(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 235654, 5560]*) (*NotebookOutlinePosition[ 236419, 5587]*) (* CellTagsIndexPosition[ 236375, 5583]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" Pearls", Evaluatable->False, AspectRatioFixed->True] }], "Title", Evaluatable->False, AspectRatioFixed->True], Cell["Problems and Solutions\t\t\t\t\tVol 8 No 2", "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell["by Don Piele", "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Welcome back to Mathematica Pearls, the column devoted to examining \ interesting Mathematica solutions to an assortment of simple and appealing \ problems. \n\nIn April I had the pleasure of visiting Antalya, a lovely city \ in southern Turkey, on the Mediterranean coast. The purpose of the trip was \ to review the planned venue of the next international computer programming \ Olympiad. This year, in October, it will be Turkey's turn to host the 11th \ annual International Olympiad in Informatics (IOI). This is a computer \ programming competition similar in structure to the well-established \ International Mathematics Olympiad (IMO) that began in 1959. At IOI the top \ four high school computer programmers from 70 different countries compete \ over two five-hours days by writing efficient computer programs in Pascal or \ C/C++ to solve six challenging problems. For example, knowing how to program \ the shortest path on a weighted graph using Dijkstra's algorithm is often the \ key to solving an IOI type problem.\n\nI took a direct flight from Chicago to \ Istanbul with a connection to Antalya. The total time from take-off to \ landing was around 14 hours. Knowing that the deadline for this column would \ be coming up after I returned, I took a fully charged Compaq notebook \ computer along to help ease the boredom of the long flight by working on a \ new problem for this column.\n\nThe Dutch mathematician, E. Dijkstra, \ published his famous shortest path algorithm in 1959. I thought it might be \ fun to graphically illustrate this celebrated algorithm within ", StyleBox["Mathematica.", FontSlant->"Italic"], " Between two dinners, a snack, and a few hours of light sleep, this \ project filled out the remaining time nicely, and the battery, on power \ management, held out.\n" }], "Text"], Cell["\<\ I began with an example of drawing a simple weighted graph by \ placing seven vertices on the circumference of a circle, drawing edges \ between certain vertices, and assigning a weight to each edge. The assignment \ of weights was held in the matrix weightedEdges. Here each row {i,j,k} means \ there is an edge from vertex i to vertex j with weight k.\ \>", "Text"], Cell[BoxData[ \(n = 7; \n weightedEdges = {{1, 2, 5}, {1, 3, 2}, {1, 5, 7}, {2, 3, 1}, {2, 4, 5}, { 3, 4, 8}, {3, 5, 10}, {4, 5, 2}, {4, 6, 10}, {5, 6, 2}, {2, 6, 6}, { 1, 4, 12}, {6, 7, 5}, {5, 7, 20}}; \)], "Input"], Cell["\<\ The weights and the edges were easily picked off from this matrix. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(weights = weightedEdges\ /. \ {i_, j_, k_} :> k\), \(edges = weightedEdges\ /. \ {i_, j_, k_} :> {i, j}\)}], "Input"], Cell[BoxData[ \({5, 2, 7, 1, 5, 8, 10, 2, 10, 2, 6, 12, 5, 20}\)], "Output"], Cell[BoxData[ \({{1, 2}, {1, 3}, {1, 5}, {2, 3}, {2, 4}, {3, 4}, {3, 5}, {4, 5}, {4, 6}, {5, 6}, {2, 6}, {1, 4}, {6, 7}, {5, 7}}\)], "Output"] }, Open ]], Cell["\<\ Now the weighted graph was drawn on the unit circle with the \ vertices numbered from 1 to 7 and the beginning and ending vertex colored \ green.