(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[      9793,        260]
NotebookOptionsPosition[      9502,        246]
NotebookOutlinePosition[      9844,        261]
CellTagsIndexPosition[      9801,        258]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"Weight", "[", "M_", "]"}], ":=", 
   RowBox[{
    UnderoverscriptBox["\[Product]", 
     RowBox[{"i", "=", "1"}], 
     RowBox[{"Length", "[", 
      RowBox[{"Flatten", "[", "M", "]"}], "]"}]], 
    RowBox[{
     SuperscriptBox["p", 
      RowBox[{
       RowBox[{"Flatten", "[", "M", "]"}], "[", 
       RowBox[{"[", "i", "]"}], "]"}]], 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"1", "-", "p"}], ")"}], 
      RowBox[{"1", "-", 
       RowBox[{
        RowBox[{"Flatten", "[", "M", "]"}], "[", 
        RowBox[{"[", "i", "]"}], "]"}]}]]}]}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"RandM", "[", 
    RowBox[{"d_", ",", "D_"}], "]"}], ":=", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"L", ",", "i", ",", "j"}], "}"}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"L", "=", 
       RowBox[{"IntegerDigits", "[", 
        RowBox[{
         RowBox[{"Random", "[", 
          RowBox[{"Integer", ",", 
           RowBox[{"{", 
            RowBox[{"0", ",", 
             RowBox[{
              SuperscriptBox["2", 
               RowBox[{"D", " ", "d"}]], "-", "1"}]}], "}"}]}], "]"}], ",", 
         "2", ",", 
         RowBox[{"D", " ", "d"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"Return", "[", 
       RowBox[{"Table", "[", 
        RowBox[{
         RowBox[{"L", "[", 
          RowBox[{"[", 
           RowBox[{"i", "+", " ", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"j", "-", "1"}], ")"}], "d"}]}], "]"}], "]"}], ",", 
         RowBox[{"{", 
          RowBox[{"i", ",", "1", ",", "D"}], "}"}], ",", 
         RowBox[{"{", 
          RowBox[{"j", ",", "1", ",", "d"}], "}"}]}], "]"}], "]"}], ";"}]}], 
    "\[IndentingNewLine]", "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"IsSolvable", "[", "M_", "]"}], ":=", 
   RowBox[{
    RowBox[{"IsSolvable", "[", "M", "]"}], "=", 
    RowBox[{"Module", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "A", ",", "B", ",", "D", ",", "d", ",", "i", ",", "j", ",", "k"}], 
       "}"}], ",", "\[IndentingNewLine]", 
      RowBox[{
       RowBox[{
        RowBox[{"{", 
         RowBox[{"D", ",", "d"}], "}"}], "=", 
        RowBox[{"Dimensions", "[", "M", "]"}]}], ";", "\[IndentingNewLine]", 
       RowBox[{"A", "=", 
        RowBox[{"Range", "[", "d", "]"}]}], ";", "\[IndentingNewLine]", 
       RowBox[{"For", "[", 
        RowBox[{
         RowBox[{"i", " ", "=", "1"}], ",", 
         RowBox[{"i", "\[LessEqual]", "D"}], ",", 
         RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
         RowBox[{"(*", " ", 
          RowBox[{
           RowBox[{"Print", "[", "A", "]"}], ";"}], " ", "*)"}], 
         "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"B", "=", 
           RowBox[{"Cases", "[", 
            RowBox[{
             RowBox[{"Table", "[", 
              RowBox[{
               RowBox[{"If", "[", 
                RowBox[{
                 RowBox[{
                  RowBox[{"M", "[", 
                   RowBox[{"[", 
                    RowBox[{"i", ",", "j"}], "]"}], "]"}], "\[Equal]", "1"}], 
                 ",", "j", ",", "0"}], "]"}], ",", 
               RowBox[{"{", 
                RowBox[{"j", ",", "1", ",", "d"}], "}"}]}], "]"}], ",", 
             RowBox[{"x_", "/;", 
              RowBox[{"x", ">", "0"}]}]}], "]"}]}], ";", 
          "\[IndentingNewLine]", 
          RowBox[{"A", "=", 
           RowBox[{"A", "\[Intersection]", "B"}]}], ";", 
          "\[IndentingNewLine]", 
          RowBox[{"For", "[", 
           RowBox[{
            RowBox[{"k", "=", "1"}], ",", 
            RowBox[{"k", "\[LessEqual]", "d"}], ",", 
            RowBox[{"k", "++"}], ",", "\[IndentingNewLine]", 
            RowBox[{
             RowBox[{"For", "[", 
              RowBox[{
               RowBox[{"j", "=", "1"}], ",", 
               RowBox[{"j", "\[LessEqual]", 
                RowBox[{"Length", "[", "A", "]"}]}], ",", 
               RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", 
               RowBox[{
                RowBox[{"If", "[", 
                 RowBox[{
                  RowBox[{"MemberQ", "[", 
                   RowBox[{"B", ",", 
                    RowBox[{
                    RowBox[{"A", "[", 
                    RowBox[{"[", "j", "]"}], "]"}], "-", "1"}]}], "]"}], ",", 
                  
                  RowBox[{"A", "=", 
                   RowBox[{"A", "\[Union]", 
                    RowBox[{"{", 
                    RowBox[{
                    RowBox[{"A", "[", 
                    RowBox[{"[", "j", "]"}], "]"}], "-", "1"}], "}"}]}]}]}], 
                 "]"}], ";", "\[IndentingNewLine]", 
                RowBox[{"If", "[", 
                 RowBox[{
                  RowBox[{"MemberQ", "[", 
                   RowBox[{"B", ",", 
                    RowBox[{
                    RowBox[{"A", "[", 
                    RowBox[{"[", "j", "]"}], "]"}], "+", "1"}]}], "]"}], ",", 
                  
