▁sharing ▁a ▁square ▁c ake ▁and ▁wanting ▁square ▁pieces ▁the ▁proportional ity ▁is ▁, ▁and ▁this ▁is ▁the ▁best ▁that ▁can ▁be ▁guaranteed ▁even ▁without ▁env y . ▁▁ ▁For ▁three ▁partners ▁sharing ▁a ▁square ▁c ake ▁and ▁wanting ▁square ▁pieces ▁the ▁proportional ity ▁is ▁; ▁the ▁best ▁that ▁can ▁be ▁guaranteed ▁without ▁env y ▁is ▁. ▁ ▁Dis connected ▁pieces ▁ ▁Pro ced ures ▁▁ ▁For ▁three ▁partners , ▁the ▁Self ridge – Con way ▁discrete ▁procedure ▁makes ▁an ▁env y - free ▁division ▁with ▁at ▁most ▁ 5 ▁cut s . ▁ ▁Other ▁procedures ▁using ▁moving ▁kn ives ▁require ▁fewer ▁cut s : ▁ ▁The ▁Lev more – Cook ▁moving - kn ives ▁procedure ▁requires ▁at ▁most ▁ 4 ▁cut s ; ▁ ▁The ▁Br ams – T aylor – Z w icker ▁rot ating - kn ives ▁procedure ▁for ▁pie - cut ting ▁requires ▁at ▁most ▁ 3 ▁cut s ▁( this ▁results ▁in ▁three ▁connected ▁pieces ▁when ▁the ▁c ake ▁is ▁a ▁circle , ▁but ▁when ▁the ▁c ake ▁is ▁an ▁interval ▁there ▁will ▁be ▁four ▁pieces ); ▁ ▁Using ▁the ▁Austin ▁moving - kn ife ▁procedure ▁for ▁two ▁partners , ▁it ▁is ▁possible ▁to ▁get ▁an ▁env y - free ▁division ▁for ▁ 3 ▁partners ▁using ▁at ▁most ▁ 3 ▁cut s . ▁This ▁idea ▁is ▁attributed ▁to ▁William ▁Web b . ▁ ▁For ▁four ▁partners , ▁The ▁Br ams – T aylor – Z w icker ▁procedure ▁makes ▁an ▁env y - free ▁division ▁with ▁at ▁most