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Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"other2", "=", RowBox[{"findOtherSolutions", "[", RowBox[{"start", ",", "end", ",", "soln2"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.513838066625*^9, 3.5138380733125*^9}, { 3.5142204238466206`*^9, 3.5142204315180597`*^9}, 3.524208590984375*^9}, FontSize->18], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Number of alternative solutions found: \"\>", "\[InvisibleSpace]", "2"}], SequenceForm["Number of alternative solutions found: ", 2], Editable->False]], "Print", CellChangeTimes->{3.5242087738125*^9, 3.524209072296875*^9, 3.527254895734375*^9}] }, Open ]], Cell[TextData[{ Cell[BoxData["showOtherSolutions"]], " returns a demo which allows an easy comparison between the original \ solution (shown in the left part) with alternative solutions found by ", Cell[BoxData["findOtherSolutions"]], " (shown on the right). The alternative solution avoids the yellow edge \ highlighted in the left part. 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True, ControlPlacement -> Top}, "DefaultOptions" :> {}], ImageSizeCache->{969., {333., 341.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`showOneSolutionAnimation[ Pattern[$CellContext`s, Blank[]], Optional[ Pattern[$CellContext`highlight, Blank[]], {}], Optional[ Pattern[$CellContext`tmax, Blank[]], 1], Pattern[$CellContext`opts, BlankNullSequence[]]] := With[{$CellContext`h = Part[ ImageDimensions[$CellContext`railwayMaze], 2], $CellContext`img = $CellContext`railwayMaze}, Manipulate[ Show[$CellContext`img, Graphics[{Red, Point[ Map[ $CellContext`transform[$CellContext`h], Take[$CellContext`s, $CellContext`t]]], Yellow, Point[ Map[ $CellContext`transform[$CellContext`h], \ $CellContext`highlight]], PointSize[0.02], Lighter[Blue], Point[ $CellContext`transform[$CellContext`h][ Part[$CellContext`s, $CellContext`t]]]}], $CellContext`opts], \ {{$CellContext`t, $CellContext`tmax}, 1, Length[$CellContext`s], 1, ControlPlacement 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Thus, although solution 2 looks \ identical to the original solution, it traverses the loop on the right in the \ opposite direction. 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287}, {708, 287}, {709, 286}, {710, 286}, {711, 286}, {712, 286}, {713, 286}, {714, 285}, {715, 285}, {716, 285}, {717, 285}, {718, 285}, {719, 285}, {720, 285}, {721, 284}, {722, 284}, {723, 284}, {724, 284}}}}, $CellContext`i$$, 1]], ImageSize -> Scaled[2/5]]}}], "Specifications" :> {{{$CellContext`i$$, 1, "alternative solution number"}, 1, 10, 1, Appearance -> "Open"}}, "Options" :> { SynchronousInitialization -> False, SynchronousUpdating -> False, TrackedSymbols -> True, ControlPlacement -> Top}, "DefaultOptions" :> {}], ImageSizeCache->{693., {244., 252.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`showOneSolutionAnimation[ Pattern[$CellContext`s, Blank[]], Optional[ Pattern[$CellContext`highlight, Blank[]], {}], Optional[ Pattern[$CellContext`tmax, Blank[]], 1], Pattern[$CellContext`opts, BlankNullSequence[]]] := With[{$CellContext`h = Part[ ImageDimensions[$CellContext`railwayMaze], 2], $CellContext`img = 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Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`y, Blank[]]}] := {$CellContext`y, $CellContext`h - $CellContext`x}}; Typeset`initDone$$ = True), SynchronousInitialization->False, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.5242099526875*^9, 3.527255694109375*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Solutions 1 and 2 show two other possible ways to pass through the upper \ right region.