How to do the hexagonal cases, no upright quarters.

hh = Select[values,#[[1]]>0.4429&]
shortHex=Union[Map[#[[3]]&,hh]];   Length[shortHex]==14 
noups = PSelect[AllHex,Min[#]>0&] == Join[Range[31],Range[47,60]];
NoUprightTypes = Select[hh,Inside[Last[#],noups]&]

(* try heights *)
vvh=Map[(Initialize[6,#[[3]]];
	Table[ LPmax[
	"SHORT/HEX/LP/cplexE.lp6."<>S[#[[3]]]<>".F"<>S[#[[4]]]<>".C"<>S[#[[5]]],
	StringVar["y",i]
		],{i,1,Max[GBLregions]}]//Max)&,
	NoUprightTypes
	];
vvht213=PSelect[vvh,#<2.13&] == {};  (* wasn't worth the effort *)
vvhtA = {};


branchText[u_]:= Module[{j,h},
	Table[{
		"read ",fileText["cplexE.lp",6,u[[3]],u[[4]],u[[5]]]," lp\n",
		"add \n",
		Branch[Band[6,u[[3]]],j],
		"end \n",
		"opt \n"
		}//StringJoin,{j,1,2^Length[Band[6,u[[3]]]]}] //StringJoin
	];


Module[{stream,vv},
vv=Map[branchText,NoUprightTypes]//StringJoin;
stream=OpenWrite["SHORT/HEX/cplexBn.exec"];
        WriteString[stream,
        "set logfile cplexBn.log\n"<>
        "set preprocessing presolve no\n"
        ];
WriteString[stream,vv];
Close[stream];
];

<< SHORT/HEX/cplexBn.sum;
valHexn = values;
valHexnHi=Select[valHexn,#[[1]]>0.4429&]; Length[valHexnHi]==1262;
Map[#[[3]]&,valHexnHi]//Union == {59, 70, 131, 146, 248};
pHHn = Map[Drop[#,1]&,valHexnHi]//Union == 
	  {{6, 59, 17, 8}, {6, 59, 17, 24}, {6, 59, 17, 28}, {6, 59, 17, 53}, 
   {6, 70, 19, 9}, {6, 131, 17, 1}, {6, 131, 17, 10}, {6, 131, 17, 20}, 
   {6, 131, 17, 21}, {6, 131, 17, 29}, {6, 131, 17, 58}, {6, 146, 17, 57}, 
   {6, 248, 16, 9}}