(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 30267, 798] NotebookOptionsPosition[ 28050, 723] NotebookOutlinePosition[ 28667, 744] CellTagsIndexPosition[ 28624, 741] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["\nDie Jongleur-Folge \n", "Title"], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztmUFKA0EQRQfNQnArLkUv4cKd4gEkniAB3QpRkJzHC3i3HGCcgAEZuqt+ dfXYnZ6/eCGZzGSq3vya0MnN+m35etJ13fvZ8LBcfd5vNqvt82J48bD9eFmf Dk8ef9lv7PueEEIIIYQQQiblavfUHyhdSw3QR9gFfdCH5II+6ENygfho1V2K j5az5HXRko9QX1J/LbuI9UcfWDZCPkrXb2H/N8iBHNk4Zv66iDmZi4+QC9TH f9fo3SfVRU0+tDnW6vacK/a5pWZF6xW5jh7vNWVD6xXNde5seHykukN6RX1Y 7j2Ii5APiwurE+2eJr0fyjqSHWvWPGtZiw+tV8QP4hQ575QukONSe0rNTw4X KWtZ9Fgk72g/6Nx452QqH9p192Y9ZQ7HXHzdwj3FfgfQjkUc5co6ev+Jsbg+ h3xI613pWMlvajak7Gr7I04uv+/gbKTMD+oD6U9zYXUVq8HjQ3Pi6TOnC/Q6 WGbF6gOpy7tPbnL4QDJ2LOTy0QrI9wN9zDMb9CH7QLa3DrNBH1Yfc52Vce/j dWvp2mrwMeds0AfupHRN9FEPdBF3UroOQgghhBBCCCnND0Go6oM= "], {{0, 68}, {68, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], BaseStyle->"ImageGraphics", ImageSizeRaw->{68, 68}, PlotRange->{{0, 68}, {0, 68}}]], "Input"], "\n", StyleBox["Albert A. G\[ADoubleDot]chter, 2011\n\n\n", "Subsubtitle"], StyleBox["Die Jongleur-Folge (juggler sequence) wurde von C.A. Pickover \ erfunden und in seinem 1991 erschienenen Buch ", FontFamily->"Times New Roman"], StyleBox["Computers and the Imagination ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["beschrieben. So wie die B\[ADoubleDot]lle eines Jongleurs sich auf \ und ab bewegen, geb\[ADoubleDot]rden sich die Glieder dieser Folge.\n\ Beispiel: \t9, 27, 140, 11, 36, 6, 2, 1\nEs ist nicht ganz einfach, das \ Bildungsgesetz selber heraus zu finden. Es lautet:\n\n\t", FontFamily->"Times New Roman"], Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]], FormatType->"TraditionalForm"], "= ", Cell[BoxData[ FormBox[ RowBox[{"\[Piecewise]", GridBox[{ { RowBox[{"[", RowBox[{"\[Sqrt]", "n"}], "]"}], RowBox[{"f\[UDoubleDot]r", " ", "n", " ", "gerade"}]}, { RowBox[{"[", RowBox[{"\[Sqrt]", SuperscriptBox["n", "3"]}], "]"}], RowBox[{"f\[UDoubleDot]r", " ", "n", " ", "ungerade"}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], TraditionalForm]], FormatType->"TraditionalForm"], StyleBox["\n\n(Die eckigen Klammern bedeuten die gr\[ODoubleDot]sste ganze \ Zahl ", FontFamily->"Times New Roman"], "kleiner oder gleich des in der Klammer berechneten Wertes). \nIn ", StyleBox["mathematica ", FontSlant->"Italic"], "erh\[ADoubleDot]lt man:" }], "Text", CellChangeTimes->{{3.503633635484375*^9, 3.503633693703125*^9}, 3.503634824921875*^9, {3.50363500084375*^9, 3.503635038109375*^9}, { 3.503635072296875*^9, 3.503635103359375*^9}, {3.50363519571875*^9, 3.503635220953125*^9}, {3.503635324640625*^9, 3.503635350046875*^9}, { 3.503640799453125*^9, 3.503640821375*^9}, 3.50364085584375*^9, 3.50364138846875*^9, {3.503651827171875*^9, 3.50365193190625*^9}, { 3.503664043578125*^9, 3.503664133*^9}, {3.503664184734375*^9, 3.503664248625*^9}, {3.503664291390625*^9, 3.503664297296875*^9}, { 3.50366517865625*^9, 3.503665214734375*^9}, {3.50366561571875*^9, 3.503665660671875*^9}, 3.503665692015625*^9, {3.503665739015625*^9, 3.503665807265625*^9}, {3.5036658461875*^9, 3.503665921625*^9}, { 3.503666532671875*^9, 3.503666535390625*^9}, {3.5037182916315002`*^9, 3.5037182937721252`*^9}, {3.5037183372096252`*^9, 