                  RowBox[{"A", "=", 
                   RowBox[{"A", "\[Union]", 
                    RowBox[{"{", 
                    RowBox[{
                    RowBox[{"A", "[", 
                    RowBox[{"[", "j", "]"}], "]"}], "+", "1"}], "}"}]}]}]}], 
                 "]"}], ";"}]}], "\[IndentingNewLine]", "]"}], ";"}]}], 
           "\[IndentingNewLine]", "]"}], ";"}]}], "\[IndentingNewLine]", 
        "]"}], ";", "\[IndentingNewLine]", 
       RowBox[{"If", "[", 
        RowBox[{
         RowBox[{
          RowBox[{"Length", "[", "A", "]"}], ">", "0"}], ",", 
         RowBox[{"Return", "[", "True", "]"}], ",", 
         RowBox[{"Return", "[", "False", "]"}]}], "]"}], ";"}]}], 
     "\[IndentingNewLine]", "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"MonteCarlo", "[", 
    RowBox[{"d_", ",", "D_", ",", "K_"}], "]"}], ":=", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"num", ",", "denom", ",", "M", ",", "W", ",", "i"}], "}"}], ",",
      "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"num", "=", "0"}], ";", 
      RowBox[{"denom", "=", "0"}], ";", "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "1"}], ",", 
        RowBox[{"i", "\[LessEqual]", "K"}], ",", 
        RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{"M", "=", 
          RowBox[{"RandM", "[", 
           RowBox[{"d", ",", "D"}], "]"}]}], ";", 
         RowBox[{"W", "=", 
          RowBox[{"Weight", "[", "M", "]"}]}], ";", "\[IndentingNewLine]", 
         RowBox[{"If", "[", 
          RowBox[{
           RowBox[{"IsSolvable", "[", "M", "]"}], ",", 
           RowBox[{"num", "+=", "W"}]}], "]"}], ";", "\[IndentingNewLine]", 
         RowBox[{"denom", "+=", "W"}], ";"}]}], "\[IndentingNewLine]", "]"}], 
      ";", "\[IndentingNewLine]", "\[IndentingNewLine]", 
      RowBox[{"(*", 
       RowBox[{
        RowBox[{"Print", "[", 
         RowBox[{"Plot", "[", 
          RowBox[{
           RowBox[{"num", "/", "denom"}], ",", 
           RowBox[{"{", 
            RowBox[{"p", ",", "0", ",", "1"}], "}"}]}], "]"}], "]"}], ";"}], 
       "*)"}], "\[IndentingNewLine]", 
      RowBox[{"Return", "[", 
       RowBox[{
        RowBox[{"Expand", "[", "num", "]"}], "/", 
        RowBox[{"Expand", "[", "denom", "]"}]}], "]"}], ";"}]}], 
    "\[IndentingNewLine]", "]"}]}], ";"}]}], "Input",
 CellChangeTimes->{{3.449221755765625*^9, 3.449221764359375*^9}, {
   3.449221920109375*^9, 3.449221945359375*^9}, {3.449222068734375*^9, 
   3.44922207909375*^9}, {3.44922211109375*^9, 3.4492221150625*^9}, {
   3.44922220725*^9, 3.449222223421875*^9}, {3.449222262390625*^9, 
   3.44922231546875*^9}, {3.44922243703125*^9, 3.4492224616875*^9}, {
   3.449222535046875*^9, 3.44922254415625*^9}, {3.44922258075*^9, 
   3.449222593953125*^9}, {3.4492226355625*^9, 3.449222648734375*^9}, {
   3.449223082984375*^9, 3.449223229734375*^9}, {3.449223320734375*^9, 
   3.44922333371875*^9}, 3.44922337375*^9, {3.449223465640625*^9, 
   3.44922350509375*^9}, {3.449223964671875*^9, 3.449224068171875*^9}, {
   3.449224102703125*^9, 3.44922410903125*^9}, {3.4492241595625*^9, 
   3.449224215671875*^9}, {3.44924192346875*^9, 3.449241927890625*^9}, {
   3.449242043046875*^9, 3.44924206115625*^9}, {3.449242174890625*^9, 
   3.449242230453125*^9}, {3.449412030765625*^9, 3.449412034171875*^9}, {
   3.449412112671875*^9, 3.44941211446875*^9}, {3.44941443059375*^9, 
   3.44941444278125*^9}, {3.449414617859375*^9, 3.44941467140625*^9}, {
   3.449415013296875*^9, 3.449415026796875*^9}, {3.449415174671875*^9, 
   3.44941517928125*^9}, {3.44941673971875*^9, 3.449416743578125*^9}}],

Cell["\<\
Released under Kellen's license: \
http://math.rutgers.edu/~kellenm/license.html\
\>", "Text",
 CellChangeTimes->{{3.474378710914875*^9, 3.474378724446125*^9}}]
},
WindowSize->{1230, 668},
WindowMargins->{{0, Automatic}, {Automatic, -1}},
FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (April 28, 2007)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[568, 21, 8757, 217, 759, "Input"],
Cell[9328, 240, 170, 4, 34, "Text"]
}
]
*)

(* End of internal cache information *)