(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1929203, 81110]*) (*NotebookOutlinePosition[ 1929884, 81134]*) (* CellTagsIndexPosition[ 1929840, 81130]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The theory of Russian Roulette and the Sierpinski Gasket", "Subtitle"], Cell["\<\ Tomohide Hashiba Yuta Nakagawa Akihiro Hyogu Hiroshi Matsui Ryohei Miyadera \ \>", "Text", FontFamily->"Courier New"], Cell[CellGroupData[{ Cell["1. Introduction", "Subsubsection", FontFamily->"Courier New"], Cell[TextData[{ "As we presented before (See R. Miyadera [1]), the triangle of the list of \ probabilities has the Pascal's triangle like property. If we make a triangle \ using the least nonnegative residues of the ", StyleBox["numerators", FontFamily->"Times New Roman"], " of the fractions in the triangle taken modulo 2, then we get very \ interesting figures. When the number of the player is 2, then it is very \ similar to the Sierpinski Gasket. When the number of players is 3, it is \ slightly similar to the Sierpinski Gasket. \n4 freshmen (10th grade) in a \ high school studied the theory of Russian roulette, and discovered these \ figures. They and their teacher also made a very interesting animation of the \ figures based on the theory of Russian roulette." }], "Text", FontFamily->"Times New Roman"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2. Mathematical theory of general Russian roulette.(The result of this \ section is included in Miyadera[1].)\ \>", "Subsubsection", FontFamily->"Courier New"], Cell[TextData[StyleBox["In the Russian roulette game we are going to study \ here, one does not rotate the cylinder before he pulls the trigger. We \ denote by F[p,n,m,v] the probability of death of the v-th player in this \ game.", FontFamily->"Times New Roman"]], "Text", FontFamily->"Courier New", FontSize->12], Cell[TextData[{ StyleBox["We define ", FontFamily->"Times New Roman"], " ", Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[\(U[p, n, m, v]\), FontSize->10], StyleBox[" ", FontSize->10], StyleBox["=", FontSize->12], StyleBox[" ", FontSize->10], StyleBox[\(\[Sum]\+\(z = 0\)\%\(t\ \ - 1\)\(\ \[InvisiblePrefixScriptBase]\_\(n - v - pz\)\)\(C\_\(m - 1\)\)\), FontSize->10]}], StyleBox[",", FontSize->10]}]]], StyleBox["where", FontFamily->"Times New Roman"], "\n", Cell[BoxData[ \(\(\(\ \)\(t\ = \ \(\([\(n - m + p - v + 1\)\/p]\)\(.\)\)\)\)\)]], "\n", StyleBox["Note that this number U[p,n,m,v] is an important number in this \ article.", FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[ RowBox[{ StyleBox["T", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox["heor", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], StyleBox["em", FontFamily->"Times New Roman", FontWeight->"Bold"]}]], StyleBox[" ", FontFamily->"Times New Roman", FontSize->10, FontWeight->"Bold"], StyleBox[ RowBox[{ StyleBox["1", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], StyleBox[".", FontFamily->"Times New Roman", FontSize->10, FontWeight->"Bold"]}]], StyleBox[" ", FontSize->10], \(F[p, n, m, v]\)}], "=", StyleBox[" "], StyleBox[\(\(U[p, n, m, v]\/\(\[InvisiblePrefixScriptBase]\_n\)\(C\_m\)\)\(.\)\)]}]], \ "Text", FontFamily->"Courier New", FontSize->12], Cell[TextData[StyleBox["For a proof see Miyadera [1].", FontFamily->"Times New Roman"]], "Text", FontFamily->"Courier New"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["1. ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" code for F[p,n,m,v].", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ StyleBox[\(F[p2_, n2_, m2_, v2_] := Block[{t2, u}, t2 = IntegerPart[\(1 + p2 - m2 + n2 - v2\)\/p2]; u = \((\[Sum]\+\(z = 0\)\%\(t2 - 1\)Binomial[n2 - v2 - p2*z, m2 - 1])\); FractionBox[u, Binomial[n2, m2]] // DisplayForm]\), FontSize->12]], "Input", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[BoxData[ \(F[2, 6, 4, 1]\)], "Input", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ TagBox[\(11\/15\), DisplayForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 1", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[". The purpose of this example is to observe a Pascal's \ triangle-like pattern of fractions. ", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], Cell[TextData[{ StyleBox["To do this we have to make a triangle. Here we let p = 2 and v = \ 1, and make F[p,n,m,v] for any natural number n,m with n \[LessEqual] 7 and m \ \[LessEqual] n. Then we get a triangle(See Figure 1). If we calculate \ F[2,n,m,1] for each n and m, then we have the following triangle of fractions \ (See Figure 2).\:3000\n The pattern is quite obvious in Figure (2). For \ example look at 6th row. F[2,6,2,1]= ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`9\/\(\(15\)\(\ \)\)\)], FontFamily->"Times New Roman"], StyleBox[" and F[2,6,3,1]= ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`13\/20\)], FontFamily->"Times New Roman"], StyleBox[" are the second and third ones in the row. F[2,7,3,1]", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(22\/35\)\)\)], FontFamily->"Times New Roman"], StyleBox["= ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(9 + 13\)\/\(15 + 20\)\)], FontFamily->"Times New Roman"], StyleBox[", which is the third one in the 7th row. This reminds us of \ Pascal's triangle. By Theorem 1 and 2 there exists the same kind of relation \ among F[p,n,m,v] ,F[p,n,m+1,v], F[p,n+1,m+1,v] for any natural number p,v,n \ and m with n \[GreaterEqual] m. ", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], Cell[TextData[StyleBox[" \ Figure(1)", FontFamily->"Times New Roman"]], "Text"], Cell[TextData[StyleBox[" \:3000\:3000\:3000 \ {F[2,1,1,1]} \n {F[2,2,1,1],F[2,2,2,1]}\n \ {F[2,3,1,1],F[2,3,2,1],F[2,3,3,1]}\n \ {F[2,4,1,1],F[2,4,2,1],F[2,4,3,1],F[2,4,4,1]}\n \ {F[2,5,1,1],F[2,5,2,1],F[2,5,3,1],F[2,5,4,1],F[2,5,5,1]}\n \ {F[2,6,1,1],F[2,6,2,1],F[2,6,31,1],F[2,6,4,1],F[2,6,5,1],F[2,6,6,1]}\n\ {F[2,7,1,1],F[2,7,1,1],F[2,7,3,1],F[2,7,4,1],F[2,7,5,1],F[2,7,6,1],F[2,7,7,1]}\ ", FontFamily->"Courier New"]], "Text", Background->GrayLevel[0.900008]], Cell[TextData[StyleBox[" \ Figure(2)", FontFamily->"Times New Roman"]], "Text"], Cell[TextData[{ " ", StyleBox[" ", FontSize->12], StyleBox["{1} ", FontFamily->"Courier New", FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ StyleBox[\({1\/2, 1\/1}\), FontFamily->"Courier New"]], FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ \({2\/3, 2\/3, 1\/1}\)], FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ \({2\/4, 4\/6, 3\/4, 1\/1}\)], FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ \({3\/5, 6\/10, 7\/10, 4\/5, 1\/1}\)], FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ RowBox[{ StyleBox["{", FontColor->GrayLevel[0]], RowBox[{ StyleBox[\(3\/6\), FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[\(9\/15\), FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->RGBColor[0, 0, 1]], StyleBox[\(13\/20\), FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontColor->GrayLevel[0]], \(11\/15\), ",", \(5\/6\), ",", \(1\/1\)}], "}"}]], FontSize->12], StyleBox["\n ", FontSize->12], Cell[BoxData[ RowBox[{"{", RowBox[{\(4\/7\), ",", \(12\/21\), ",", StyleBox[\(22\/35\), FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ",", \(24\/35\), ",", \(16\/21\), ",", \(6\/7\), ",", "1"}], "}"}]], FontSize->12], StyleBox["\n ", FontSize->14] }], "Text", FontFamily->"Helvetica", Background->GrayLevel[0.900008]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" 2. ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" code to make the above triangle. ", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[{ StyleBox[\(Clear[triangle, \ \ death];\), FontSize->10], "\[IndentingNewLine]", StyleBox[\(triangle[t_, p_, v_] := Block[{death}, death[p2_, n2_, m2_, v2_] := Block[{t2, u}, t2 = IntegerPart[\(1 + p2 - m2 + n2 - v2\)\/p2]; u = \((\[Sum]\+\(z = 0\)\%\(t2 - 1\)Binomial[n2 - v2 - p2*z, m2 - 1])\); FractionBox[u, Binomial[n2, m2]] // DisplayForm]; ColumnForm[Table[Table[death[p, n, m, v], {m, 1, n}], {n, 1, t}], Center]];\), FontSize->10]}], "Input", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(triangle[7, 2, 1]\)], "Input", FontColor->RGBColor[0, 0, 1]] }, Open ]] }, Open ]] }, Open ]], Cell["3. New facts about Russian roulette.", "Subsubsection", FontFamily->"Courier New"], Cell[TextData[{ StyleBox["Example 2.", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" Here we calculate the probabilities of the 1st person's death \ when 3 persons play the game, and we make a triangle for the number of \ cylinders n = 1,2,3,..., 12.", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], Cell[TextData[{ " ", StyleBox[" Figure(3)", FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({1\/1}\)}, {\({1\/2, 1\/1}\)}, {\({1\/3, 2\/3, 1\/1}\)}, {\({2\/4, 3\/6, 3\/4, 1\/1}\)}, {\({2\/5, 5\/10, 6\/10, 4\/5, 1\/1}\)}, {\({2\/6, 7\/15, 11\/20, 10\/15, 5\/6, 1\/1}\)}, {\({3\/7, 9\/21, 18\/35, 21\/35, 15\/21, 6\/7, 1\/1}\)}, {\({3\/8, 12\/28, 27\/56, 39\/70, 36\/56, 21\/28, 7\/8, 1\/1}\)}, {\({3\/9, 15\/36, 39\/84, 66\/126, 75\/126, 57\/84, 28\/36, 8\/9, 1\/1}\)}, {\({4\/10, 18\/45, 54\/120, 105\/210, 141\/252, 132\/210, 85\/120, 36\/45, 9\/10, 1\/1}\)}, {\({4\/11, 22\/55, 72\/165, 159\/330, 246\/462, 273\/462, 217\/330, 121\/165, 45\/55, 10\/11, 1\/1}\)}, {\({4\/12, 26\/66, 94\/220, 231\/495, 405\/792, 519\/924, 490\/792, 338\/495, 166\/220, 55\/66, 11\/12, 1\/1}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Center}], ColumnForm[ {{ DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 1, 2]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 1, 3]], DisplayForm[ FractionBox[ 2, 3]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 2, 4]], DisplayForm[ FractionBox[ 3, 6]], DisplayForm[ FractionBox[ 3, 4]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 2, 5]], DisplayForm[ FractionBox[ 5, 10]], DisplayForm[ FractionBox[ 6, 10]], DisplayForm[ FractionBox[ 4, 5]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 2, 6]], DisplayForm[ FractionBox[ 7, 15]], DisplayForm[ FractionBox[ 11, 20]], DisplayForm[ FractionBox[ 10, 15]], DisplayForm[ FractionBox[ 5, 6]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 3, 7]], DisplayForm[ FractionBox[ 9, 21]], DisplayForm[ FractionBox[ 18, 35]], DisplayForm[ FractionBox[ 21, 35]], DisplayForm[ FractionBox[ 15, 21]], DisplayForm[ FractionBox[ 6, 7]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 3, 8]], DisplayForm[ FractionBox[ 12, 28]], DisplayForm[ FractionBox[ 27, 56]], DisplayForm[ FractionBox[ 39, 70]], DisplayForm[ FractionBox[ 36, 56]], DisplayForm[ FractionBox[ 21, 28]], DisplayForm[ FractionBox[ 7, 8]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 3, 9]], DisplayForm[ FractionBox[ 15, 36]], DisplayForm[ FractionBox[ 39, 84]], DisplayForm[ FractionBox[ 66, 126]], DisplayForm[ FractionBox[ 75, 126]], DisplayForm[ FractionBox[ 57, 