3.5037183967877502`*^9}, {3.5037184639908752`*^9, 3.5037184640065002`*^9}, {3.5037185287565002`*^9, 3.5037187034127502`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"J", "[", "n_", "]"}], ":=", RowBox[{"If", "[", RowBox[{ RowBox[{"EvenQ", "[", "n", "]"}], ",", RowBox[{"Floor", "[", RowBox[{"Sqrt", "[", "n", "]"}], "]"}], ",", RowBox[{"Floor", "[", RowBox[{"Sqrt", "[", RowBox[{"n", "^", "3"}], "]"}], "]"}]}], "]"}]}], ";"}]], "Input"], Cell["\<\ So wird J[9] = 27 und J[27] =140. \ \>", "Text", CellChangeTimes->{{3.50366605825*^9, 3.503666146296875*^9}, 3.50366619421875*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"J", "[", RowBox[{"J", "[", RowBox[{"J", "[", RowBox[{"J", "[", "48443", "]"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.503640491109375*^9, 3.503640518609375*^9}}], Cell[BoxData["523578821252958052233532"], "Output", CellChangeTimes->{3.503666837578125*^9}] }, Open ]], Cell["\<\ Die ganze Folge ergibt sich mit:\ \>", "Text", CellChangeTimes->{3.5036662301875*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"jongleur", "[", "n_", "]"}], ":=", RowBox[{"NestWhileList", "[", RowBox[{"J", ",", "n", ",", RowBox[{ RowBox[{"#", "\[NotEqual]", "1"}], "&"}]}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.50364028628125*^9, 3.50364029428125*^9}, 3.503665832078125*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"jongleur", "[", "9", "]"}]], "Input", CellChangeTimes->{{3.50364030990625*^9, 3.50364033196875*^9}, 3.503664150546875*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ "9", ",", "27", ",", "140", ",", "11", ",", "36", ",", "6", ",", "2", ",", "1"}], "}"}]], "Output", CellChangeTimes->{ 3.50364033325*^9, {3.50366416590625*^9, 3.503664179578125*^9}, 3.503666837671875*^9}] }, Open ]], Cell[TextData[{ "Nach der Ver\[ODoubleDot]ffentlichung des Buches besch\[ADoubleDot]ftigten \ sich zahlreiche Interessenten mit der Jongleur-Folge. Harry J. Smith \ schrieb am 27.6.1992 an Pickover:\n", StyleBox["I have now computed a larger juggler number. It is a \ 972,463-digit super giant for the starting number 48443. The entire juggler \ sequence starting from 48443 was computed in 28 hours.", FontSlant->"Italic"], "\nUm diese grosse Zahl mit dem Computer zu berechnen, schrieb er ein \ eigenes Programm. Heute erledigt ", StyleBox["mathematica ", FontSlant->"Italic"], "dies in ca. 10 Minuten auf einem mittelm\[ADoubleDot]ssigen Rechner.\n" }], "Text", CellChangeTimes->{{3.503666270765625*^9, 3.503666321828125*^9}, { 3.50366642040625*^9, 3.5036665183125*^9}, {3.503666602*^9, 3.5036667034375*^9}, {3.5036692651875*^9, 3.5036692661875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"jongleur", "[", "48443", "]"}]], "Input", CellChangeTimes->{{3.503570128328125*^9, 3.503570135578125*^9}, { 3.50364057821875*^9, 3.503640581296875*^9}}], Cell[BoxData[ InterpretationBox[ TagBox[ PanelBox[GridBox[{ { StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeExplanation"], StandardForm], ImageSizeCache->{280., {2., 9.