\ \>", "Item", CellChangeTimes->{{3.52395546294274*^9, 3.5239554974927163`*^9}, { 3.5239555942722516`*^9, 3.5239555942782516`*^9}}, FontSize->18], Cell["\<\ Solution 3 traverses the middle bottom loop in the opposite direction.\ \>", "Item", CellChangeTimes->{{3.52395559459227*^9, 3.5239556075520115`*^9}, { 3.52395564162396*^9, 3.5239556998522906`*^9}, {3.524210044015625*^9, 3.5242100553125*^9}}, FontSize->18], Cell["\<\ Solutions 4\[Dash]8 show alternative ways of going through the upper left \ region.\ \>", "Item", CellChangeTimes->{{3.523955700224312*^9, 3.52395570386452*^9}, { 3.5239557644339848`*^9, 3.523955793859668*^9}, {3.52421014665625*^9, 3.52421015259375*^9}}, FontSize->18], Cell["\<\ Solution 9 is identical to the original solution (it uses a small artifact in \ the image).\ \>", "Item", CellChangeTimes->{{3.5239557995449924`*^9, 3.5239558038002357`*^9}, { 3.5239558536960897`*^9, 3.5239559016808343`*^9}, {3.524210167171875*^9, 3.524210197296875*^9}}, FontSize->18], Cell["\<\ Solution 10 is incorrect (illegal turn in the bottom region).\ \>", "Item", CellChangeTimes->{{3.523955498025747*^9, 3.523955528756504*^9}, { 3.5239559076401753`*^9, 3.5239559356157756`*^9}, {3.524210228140625*^9, 3.5242102365625*^9}}, FontSize->18] }, Open ]], Cell["\<\ Assume that we don\[CloseCurlyQuote]t count simple loop-reversals as being \ different. Then solutions 4 and 5 are the same, and also 7 and 8 represent \ the same solution. To sum up, we have three distinct ways to pass through the upper right \ region, and four distinct ways to go through the upper left region. These \ possibilities can be combined against each other, which gives at least 12 \ distinct solutions of the current maze (note that there might be other \ solutions that we haven\[CloseCurlyQuote]t found).\ \>", "Text", CellChangeTimes->{{3.52421371821875*^9, 3.524213973421875*^9}}, FontSize->18] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ The necessity of manual intervention\ \>", "Section", CellChangeTimes->{ 3.483202458955147*^9, {3.514308863196991*^9, 3.5143088633311243`*^9}, { 3.525968008491189*^9, 3.525968018244747*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"original4", "=", GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzsvXtwVdd9PX4T7ARh8/IDHAzFlkHCkJdhTJNvzMvk0XomyeT7nfbfNG4t JzYGBdkEMELY+EHMwyKG2OBxpzOdTB9pO+00mQYP2Bg3aawUSIKJEQSQAQPC BoFsUAiJ7m/97hqt2dr73HOv7lvSZ/1x5+jq3HP22Xufc9b+7LXX59Z7F/3f ug8nEomGofj4v9/4zrzFi7+x7P9dhT/uXvbwffcOwcYE/LUfG///dtJgMBgM 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{200, 499}, {201, 499}, {202, 499}, {203, 499}, {204, 500}, {205, 499}, {206, 499}, {207, 499}, {208, 500}, {209, 499}}}}, $CellContext`i$$, 1]], ImageSize -> Scaled[2/5]]}}], "Specifications" :> {{{$CellContext`i$$, 1, "alternative solution number"}, 1, 13, 1, Appearance -> "Open"}}, "Options" :> { SynchronousInitialization -> False, SynchronousUpdating -> False, TrackedSymbols -> True, ControlPlacement -> Top}, "DefaultOptions" :> {}], ImageSizeCache->{691., {262., 270.