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(edgesG = Map[Line[{{Cos[\ \[Pi]\ \(#[\([1]\)] - 1\)\/n], \(\((\(-1\))\)\^#[\([1]\)]\) Sin[\ \[Pi]\ \(#[\([1]\)] - 1\)\/n]}, { Cos[\ \[Pi]\ \(#[\([2]\)] - 1\)\/n], \(\((\(-1\))\)\^#[\([2]\)]\) Sin[\ \[Pi]\ \(#[\([2]\)] - 1\)\/n]}}]&, edges]; \n weightsPos = \(edgesG /. 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the same as the weight from vertex j to \ vertex i. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[adjM]; \nadjM[i_, j_] := \[Infinity]\ ; \n adjM[i_, j_] := adjM[j, i]\ /; \ i > j; \n adjM[i_, j_] := 0\ /; \ i == j; \t\t\n weightedEdges\ /. \ {i_, j_, k_} :> \((adjM[i, j] = k)\); \n Array[adjM, {n, n}] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"0", "5", "2", "12", "7", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}, {"5", "0", "1", "5", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], "6", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}, {"2", "1", "0", "8", "10", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}, {"12", "5", "8", "0", "2", "10", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]}, {"7", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], "10", "2", "0", "2", "20"}, { InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], "6", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], "10", "2", "0", "5"}, { InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]], "20", "5", "0"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[ "Dijkstra's algorithm begins by assigning 0 to vertex 1 and \[Infinity] to \ the remaining vertices. 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Notice that \ 0 is the length of the shortest path from 1 to itself. Let S be the set of \ vertices whose shortest path is known. We know that vertex 1 belongs to S \ since the shortest path from vertex 1 to vertex 1 is zero. ", StyleBox[ "S={1}, distance={0,\[Infinity],\[Infinity],\[Infinity],\[Infinity],\ \[Infinity],\[Infinity]}.", FontWeight->"Bold"], "\n\nNow the distance list was updated as follows: For each vertex \ connected by an edge to vertex 1, replace the distance to that vertex by the \ weight of the edge. Those vertices that are not connected directly to vertex \ 1 by an edge are left alone. This is coded as follows:\n\n", StyleBox[ "If [distance[[1]]+adjM[1,v]"Bold"], "\n\nThis leads to a new distance list as shown below. Now pick the vertex \ with the shortest distance to vertex 1 and add it to the set S of vertices \ whose shortest distance from vertex 1 is known. Vertex 3 has the shortest \ distance of 2. 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o`00T7oo0P001goo0P000Woo0P00EWoo002IOol01Sg_001Nmekg000mkeIoo`00VGoo00INmcg_0000 03g_G_MFOol00?Eoo`00mGoo003eOol00?Eoo`00mGoo003eOol00?Eoo`00mGoo003eOol00?Eoo`00 mGoo003eOol00?Eoo`00mGoo003eOol00?Eoo`00mGoo0000\ \>"], ImageRangeCache->{{{73.4375, 269.188}, {561.438, 365.688}} -> {-2.1004, 3.28356, 0.00980852, 0.00980852}}], Cell[TextData[StyleBox[ "Write a program that will take a weighted graph defined on n vertices and \ display a shortest path between vertex 1 and vertex n. Shown above is a path \ of shortest length from vertex 1 to vertex 7 for the weighted graph described \ above. Test your program with:"]], "Text"], Cell[BoxData[ \(n = 21; \n weightedEdges = {{1, 2, 2}, {1, 3, 4}, {1, 4, 1}, {2, 3, 3}, {2, 5, 1}, { 3, 5, 2}, {4, 6, 5}, {4, 7, 4}, {5, 8, 3}, {6, 7, 3}, {6, 8, 3}, { 6, 9, 2}, {6, 10, 4}, {7, 11, 2}, {8, 12, 1}, {8, 15, 8}, {9, 12, 3}, {9, 14, 2}, {10, 11, 6}, {10, 13, 3}, {10, 14, 6}, {11, 13, 4}, {11, 18, 2}, {12, 15, 6}, {12, 14, 3}, {13, 14, 5}, {13, 17, 2}, {13, 18, 1}, {14, 15, 4}, {14, 16, 2}, {15, 19, 6}, {15, 16, 2}, {16, 17, 1}, {16, 20, 1}, {17, 20, 3}, {17, 18, 8}, {18, 20, 5}, {19, 21, 2}, {20, 21, 8}}; \)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Pearl Harvest I", "Section"], Cell[TextData[StyleBox[ "In the last column, three problems were presented. Here is the solution to \ one of them."]], "Text"], Cell[CellGroupData[{ Cell["Zero Sum ('89)", "Subsection"], Cell[TextData[{ "Create the function ", StyleBox["zeroSum[n_] ", FontWeight->"Bold"], "that finds all solutions to the expression:\n\n\t \ \t1 @ 2 @ 3 @......