}}], StripOnInput->False, DynamicUpdating->True], "Panel", StripOnInput->False, Background->None]}, { ItemBox[ TagBox[ RowBox[{"{", RowBox[{ "48443", ",", "10662193", ",", "34815273349", ",", "6496130099313865", ",", "523578821252958052233532", ",", "723587466207", ",", "615512041010804067", ",", "482897358660562651148793788", ",", "21974925680433", ",", "103012783516625098121", ",", "1045530445028727953685811220915", ",", RowBox[{"\[LeftSkeleton]", "136", "\[RightSkeleton]"}], ",", "1544131", ",", "1918784550", ",", "43803", ",", "9167602", ",", "3027", ",", "166540", ",", "408", ",", "20", ",", "4", ",", "2", ",", "1"}], "}"}], Short[#, 5]& ], Background->GrayLevel[1], BaseStyle->{Deployed -> False}, Frame->True, FrameStyle->GrayLevel[0, 0.2], StripOnInput->False]}, { RowBox[{ ButtonBox[ StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowLess"], StandardForm], ImageSizeCache->{50., {1., 9.}}], StripOnInput->False, DynamicUpdating->True], "Panel", StripOnInput->False, Background->None], Appearance->Automatic, ButtonFunction:>OutputSizeLimit`ButtonFunction[ Identity, 8, 22961486582571614970, 5/2], Enabled->True, Evaluator->Automatic, Method->"Queued"], "\[ThinSpace]", ButtonBox[ StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowMore"], StandardForm], ImageSizeCache->{53., {1., 9.}}], StripOnInput->False, DynamicUpdating->True], "Panel", StripOnInput->False, Background->None], Appearance->Automatic, ButtonFunction:>OutputSizeLimit`ButtonFunction[ Identity, 8, 22961486582571614970, 5 2], Enabled->True, Evaluator->Automatic, Method->"Queued"], "\[ThinSpace]", ButtonBox[ StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowAll"], StandardForm], ImageSizeCache->{82., {2., 9.}}], StripOnInput->False, DynamicUpdating->True], "Panel", StripOnInput->False, Background->None], Appearance->Automatic, ButtonFunction:>OutputSizeLimit`ButtonFunction[ Identity, 8, 22961486582571614970, Infinity], Enabled->True, Evaluator->Automatic, Method->"Queued"], "\[ThinSpace]", ButtonBox[ StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeChangeLimit"], StandardForm], ImageSizeCache->{74., {1., 8.}}], StripOnInput->False, DynamicUpdating->True], "Panel", StripOnInput->False, Background->None], Appearance->Automatic, ButtonFunction:>FrontEndExecute[{ FrontEnd`SetOptions[ FrontEnd`$FrontEnd, FrontEnd`PreferencesSettings -> {"Page" -> "Evaluation"}], FrontEnd`FrontEndToken["PreferencesDialog"]}], Evaluator->None, Method->"Preemptive"]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxDividers->{ "Columns" -> {{False}}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[1.2]}, Offset[0.2]}, "RowsIndexed" -> {}}], DefaultBaseStyle->{}, FrameMargins->5], Deploy, DefaultBaseStyle->{Deployed -> True}], Out[8]]], "Output", CellChangeTimes->{3.503570618078125*^9, 3.503571793859375*^9, 3.503667451765625*^9}] }, Open ]], Cell["\<\ Die gr\[ODoubleDot]sste Zahl in der Folge hat wie erw\[ADoubleDot]hnt 972 463 \ Ziffern. Der folgende output wird deshalb unterdr\[UDoubleDot]ckt.\ \>", "Text", CellChangeTimes->{{3.5037187868346252`*^9, 3.5037188336940002`*^9}, { 3.5037191580221252`*^9, 3.5037191751315002`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Max", "[", RowBox[{"jongleur", "[", "48443", "]"}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.503570843546875*^9, 3.50357085140625*^9}, 3.503580380359375*^9, {3.503640592484375*^9, 3.503640596265625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"IntegerLength", "[", RowBox[{"Max", "[", RowBox[{"jongleur", "[", "48443", "]"}], "]"}], "]"}], "//", "Timing"}]], "Input", CellChangeTimes->{{3.503640697140625*^9, 