▁division . ▁William ▁Gas arch ▁has ▁compared ▁the ▁three ▁un bounded ▁algorithms ▁using ▁ord inal ▁numbers . ▁ ▁The ▁question ▁of ▁whether ▁env y - free ▁c ake - cut ting ▁can ▁be ▁done ▁in ▁bounded ▁time ▁for ▁four ▁or ▁more ▁partners ▁had ▁been ▁open ▁for ▁many ▁years . ▁It ▁was ▁finally ▁solved ▁in ▁ 2 0 1 6 ▁by ▁Har iz ▁Az iz ▁and ▁Simon ▁Mack en zie . ▁Initial ly ▁they ▁developed ▁a ▁bounded - time ▁algorithm ▁for ▁four ▁partners . ▁Then ▁they ▁extended ▁their ▁algorithm ▁to ▁handle ▁any ▁number ▁of ▁partners . ▁Their ▁algorithm ▁requires ▁at ▁most ▁ ▁queries . ▁While ▁this ▁number ▁is ▁bounded , ▁it ▁is ▁very ▁far ▁from ▁the ▁lower ▁bound ▁of ▁. ▁So ▁the ▁question ▁of ▁how ▁many ▁queries ▁are ▁required ▁for ▁env y - free ▁c ake - cut ting ▁is ▁still ▁open . ▁ ▁Appro xim ations ▁and ▁partial ▁solutions ▁ ▁A ▁re entr ant ▁variant ▁of ▁the ▁last ▁dimin isher ▁protocol ▁finds ▁an ▁add itive ▁approximation ▁to ▁an ▁env y - free ▁division ▁in ▁bounded ▁time . ▁Specifically , ▁for ▁every ▁constant ▁, ▁it ▁returns ▁a ▁division ▁in ▁which ▁the ▁value ▁of ▁each ▁partner ▁is ▁at ▁least ▁the ▁largest ▁value ▁minus ▁, ▁in ▁time ▁. ▁ ▁If ▁all ▁value ▁measures ▁are ▁piece wise ▁linear , ▁there ▁is ▁an ▁algorithm ▁which ▁is ▁polynomial ▁in ▁the ▁size ▁of ▁the ▁representation ▁of ▁the ▁value ▁functions . ▁The ▁number ▁of ▁queries ▁is ▁, ▁where ▁ ▁is ▁the ▁number ▁of ▁dis continu ities ▁in ▁the ▁derivatives ▁of ▁the