@n == 0 \n\t \t\nwhere n is a \ single digit and each occurance of @ can be either a (+) or (-) or () which \ means to concatenate the digits.\n\nOne solution for n = 7, 1 - 23 - 45 + 67\ \t \t" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Solution by Eric Rimbey (erimbey@wolfram.com)\ \>", "Subsection"], Cell[TextData[{ StyleBox[ "This solution by Eric is a good example of how clean a solution can be if \ constructed functionally. I will attempt to deconstruct his solution to see \ what makes it tick. Since there are only three possibilities for a character \ between each of the digits, the maximum number of expressions that will need \ to be examined when n=9 is "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(3\^8\ \ \ or\ \)\)]]], StyleBox[ " 6561. So a brute force approach will easily work. Begin by defining,\n"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(chars = {"\<+\>", "\<-\>", "\<\>"}\), \(digitsequence = ToString/@Range[9]\)}], "Input"], Cell[BoxData[ \({"+", "-", ""}\)], "Output"], Cell[BoxData[ \({"1", "2", "3", "4", "5", "6", "7", "8", "9"}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Form all the character sequences."]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(charsequences = Distribute[Join[Table[chars, {8}], {"\<\>"}], List]; \n Short[charsequences, 6]\)], "Input"], Cell[BoxData[ TagBox[ \({{"+", "+", "+", "+", "+", "+", "+", "+", ""}, {"+", "+", "+", "+", "+", "+", "+", "-", ""}, {"+", "+", "+", "+", "+", "+", "+", "", ""}, {"+", "+", "+", "+", "+", "+", "-", "+", ""}, {"+", "+", "+", "+", "+", "+", "-", "-", ""}, {"+", "+", "+", "+", "+", "+", "-", "", ""}, {"+", "+", "+", "+", "+", "+", "", "+", ""}, {"+", "+", "+", "+", "+", "+", "", "-", ""}, \[LeftSkeleton]6546\[RightSkeleton], {"", "", "", "", "", "", "+", "", ""}, {"", "", "", "", "", "", "-", "+", ""}, {"", "", "", "", "", "", "-", "-", ""}, {"", "", "", "", "", "", "-", "", ""}, {"", "", "", "", "", "", "", "+", ""}, {"", "", "", "", "", "", "", "-", ""}, {"", "", "", "", "", "", "", "", ""}}\), (Short[ #, 6]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[""], "Input"], Cell[BoxData[ TagBox[ \({{"+", "+", "+", "+", "+", "+", "+", "+", ""}, {"+", "+", "+", "+", "+", "+", "+", "-", ""}, {"+", "+", "+", "+", "+", "+", "+", "", ""}, {"+", "+", "+", "+", "+", "+", "-", "+", ""}, \[LeftSkeleton]6554\[RightSkeleton], {"", "", "", "", "", "", "", "+", ""}, {"", "", "", "", "", "", "", "-", ""}, {"", "", "", "", "", "", "", "", ""}}\), (Short[ #, 6]&)]], "Output"] }, Open ]], Cell["\<\ Place the characters between the digits and form the possible \ strings.