84]], DisplayForm[ FractionBox[ 28, 36]], DisplayForm[ FractionBox[ 8, 9]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 4, 10]], DisplayForm[ FractionBox[ 18, 45]], DisplayForm[ FractionBox[ 54, 120]], DisplayForm[ FractionBox[ 105, 210]], DisplayForm[ FractionBox[ 141, 252]], DisplayForm[ FractionBox[ 132, 210]], DisplayForm[ FractionBox[ 85, 120]], DisplayForm[ FractionBox[ 36, 45]], DisplayForm[ FractionBox[ 9, 10]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 4, 11]], DisplayForm[ FractionBox[ 22, 55]], DisplayForm[ FractionBox[ 72, 165]], DisplayForm[ FractionBox[ 159, 330]], DisplayForm[ FractionBox[ 246, 462]], DisplayForm[ FractionBox[ 273, 462]], DisplayForm[ FractionBox[ 217, 330]], DisplayForm[ FractionBox[ 121, 165]], DisplayForm[ FractionBox[ 45, 55]], DisplayForm[ FractionBox[ 10, 11]], DisplayForm[ FractionBox[ 1, 1]]}, { DisplayForm[ FractionBox[ 4, 12]], DisplayForm[ FractionBox[ 26, 66]], DisplayForm[ FractionBox[ 94, 220]], DisplayForm[ FractionBox[ 231, 495]], DisplayForm[ FractionBox[ 405, 792]], DisplayForm[ FractionBox[ 519, 924]], DisplayForm[ FractionBox[ 490, 792]], DisplayForm[ FractionBox[ 338, 495]], DisplayForm[ FractionBox[ 166, 220]], DisplayForm[ FractionBox[ 55, 66]], DisplayForm[ FractionBox[ 11, 12]], DisplayForm[ FractionBox[ 1, 1]]}}, Center], Editable->False]], "Text", Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["3.", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" code for the above triangle.", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[{ StyleBox[\(Clear[triangle, \ \ death];\), FontSize->10], "\[IndentingNewLine]", StyleBox[\(triangle[t_, p_, v_] := Block[{death}, death[p2_, n2_, m2_, v2_] := Block[{t2, u}, t2 = IntegerPart[\(1 + p2 - m2 + n2 - v2\)\/p2]; u = \((\[Sum]\+\(z = 0\)\%\(t2 - 1\)Binomial[n2 - v2 - p2*z, m2 - 1])\); FractionBox[u, Binomial[n2, m2]] // DisplayForm]; ColumnForm[Table[Table[death[p, n, m, v], {m, 1, n}], {n, 1, t}], Center]];\), FontSize->10]}], "Input", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(triangle[12, 3, 1]\)], "Input", FontColor->RGBColor[0, 0, 1]] }, Closed]], Cell[TextData[{ StyleBox["Example 3.", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" Next we are going to make a new triangle using the numerators of \ the fractions in the above triangle.", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], Cell[TextData[{ " \:3000", StyleBox[" Figure(4)", FontFamily->"Times New Roman"] }], "Text"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({1}\)}, {\({1, 1}\)}, {\({1, 2, 1}\)}, {\({2, 3, 3, 1}\)}, {\({2, 5, 6, 4, 1}\)}, {\({2, 7, 11, 10, 5, 1}\)}, {\({3, 9, 18, 21, 15, 6, 1}\)}, {\({3, 12, 27, 39, 36, 21, 7, 1}\)}, {\({3, 15, 39, 66, 75, 57, 28, 8, 1}\)}, {\({4, 18, 54, 105, 141, 132, 85, 36, 9, 1}\)}, {\({4, 22, 72, 159, 246, 273, 217, 121, 45, 10, 1}\)}, {\({4, 26, 94, 231, 405, 519, 490, 338, 166, 55, 11, 1}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Center}], ColumnForm[ {{1}, {1, 1}, {1, 2, 1}, {2, 3, 3, 1}, {2, 5, 6, 4, 1}, {2, 7, 11, 10, 5, 1}, {3, 9, 18, 21, 15, 6, 1}, {3, 12, 27, 39, 36, 21, 7, 1}, {3, 15, 39, 66, 75, 57, 28, 8, 1}, {4, 18, 54, 105, 141, 132, 85, 36, 9, 1}, {4, 22, 72, 159, 246, 273, 217, 121, 45, 10, 1}, {4, 26, 94, 231, 405, 519, 490, 338, 166, 55, 11, 1}}, Center], Editable->False]], "Output", Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["4.", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " code to create the above triangle. 