3.5036406995625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"600.8910000000001`", ",", "972463"}], "}"}]], "Output", CellChangeTimes->{3.503669261703125*^9}] }, Open ]], Cell["\<\ Mein Rechner ben\[ODoubleDot]tigt etwa 10 Minuten, um die Zahl und deren L\ \[ADoubleDot]nge zu berechnen. Die Jongleur-Folge zur Startzahl 48443 besitzt \ 158 Glieder. \ \>", "Text", CellChangeTimes->{{3.5037192562565002`*^9, 3.5037192700065002`*^9}, { 3.5037193276940002`*^9, 3.5037193988971252`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Length", "[", RowBox[{"jongleur", "[", "48443", "]"}], "]"}]], "Input", CellChangeTimes->{{3.5035710385625*^9, 3.50357105140625*^9}, { 3.5036406378125*^9, 3.50364064046875*^9}}], Cell[BoxData["158"], "Output", CellChangeTimes->{3.503571409734375*^9, 3.50357234740625*^9, 3.503669866609375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"jongleur", "/@", RowBox[{"Range", "[", "5", "]"}]}], "//", "ColumnForm"}]], "Input", CellChangeTimes->{{3.503576942828125*^9, 3.503577011546875*^9}, { 3.503640654421875*^9, 3.5036406570625*^9}}], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{"{", "1", "}"}]}, { RowBox[{"{", RowBox[{"2", ",", "1"}], "}"}]}, { RowBox[{"{", RowBox[{ "3", ",", "5", ",", "11", ",", "36", ",", "6", ",", "2", ",", 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Einige Definitionen lassen sich problemlos \ \[UDoubleDot]bernehmen. Wegen der Wurzelausdr\[UDoubleDot]cke ist aber eine \ unkomplizierte ", Cell[BoxData[ FormBox["Zahlbereichs", TraditionalForm]]], "erweiterung nicht m\[ODoubleDot]glich. Daher gibt es keine \ Jongleur-Fraktale!\n", StyleBox["stoppingnumber", FontSlant->"Italic"], " ist das erste Glied in der Folge, welches kleiner als die Startzahl ist." }], "Text", CellChangeTimes->{{3.50367010234375*^9, 3.503670372453125*^9}, { 3.50367041425*^9, 3.503670421734375*^9}, {3.503670515234375*^9, 3.50367051565625*^9}, {3.5037195332408752`*^9, 3.5037195851471252`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"stoppingnumber", "[", "n_", "]"}], ":=", RowBox[{"First", "[", RowBox[{"Select", "[", RowBox[{ RowBox[{"jongleur", "[", "n", "]"}], ",", RowBox[{ RowBox[{"#", "<", "n"}], "&"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.50357913753125*^9, 3.503579144578125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"stoppingtime", "[", "n_", "]"}], ":=", RowBox[{"Length", "[", RowBox[{"NestWhileList", "[", RowBox[{"js", ",", "n", ",", RowBox[{ RowBox[{"#", ">", RowBox[{"stoppingnumber", "[", "n", "]"}]}], "&"}]}], "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.503579193328125*^9, 3.5035791938125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"stoppingnumber", "[", "27", "]"}]], "Input", CellChangeTimes->{{3.503580127890625*^9, 3.5035801301875*^9}}], Cell[BoxData["11"], "Output", CellChangeTimes->{3.5035801318125*^9, 3.50366987071875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"stoppingtime", "[", "27", "]"}]], "Input", CellChangeTimes->{{3.503580143109375*^9, 3.503580145140625*^9}}], Cell[BoxData["2"], "Output", CellChangeTimes->{3.5035801468125*^9, 3.503669870765625*^9}] }, Open ]], Cell[TextData[{ "\nBis heute ist nicht gekl\[ADoubleDot]rt, ob jede Jongleur-Folge bei 1 \ endet. \nWegen der grossen Zahlen braucht es ein potentes Werkzeug, wenn man \ Jongleur-Folgen untersuchen will. Daher eignen sich Taschenrechner oder \ Tabellenkalkulationen wenig.