▁Mack en zie ▁presented ▁a ▁discrete ▁env y - free ▁protocol ▁that ▁requires ▁at ▁most ▁ ▁queries . ▁There ▁is ▁still ▁a ▁very ▁large ▁gap ▁between ▁the ▁lower ▁bound ▁and ▁the ▁procedure . ▁As ▁of ▁August ▁ 2 0 1 6 , ▁the ▁exact ▁run - time ▁complexity ▁of ▁env y - f reen ess ▁is ▁still ▁unknown . ▁ ▁For ▁the ▁case ▁of ▁connected ▁pieces , ▁it ▁was ▁noted ▁that ▁the ▁hard ness ▁result ▁assumes ▁that ▁the ▁entire ▁c ake ▁must ▁be ▁divided . ▁If ▁this ▁requirement ▁is ▁replaced ▁by ▁the ▁we aker ▁requirement ▁that ▁every ▁partner ▁receives ▁a ▁proportional ▁value ▁( at ▁least ▁ 1 / n ▁of ▁the ▁total ▁c ake ▁value ▁according ▁to ▁their ▁own ▁valu ation ), ▁then ▁a ▁bounded ▁procedure ▁for ▁three ▁partners ▁is ▁known , ▁but ▁it ▁has ▁remained ▁an ▁open ▁problem ▁whether ▁there ▁exist ▁bounded - time ▁procedures ▁for ▁four ▁or ▁more ▁partners . ▁ ▁Con ne cted ▁pieces ▁ ▁Ex istence ▁proof ▁▁ ▁An ▁env y - free ▁division ▁for ▁n ▁agents ▁with ▁connected ▁pieces ▁always ▁exists ▁under ▁the ▁following ▁m ild ▁assumptions : ▁ ▁No ▁agent ▁pre fers ▁an ▁empty ▁piece ▁over ▁a ▁non - empty ▁piece . ▁ ▁The ▁prefer ences ▁of ▁the ▁agents ▁are ▁continuous . ▁Note ▁that ▁it ▁is ▁not ▁required ▁that ▁the ▁prefer ences ▁of ▁the ▁agents ▁are ▁represented ▁by ▁an ▁add itive ▁function . ▁ ▁The ▁main ▁concept ▁in ▁the ▁proof ▁is ▁the ▁simple x ▁of ▁partitions . ▁Suppose ▁the ▁c ake ▁is ▁the ▁interval ▁[ 0 ,