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[(0)] .35222 .59083 0 0 Mshowa .5 0 1 r [(1)] .38506 .59083 0 0 Mshowa [(1)] .4179 .59083 0 0 Mshowa 0 1 1 r [(0)] .45074 .59083 0 0 Mshowa .5 0 1 r [(1)] .48358 .59083 0 0 Mshowa [(1)] .51642 .59083 0 0 Mshowa 0 1 1 r [(0)] .54926 .59083 0 0 Mshowa .5 0 1 r [(1)] .5821 .59083 0 0 Mshowa [(1)] .61494 .59083 0 0 Mshowa 0 1 1 r [(0)] .64778 .59083 0 0 Mshowa .5 0 1 r [(1)] .68062 .59083 0 0 Mshowa [(1)] .71346 .59083 0 0 Mshowa [(1)] .27011 .55556 0 0 Mshowa 0 1 1 r [(0)] .30296 .55556 0 0 Mshowa .5 0 1 r [(1)] .3358 .55556 0 0 Mshowa [(1)] .36864 .55556 0 0 Mshowa 0 1 1 r [(0)] .40148 .55556 0 0 Mshowa .5 0 1 r [(1)] .43432 .55556 0 0 Mshowa [(1)] .46716 .55556 0 0 Mshowa 0 1 1 r [(0)] .5 .55556 0 0 Mshowa .5 0 1 r [(1)] .53284 .55556 0 0 Mshowa [(1)] .56568 .55556 0 0 Mshowa 0 1 1 r [(0)] .59852 .55556 0 0 Mshowa .5 0 1 r [(1)] .63136 .55556 0 0 Mshowa [(1)] .6642 .55556 0 0 Mshowa 0 1 1 r [(0)] .69704 .55556 0 0 Mshowa .5 0 1 r [(1)] .72989 .55556 0 0 Mshowa 0 1 1 r [(0)] .25369 .52028 0 0 Mshowa .5 0 1 r [(1)] .28654 .52028 0 0 Mshowa [(1)] .31938 .52028 0 0 Mshowa 0 1 1 r [(0)] .35222 .52028 0 0 Mshowa .5 0 1 r [(1)] .38506 .52028 0 0 Mshowa [(1)] .4179 .52028 0 0 Mshowa 0 1 1 r [(0)] .45074 .52028 0 0 Mshowa .5 0 1 r [(1)] .48358 .52028 0 0 Mshowa [(1)] .51642 .52028 0 0 Mshowa 0 1 1 r [(0)] .54926 .52028 0 0 Mshowa .5 0 1 r [(1)] .5821 .52028 0 0 Mshowa [(1)] .61494 .52028 0 0 Mshowa 0 1 1 r [(0)] .64778 .52028 0 0 Mshowa .5 0 1 r [(1)] .68062 .52028 0 0 Mshowa [(1)] .71346 .52028 0 0 Mshowa [(1)] .74631 .52028 0 0 Mshowa 0 1 1 r [(0)] .23727 .48501 0 0 Mshowa .5 0 1 r [(1)] .27011 .48501 0 0 Mshowa 0 1 1 r [(0)] .30296 .48501 0 0 Mshowa .5 0 1 r [(1)] .3358 .48501 0 0 Mshowa [(1)] .36864 .48501 0 0 Mshowa 0 1 1 r [(0)] .40148 .48501 0 0 Mshowa .5 0 1 r [(1)] .43432 .48501 0 0 Mshowa [(1)] .46716 .48501 0 0 Mshowa 0 1 1 r [(0)] .5 .48501 0 0 Mshowa .5 0 1 r [(1)] .53284 .48501 0 0 Mshowa [(1)] .56568 .48501 0 0 Mshowa 0 1 1 r [(0)] .59852 .48501 0 0 Mshowa .5 0 1 r [(1)] .63136 .48501 0 0 Mshowa [(1)] .6642 .48501 0 0 Mshowa 0 1 1 r [(0)] .69704 .48501 0 0 Mshowa [(0)] .72989 .48501 0 0 Mshowa .5 0 1 r [(1)] .76273 .48501 0 0 Mshowa 0 1 1 r [(0)] .22085 .44974 0 0 Mshowa .5 0 1 r [(1)] .25369 .44974 0 0 Mshowa [(1)] .28654 .44974 0 0 Mshowa [(1)] .31938 .44974 0 0 Mshowa 0 1 1 r [(0)] .35222 .44974 0 0 Mshowa .5 0 1 r [(1)] .38506 .44974 0 0 Mshowa [(1)] .4179 .44974 0 0 Mshowa 0 1 1 r [(0)] .45074 .44974 0 0 Mshowa .5 0 1 r [(1)] .48358 .44974 0 0 Mshowa [(1)] .51642 .44974 0 0 Mshowa 0 1 1 r [(0)] .54926 .44974 0 0 Mshowa .5 0 1 r [(1)] .5821 .44974 0 0 Mshowa [(1)] .61494 .44974 0 0 Mshowa 0 1 1 r [(0)] .64778 .44974 0 0 Mshowa .5 0 1 r [(1)] .68062 .44974 0 0 Mshowa 0 1 1 r [(0)] .71346 .44974 0 0 Mshowa .5 0 1 r [(1)] .74631 .44974 0 0 Mshowa [(1)] .77915 .44974 0 0 Mshowa [(1)] .20443 .41446 0 0 Mshowa [(1)] .23727 .41446 0 0 Mshowa 0 1 1 r [(0)] .27011 .41446 0 0 Mshowa [(0)] .30296 .41446 0 0 Mshowa .5 0 1 r [(1)] .3358 .41446 0 0 Mshowa [(1)] .36864 .41446 0 0 Mshowa 0 1 1 r [(0)] .40148 .41446 0 0 Mshowa .5 0 1 r [(1)] .43432 .41446 0 0 Mshowa [(1)] .46716 .41446 0 0 Mshowa 0 1 1 r [(0)] .5 .41446 0 0 Mshowa .5 0 1 r [(1)] .53284 .41446 0 0 Mshowa [(1)] .56568 .41446 0 0 Mshowa 0 1 1 r [(0)] .59852 .41446 0 0 Mshowa .5 0 1 r [(1)] .63136 .41446 0 0 Mshowa [(1)] .6642 .41446 0 0 Mshowa [(1)] .69704 .41446 0 0 Mshowa [(1)] .72989 .41446 0 0 Mshowa 0 1 1 r [(0)] .76273 .41446 0 0 Mshowa .5 0 1 r [(1)] .79557 .41446 0 0 Mshowa [(1)] .18801 .37919 0 0 Mshowa 0 1 1 r [(0)] .22085 .37919 0 0 Mshowa .5 0 1 r [(1)] .25369 .37919 0 0 Mshowa 0 1 1 r [(0)] .28654 .37919 0 0 Mshowa .5 0 1 r [(1)] .31938 .37919 0 0 Mshowa 0 1 1 r [(0)] .35222 .37919 0 0 Mshowa .5 0 1 r [(1)] .38506 .37919 0 0 Mshowa [(1)] .4179 .37919 0 0 Mshowa 0 1 1 r [(0)] .45074 .37919 0 0 Mshowa .5 0 1 r [(1)] .48358 .37919 0 0 Mshowa [(1)] .51642 .37919 0 0 Mshowa 0 1 1 r [(0)] .54926 .37919 0 0 Mshowa .5 0 1 r [(1)] .5821 .37919 0 0 Mshowa [(1)] .61494 .37919 0 0 Mshowa 0 1 1 r [(0)] .64778 .37919 0 0 Mshowa [(0)] .68062 .37919 0 0 Mshowa [(0)] .71346 .37919 0 0 Mshowa .5 0 1 r [(1)] .74631 .37919 0 0 Mshowa [(1)] .77915 .37919 0 0 Mshowa [(1)] .81199 .37919 0 0 Mshowa [(1)] .17159 .34392 0 0 Mshowa [(1)] .20443 .34392 0 0 Mshowa [(1)] .23727 .34392 0 0 Mshowa [(1)] .27011 .34392 0 0 Mshowa [(1)] .30296 .34392 0 0 Mshowa [(1)] .3358 .34392 0 0 Mshowa [(1)] .36864 .34392 0 0 Mshowa 0 1 1 r [(0)] .40148 .34392 0 0 Mshowa .5 0 1 r [(1)] .43432 .34392 0 0 Mshowa [(1)] .46716 .34392 0 0 Mshowa 0 1 1 r [(0)] .5 .34392 0 0 Mshowa .5 0 1 r [(1)] .53284 .34392 0 0 Mshowa [(1)] .56568 .34392 0 0 Mshowa 0 1 1 r [(0)] .59852 .34392 0 0 Mshowa .5 0 1 r [(1)] .63136 .34392 0 0 Mshowa 0 1 1 r [(0)] .6642 .34392 0 0 Mshowa [(0)] .69704 .34392 0 0 Mshowa .5 0 1 r [(1)] .72989 .34392 0 0 Mshowa 0 1 1 r [(0)] .76273 .34392 0 0 Mshowa [(0)] .79557 .34392 0 0 Mshowa .5 0 1 r [(1)] .82841 .34392 0 0 Mshowa 0 1 1 r [(0)] .15517 .30864 0 0 Mshowa [(0)] .18801 .30864 0 0 Mshowa [(0)] .22085 .30864 0 0 Mshowa [(0)] .25369 .30864 0 0 Mshowa [(0)] .28654 .30864 0 0 Mshowa [(0)] .31938 .30864 0 0 Mshowa [(0)] .35222 .30864 0 0 Mshowa .5 0 1 r [(1)] .38506 .30864 0 0 Mshowa [(1)] .4179 .30864 0 0 Mshowa 0 1 1 r [(0)] .45074 .30864 0 0 Mshowa .5 0 1 r [(1)] .48358 .30864 0 0 Mshowa [(1)] .51642 .30864 0 0 Mshowa 0 1 1 r [(0)] .54926 .30864 0 0 Mshowa .5 0 1 r [(1)] .5821 .30864 0 0 Mshowa [(1)] .61494 .30864 0 0 Mshowa [(1)] .64778 .30864 0 0 Mshowa 0 1 1 r [(0)] .68062 .30864 0 0 Mshowa .5 0 1 r [(1)] .71346 .30864 0 0 Mshowa [(1)] .74631 .30864 0 0 Mshowa 0 1 1 r [(0)] .77915 .30864 0 0 Mshowa .5 0 1 r [(1)] .81199 .30864 0 0 Mshowa [(1)] .84483 .30864 0 0 Mshowa 0 1 1 r [(0)] .13875 .27337 0 0 Mshowa [(0)] .17159 .27337 0 0 Mshowa [(0)] .20443 .27337 0 0 Mshowa [(0)] .23727 .27337 0 0 Mshowa [(0)] .27011 .27337 0 0 Mshowa [(0)] .30296 .27337 0 0 Mshowa [(0)] .3358 .27337 0 0 Mshowa .5 0 1 r [(1)] .36864 .27337 0 0 Mshowa 0 1 1 r [(0)] .40148 .27337 0 0 Mshowa .5 0 1 r [(1)] .43432 .27337 0 0 Mshowa [(1)] .46716 .27337 0 0 Mshowa 0 1 1 r [(0)] .5 .27337 0 0 Mshowa .5 0 1 r [(1)] .53284 .27337 0 0 Mshowa [(1)] .56568 .27337 0 0 Mshowa 0 1 1 r [(0)] .59852 .27337 0 0 Mshowa [(0)] .63136 .27337 0 0 Mshowa .5 0 1 r [(1)] .6642 .27337 0 0 Mshowa [(1)] .69704 .27337 0 0 Mshowa 0 1 1 r [(0)] .72989 .27337 0 0 Mshowa .5 0 1 r [(1)] .76273 .27337 0 0 Mshowa [(1)] .79557 .27337 0 0 Mshowa 0 1 1 r [(0)] .82841 .27337 0 0 Mshowa .5 0 1 r [(1)] .86125 .27337 0 0 Mshowa 0 1 1 r [(0)] .12233 .2381 0 0 Mshowa [(0)] .15517 .2381 0 0 Mshowa [(0)] .18801 .2381 0 0 Mshowa [(0)] .22085 .2381 0 0 Mshowa [(0)] .25369 .2381 0 0 Mshowa [(0)] .28654 .2381 0 0 Mshowa [(0)] .31938 .2381 0 0 Mshowa .5 0 1 r [(1)] .35222 .2381 0 0 Mshowa [(1)] .38506 .2381 0 0 Mshowa [(1)] .4179 .2381 0 0 Mshowa 0 1 1 r [(0)] .45074 .2381 0 0 Mshowa .5 0 1 r [(1)] .48358 .2381 0 0 Mshowa [(1)] .51642 .2381 0 0 Mshowa 0 1 1 r [(0)] .54926 .2381 0 0 Mshowa .5 0 1 r [(1)] .5821 .2381 0 0 Mshowa 0 1 1 r [(0)] .61494 .2381 0 0 Mshowa .5 0 1 r [(1)] .64778 .2381 0 0 Mshowa 0 1 1 r [(0)] .68062 .2381 0 0 Mshowa .5 0 1 r [(1)] .71346 .2381 0 0 Mshowa [(1)] .74631 .2381 0 0 Mshowa 0 1 1 r [(0)] .77915 .2381 0 0 Mshowa .5 0 1 r [(1)] .81199 .2381 0 0 Mshowa [(1)] .84483 .2381 0 0 Mshowa [(1)] .87767 .2381 0 0 Mshowa [(1)] .10591 .20282 0 0 Mshowa 0 1 1 r [(0)] .13875 .20282 0 0 Mshowa [(0)] .17159 .20282 0 0 Mshowa [(0)] .20443 .20282 0 0 Mshowa [(0)] .23727 .20282 0 0 Mshowa [(0)] .27011 .20282 0 0 Mshowa [(0)] .30296 .20282 0 0 Mshowa .5 0 1 r [(1)] .3358 .20282 0 0 Mshowa 0 1 1 r [(0)] .36864 .20282 0 0 Mshowa [(0)] .40148 .20282 0 0 Mshowa .5 0 1 r [(1)] .43432 .20282 0 0 Mshowa [(1)] .46716 .20282 0 0 Mshowa 0 1 1 r [(0)] .5 .20282 0 0 Mshowa .5 0 1 r [(1)] .53284 .20282 0 0 Mshowa [(1)] .56568 .20282 0 0 Mshowa [(1)] .59852 .20282 0 0 Mshowa [(1)] .63136 .20282 0 0 Mshowa [(1)] .6642 .20282 0 0 Mshowa [(1)] .69704 .20282 0 0 Mshowa 0 1 1 r [(0)] .72989 .20282 0 0 Mshowa .5 0 1 r [(1)] .76273 .20282 0 0 Mshowa [(1)] .79557 .20282 0 0 Mshowa 0 1 1 r [(0)] .82841 .20282 0 0 Mshowa [(0)] .86125 .20282 0 0 Mshowa .5 0 1 r [(1)] .89409 .20282 0 0 Mshowa [(1)] .08949 .16755 0 0 Mshowa [(1)] .12233 .16755 0 0 Mshowa 0 1 1 r [(0)] .15517 .16755 0 0 Mshowa [(0)] .18801 .16755 0 0 Mshowa [(0)] .22085 .16755 0 0 Mshowa [(0)] .25369 .16755 0 0 Mshowa [(0)] .28654 .16755 0 0 Mshowa .5 0 1 r [(1)] .31938 .16755 0 0 Mshowa [(1)] .35222 .16755 0 0 Mshowa 0 1 1 r [(0)] .38506 .16755 0 0 Mshowa .5 0 1 r [(1)] .4179 .16755 0 0 Mshowa 0 1 1 r [(0)] .45074 .16755 0 0 Mshowa .5 0 1 r [(1)] .48358 .16755 0 0 Mshowa [(1)] .51642 .16755 0 0 Mshowa 0 1 1 r [(0)] .54926 .16755 0 0 Mshowa [(0)] .5821 .16755 0 0 Mshowa [(0)] .61494 .16755 0 0 Mshowa [(0)] .64778 .16755 0 0 Mshowa [(0)] .68062 .16755 0 0 Mshowa .5 0 1 r [(1)] .71346 .16755 0 0 Mshowa [(1)] .74631 .16755 0 0 Mshowa 0 1 1 r [(0)] .77915 .16755 0 0 Mshowa .5 0 1 r [(1)] .81199 .16755 0 0 Mshowa 0 1 1 r [(0)] .84483 .16755 0 0 Mshowa .5 0 1 r [(1)] .87767 .16755 0 0 Mshowa [(1)] .91051 .16755 0 0 Mshowa [(1)] .07307 .13228 0 0 Mshowa 0 1 1 r [(0)] .10591 .13228 0 0 Mshowa .5 0 1 r [(1)] .13875 .13228 0 0 Mshowa 0 1 1 r [(0)] .17159 .13228 0 0 Mshowa [(0)] .20443 .13228 0 0 Mshowa [(0)] .23727 .13228 0 0 Mshowa [(0)] .27011 .13228 0 0 Mshowa .5 0 1 r [(1)] .30296 .13228 0 0 Mshowa 0 1 1 r [(0)] .3358 .13228 0 0 Mshowa .5 0 1 r [(1)] .36864 .13228 0 0 Mshowa [(1)] .40148 .13228 0 0 Mshowa [(1)] .43432 .13228 0 0 Mshowa [(1)] .46716 .13228 0 0 Mshowa 0 1 1 r [(0)] .5 .13228 0 0 Mshowa .5 0 1 r [(1)] .53284 .13228 0 0 Mshowa 0 1 1 r [(0)] .56568 .13228 0 0 Mshowa [(0)] .59852 .13228 0 0 Mshowa [(0)] .63136 .13228 0 0 Mshowa [(0)] .6642 .13228 0 0 Mshowa .5 0 1 r [(1)] .69704 .13228 0 0 Mshowa 0 1 1 r [(0)] .72989 .13228 0 0 Mshowa .5 0 1 r [(1)] .76273 .13228 0 0 Mshowa [(1)] .79557 .13228 0 0 Mshowa [(1)] .82841 .13228 0 0 Mshowa [(1)] .86125 .13228 0 0 Mshowa 0 1 1 r [(0)] .89409 .13228 0 0 Mshowa .5 0 1 r [(1)] .92693 .13228 0 0 Mshowa 0 1 1 r [(0)] .05665 .097 0 0 Mshowa .5 0 1 r [(1)] .08949 .097 0 0 Mshowa [(1)] .12233 .097 0 0 Mshowa [(1)] .15517 .097 0 0 Mshowa 0 1 1 r [(0)] .18801 .097 0 0 Mshowa [(0)] .22085 .097 0 0 Mshowa [(0)] .25369 .097 0 0 Mshowa .5 0 1 r [(1)] .28654 .097 0 0 Mshowa [(1)] .31938 .097 0 0 Mshowa [(1)] .35222 .097 0 0 Mshowa 0 1 1 r [(0)] .38506 .097 0 0 Mshowa [(0)] .4179 .097 0 0 Mshowa [(0)] .45074 .097 0 0 Mshowa .5 0 1 r [(1)] .48358 .097 0 0 Mshowa [(1)] .51642 .097 0 0 Mshowa [(1)] .54926 .097 0 0 Mshowa 0 1 1 r [(0)] .5821 .097 0 0 Mshowa [(0)] .61494 .097 0 0 Mshowa [(0)] .64778 .097 0 0 Mshowa .5 0 1 r [(1)] .68062 .097 0 0 Mshowa [(1)] .71346 .097 0 0 Mshowa [(1)] .74631 .097 0 0 Mshowa 0 1 1 r [(0)] .77915 .097 0 0 Mshowa [(0)] .81199 .097 0 0 Mshowa [(0)] .84483 .097 0 0 Mshowa .5 0 1 r [(1)] .87767 .097 0 0 Mshowa [(1)] .91051 .097 0 0 Mshowa [(1)] .94335 .097 0 0 Mshowa 0 1 1 r [(0)] .04023 .06173 0 0 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code to create the above triangle." }], "Text", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(figure[p_] := Block[{death, s11}, death[p2_, n2_, m2_, v2_] := Block[{t2, u}, t2 = IntegerPart[\(1 + p2 - m2 + n2 - v2\)\/p2]; u = \((\[Sum]\+\(z = 0\)\%\(t2 - 1\)Binomial[n2 - v2 - p2*z, m2 - 1])\)]; Clear[t, s11]; t = 100; s11 = Table[Table[Mod[death[p, n, m, 1], 2], {m, 1, n}], {n, 1, t}]; Show[Graphics[ Evaluate[\[IndentingNewLine]Table[ Table[{Hue[\((6 + s11[\([x, y]\)])\)/4], Disk[{\(-0.5\) x + y, \(-x\)}, 0.5]}, \[IndentingNewLine]{y, 1, x}], {x, 1, 96}]]], AspectRatio \[Rule] 1/0.9]]\)], "Input", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(figure[3]\)], "Input", FontColor->RGBColor[0, 0, 1]] }, Closed]], Cell[TextData[{ StyleBox["Example 6.", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["Here we make triangles of 96 rows of probabilities of 1st \ person's death for 2-persons. 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p2*z, m2 - 1])\)]; Clear[t, s11]; t = 100; s11 = Table[Table[Mod[death[p, n, m, 1], 2], {m, 1, n}], {n, 1, t}]; Show[Graphics[ Evaluate[\[IndentingNewLine]Table[ Table[{Hue[\((6 + s11[\([x, y]\)])\)/4], Disk[{\(-0.5\) x + y, \(-x\)}, 0.5]}, \[IndentingNewLine]{y, 1, x}], {x, 1, 96}]]], AspectRatio \[Rule] 1/0.9]]\)], "Input", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(figure[2]\)], "Input", FontColor->RGBColor[0, 0, 1]] }, Closed]], Cell[TextData[{ StyleBox["Remark.", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" If you want to study the case of p-person for various natural \ number p, please use the ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" function figure[p] defined in the above ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" code 7.", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], 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It is interesting that we have a figure that look \ like the Sierpinski Gasket this time. 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We omit the output, because it would take much disk space.", FontFamily->"Times New Roman"] }], "Text", FontFamily->"Courier New"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["9.", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " code to create a beautiful ( ", StyleBox["color or black & white", FontColor->RGBColor[0, 1, 0]], ") movie based on the theory of Russian roulette. " }], "Text", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Movie 1.", FontColor->RGBColor[0, 0, 1]], " You can create a movie based on the Russian roulette, but you cannot make \ a good QuickTime movie from this. (We do not know why.) You had better use \ the code in Movie 2." }], "Text", FontFamily->"Times New Roman"], Cell[BoxData[ \(death[p2_, n2_, m2_, v2_] := Block[{t2, u}, t2 = IntegerPart[\(1 + p2 - m2 + n2 - v2\)\/p2]; u = \((\[Sum]\+\(z = 0\)\%\(t2 - 1\)Binomial[n2 - v2 - p2*z, m2 - 1])\)]; Clear[t, s11]; Do[t = 100; s11 = Table[Table[Mod[death[p, n, m, 1], 2], {m, 1, n}], {n, 1, t}]; Show[Graphics[ Evaluate[\[IndentingNewLine]Table[ Table[{Hue[\((6 + s11[\([x, y]\)])\)/4], Disk[{\(-0.5\) x + y, \(-x\)}, 0.5]}, \[IndentingNewLine]{y, 1, x}], {x, 1, 96}]]], AspectRatio \[Rule] 1/0.9, Background \[Rule] Hue[6/4]], {p, 2, 96}]\)], "Input", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Movie 2.", FontColor->RGBColor[0, 0, 1]], " This is different from Movie 1, because it has extra codes with it. These \ codes are colored in Green. 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