 3 ▁or ▁more ▁agents ▁cannot ▁be ▁found ▁by ▁a ▁finite ▁protocol . ▁ ▁The ▁reason ▁this ▁result ▁doesn ' t ▁contradict ▁the ▁previously ▁mentioned ▁algorithms ▁is ▁that ▁they ▁are ▁not ▁finite ▁in ▁the ▁mathematical ▁sense . ▁ ▁The ▁impos s ibility ▁proof ▁uses ▁a ▁rig id ▁measure ▁system ▁( R MS ) ▁– ▁a ▁system ▁of ▁n ▁value ▁measures , ▁for ▁which ▁an ▁env y - free ▁division ▁must ▁cut ▁the ▁c ake ▁at ▁very ▁specific ▁locations . ▁Then , ▁finding ▁an ▁env y - free ▁division ▁reduces ▁to ▁finding ▁these ▁specific ▁locations . ▁Assuming ▁the ▁c ake ▁is ▁the ▁real ▁interval ▁[ 0 , 1 ], ▁finding ▁an ▁env y - free ▁division ▁using ▁queries ▁to ▁the ▁agents ▁is ▁equivalent ▁to ▁finding ▁a ▁real ▁number ▁in ▁the