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`showOneSolutionAnimation[ Pattern[$CellContext`s, Blank[]], Optional[ Pattern[$CellContext`highlight, Blank[]], {}], Optional[ Pattern[$CellContext`tmax, Blank[]], 1], Pattern[$CellContext`opts, BlankNullSequence[]]] := With[{$CellContext`h = Part[ ImageDimensions[$CellContext`railwayMaze], 2], $CellContext`img = $CellContext`railwayMaze}, Manipulate[ Show[$CellContext`img, Graphics[{Red, Point[ Map[ $CellContext`transform[$CellContext`h], Take[$CellContext`s, $CellContext`t]]], Yellow, Point[ Map[ $CellContext`transform[$CellContext`h], \ $CellContext`highlight]], PointSize[0.02], Lighter[Blue], Point[ $CellContext`transform[$CellContext`h][ Part[$CellContext`s, $CellContext`t]]]}], $CellContext`opts], \ {{$CellContext`t, $CellContext`tmax}, 1, Length[$CellContext`s], 1, ControlPlacement -> Top}, SaveDefinitions -> True, TrackedSymbols -> True]], $CellContext`railwayMaze = Image[CompressedData[" 1:eJzsvXuQVdWdPX4TNKFRXj7AIAzaQjdCHgolk0zkJXnMWJWk8q2a+TcTZ9Im itChI0GkaRQfRB42EaJgOTVVU6l5ZGZqpiapCRYo4iQTOwOYIEqDQAsINAoN rdAhJH1/63dX9arde5977u377u7P+uPW6dvnnrPP3vucs/Znr70+N9+z6P/d MySRSEy4IpHYh43/fztpMBgMBkNvdHd3/+53v7t8+XK5C2IwGAwGg8FgMBgM BoPBYDAYDAZDAdDd3Y3Py5cv//73v/e+NBgMBoPBYDAYDAUEaPalS5ewcfr0 aff7P/zhD2UqkcFgMBgMBoPB0I8BIn0pFtR7d3R0nDlzhj8hJ+8Tfv/731vA 3GAwGAwGg8HQj9BdHGR56tJfQkHOaDAYDAaDwWAw5ICMMeqckeWpXeF3aS7B 9C0Gg8FgMBgMhpKhu7v79ylkSZJzxu7du1emsGrVqpUO+OfGjRvdnVtaWiJ3 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Then solutions 6 and 12 are identical, 7 and 11 are \ identical, 8 and 10 are identical, and 9 is the same as the original solution. To sum up, we have four distinct ways to pass through the middle left part, \ i.e. at least 4 distinct solutions of the current maze (again, there might be \ other solutions that we haven\[CloseCurlyQuote]t found).\ \>", "Text", CellChangeTimes->{{3.52421371821875*^9, 3.524213973421875*^9}, { 3.52421410853125*^9, 3.52421416125*^9}, {3.52421469896875*^9, 3.524214811*^9}, {3.524214846203125*^9, 3.52421486984375*^9}}, FontSize->18] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Possible projects", "Section", CellChangeTimes->{3.527334659122973*^9}], Cell[CellGroupData[{ Cell["\<\ Solve the last maze without manual intervention.\ \>", "ItemNumbered", CellChangeTimes->{{3.5273346725467405`*^9, 3.5273346763309574`*^9}}], Cell["\<\ Design a program that will automatically generate railway mazes.\ \>", "ItemNumbered", CellChangeTimes->{{3.5273346819132767`*^9, 3.5273346830103393`*^9}}], Cell["\<\ Find other types of mazes that are worthy of investigation and automation in \ this way.\ \>", "ItemNumbered", CellChangeTimes->{{3.52733468845065*^9, 3.527334690944793*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", CellChangeTimes->{{3.527164310135844*^9, 3.527164312108957*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "R. Penrose, ", StyleBox["Railway mazes", FontSlant->"Italic"], ", in ", StyleBox["A Lifetime of Puzzles", FontSlant->"Italic"], ", E. D. Demaine, M. L. Demaine, T. Rodgers, eds.,\nAK Peters, Wellesley, \ Mass., 2008, 133\[Dash]148." }], "ItemNumbered", CellChangeTimes->{{3.5271643587996273`*^9, 3.5271643776517057`*^9}}], Cell[TextData[{ "A. Slav\[IAcute]k and S. Wagon, ", StyleBox["Railway Mazes: From Picture to Solution", FontSlant->"Italic"], ", Journal of Recreational Mathematics vol. 36, no.\[NonBreakingSpace]3 \ (2011), 208\[Dash]221." }], "ItemNumbered", CellChangeTimes->{{3.527164413950782*^9, 3.527164430788745*^9}, { 3.527164468058877*^9, 3.5271644950334196`*^9}, {3.527248960483507*^9, 3.527248973254237*^9}}], Cell[TextData[{ "A. Slav\[IAcute]k and S. 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