\n\n\n", StyleBox["Aktivit\[ADoubleDot]ten\n\n", "Subsection"], "1. Gibt es Jongleur-Folgen, welche monoton gegen 1 fallen?\n2. Berechne die \ Jongleur-Folge f\[UDoubleDot]r n=193.\n3. Jongleur-Folgen enden auf 4 2 1, \ 6 2 1 oder 8 2 1. Gibt es noch andere?\n4. Entwickle ein \ Tabellenkalkulationsblatt, welches Jongleur-Folgen erzeugt. Welche grossen \ Zahlen lassen sich nicht mehr darstellen?\n5. Erstelle eine Liste der \ Jongleur-Folgen f\[UDoubleDot]r n = 1,...,20 mit der jeweiligen Anzahl \ Glieder und dem h\[ODoubleDot]chsten Wert.\n6. Welches ist die kleinste \ Startzahl f\[UDoubleDot]r eine Jongleur-Folge mit genau 10 Gliedern?\n" }], "Text", CellChangeTimes->{{3.503670439703125*^9, 3.503670468546875*^9}, { 3.503670533984375*^9, 3.50367060715625*^9}, {3.5037199347877502`*^9, 3.5037201221783752`*^9}, {3.5037201877721252`*^9, 3.5037202118971252`*^9}, { 3.5037202630377502`*^9, 3.5037202688971252`*^9}, {3.5037203175533752`*^9, 3.5037204094127502`*^9}, {3.5037205463815002`*^9, 3.5037206199752502`*^9}, { 3.5037207478658752`*^9, 3.5037207788033752`*^9}}] }, WindowSize->{614, 748}, WindowMargins->{{31, Automatic}, {Automatic, 4}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, FrontEndVersion->"8.0 for Microsoft Windows (32-bit) (November 7, 2010)", StyleDefinitions->Notebook[{ Cell[ StyleData[StyleDefinitions -> "Default.nb"]]}, Visible -> False, FrontEndVersion -> "8.0 for Microsoft Windows (32-bit) (November 7, 2010)", StyleDefinitions -> "PrivateStylesheetFormatting.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[557, 20, 4050, 89, 464, "Text"], Cell[4610, 111, 366, 11, 31, "Input"], Cell[4979, 124, 147, 4, 29, "Text"], Cell[CellGroupData[{ Cell[5151, 132, 211, 5, 31, "Input"], Cell[5365, 139, 93, 1, 30, "Output"] }, Open ]], Cell[5473, 143, 96, 3, 29, "Text"], Cell[5572, 148, 329, 9, 31, "Input"], Cell[CellGroupData[{ Cell[5926, 161, 151, 3, 31, "Input"], Cell[6080, 166, 257, 7, 30, "Output"] }, Open ]], Cell[6352, 176, 871, 16, 137, "Text"], Cell[CellGroupData[{ Cell[7248, 196, 179, 3, 31, "Input"], Cell[7430, 201, 4761, 131, 208, "Output"] }, Open ]], Cell[12206, 335, 293, 5, 47, "Text"], Cell[12502, 342, 251, 5, 31, "Input"], Cell[CellGroupData[{ Cell[12778, 351, 234, 6, 31, "Input"], Cell[13015, 359, 137, 3, 30, "Output"] }, Open ]], Cell[13167, 365, 316, 6, 47, "Text"], Cell[CellGroupData[{ Cell[13508, 375, 207, 4, 31, "Input"], Cell[13718, 381, 118, 2, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13873, 388, 236, 5, 31, "Input"], Cell[14112, 395, 791, 26, 90, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14940, 426, 299, 7, 31, "Input"], Cell[15242, 435, 389, 10, 241, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15668, 450, 872, 25, 52, "Input"], Cell[16543, 477, 732, 18, 236, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17312, 500, 728, 22, 72, "Input"], Cell[18043, 524, 6643, 114, 226, "Output"] }, Open ]], Cell[24701, 641, 750, 15, 101, "Text"], Cell[25454, 658, 334, 9, 31, "Input"], Cell[25791, 669, 362, 10, 31, "Input"], Cell[CellGroupData[{ Cell[26178, 683, 132, 2, 31, "Input"], Cell[26313, 687, 90, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[26440, 693, 132, 2, 31, "Input"], Cell[26575, 697, 90, 1, 30, "Output"] }, Open ]], Cell[26680, 701, 1366, 20, 319, "Text"] } ] *) (* End of internal cache information *)