\ \>", "Text"], Cell[BoxData[ \(\(digitchar = \(Flatten[Transpose[{digitsequence, #}\ ]]&\)\ /@charsequences; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Short[digitchar, 6]\)], "Input"], Cell[BoxData[ TagBox[ \({{"1", "+", "2", "+", "3", "+", "4", "+", "5", "+", "6", "+", "7", ""}, {"1", "+", "2", "+", "3", "+", "4", "+", "5", "+", "6", "-", "7", ""}, {"1", "+", "2", "+", "3", "+", "4", "+", "5", "+", "6", "", "7", ""}, {"1", "+", "2", "+", "3", "+", "4", "+", "5", "-", "6", "+", "7", ""}, \[LeftSkeleton]722\[RightSkeleton], {"1", "", "2", "", "3", "", "4", "", "5", "", "6", "+", "7", ""}, {"1", "", "2", "", "3", "", "4", "", "5", "", "6", "-", "7", ""}, {"1", "", "2", "", "3", "", "4", "", "5", "", "6", "", "7", ""}}\), (Short[ #, 6]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(possible = StringJoin\ /@\ digitchar; \nShort[possible, 4]\)], "Input"], Cell[BoxData[ TagBox[ \({"1+2+3+4+5+6+7", "1+2+3+4+5+6-7", "1+2+3+4+5+67", "1+2+3+4+5-6+7", "1+2+3+4+5-6-7", "1+2+3+4+5-67", "1+2+3+4+56+7", "1+2+3+4+56-7", "1+2+3+4+567", "1+2+3+4-5+6+7", \[LeftSkeleton]709\[RightSkeleton], "1234-567", "12345+6+7", "12345+6-7", "12345+67", "12345-6+7", "12345-6-7", "12345-67", "123456+7", "123456-7", "1234567"}\), (Short[ #, 4]&)]], "Output"] }, Open ]], Cell["Now select out the expressions that sum to zero.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(solutions = Select[possible, ToExpression[#] == 0\ &]\)], "Input"], Cell[BoxData[ \({"1+2-3+4-5-6+7", "1+2-3-4+5+6-7", "1-2+3+4-5+6-7", "1-2-3-4-5+6+7", "1-23+4+5+6+7", "1-23-45+67"}\)], "Output"] }, Open ]], Cell["Put it together and you have Eric's solution.", "Text"], Cell[BoxData[ \(ZeroSum[k_Integer?\((0 < # < 10\ &)\)] := \n\t\t\ Module[\ {chars = {"\<+\>", "\<-\>", "\<\>"}, \n\t\t\t\tdigitsequence, charsequences}, \n\t\t\tdigitsequence = ToString\ /@\ Range[k]; \n \t\t\tcharsequences = \n\t\t\t Distribute[Join[Table[chars, {k - 1}], {"\<\>"}], List]; \n\t\t Select[StringJoin\ /@ \((\(Flatten[Transpose[{digitsequence, #}]]&\)\ /@\n\t\t\t\t\t\t charsequences)\), ToExpression[#] == 0\ &]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ZeroSum[9] // Timing\)], "Input"], Cell[BoxData[ \({2.57999999999992724`\ Second, {"1+2-34-56+78+9", "1-2-34+5+6+7+8+9", "1-23-4-56-7+89", "12+3+4-5-6-7+8-9", "12+3-4+5-6+7-8-9", "12+3-45+6+7+8+9", "12+34-56-7+8+9", "12-3+4+5+6-7-8-9", "12-3+4+56-78+9", "12-3-4-5+6-7-8+9", "12-3-4-5-6+7+8-9"}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox[ "He also put it together into a oneliner, which is much slower. "]], "Text"], Cell[BoxData[ \(ZeroSum[k_Integer?\((0 < # < 10\ &)\)] := \n\t\t\t Select[StringJoin\ /@ \((\(Flatten[Transpose[{ToString\ /@\ Range[k], #}]]&\)\ /@\n \t\t\t\t\t\t Distribute[ Join[Table[{"\<+\>", "\<-\>", "\<\>"}, {k - 1}], {"\<\>"}], List])\), ToExpression[#] == 0\ &]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ZeroSum[9] // Timing\)], "Input"], Cell[BoxData[ \({15.8799999999973806`\ Second, {"1+2-34-56+78+9", "1-2-34+5+6+7+8+9", "1-23-4-56-7+89", "12+3+4-5-6-7+8-9", "12+3-4+5-6+7-8-9", "12+3-45+6+7+8+9", "12+34-56-7+8+9", "12-3+4+5+6-7-8-9", "12-3+4+56-78+9", "12-3-4-5+6-7-8+9", "12-3-4-5-6+7+8-9"}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox[ "The problem was also solved by Joseph Post .\n\n"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Send in the pearls!", "Subsection"], Cell[TextData[{ StyleBox["Have you got a "], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[ " Pearl buried in your files? I invite you to share them in this column \ and add them to our string."], StyleBox[" Send them along to don@piele.com\n"] }], "Text"], Cell[CellGroupData[{ Cell["About The Editor", "Subsubsection"], Cell[TextData[{ "Don Piele has been interested in creating programming problems since he \ began the International Computer Problem Solving Contest in the pages of ", StyleBox["Creative Computing", FontVariations->{"Underline"->True}], " in 1981. In 1992, he organized the USA Computing Olympiad (USACO), which \ selects the top four American high school computer programmers to represent \ the USA at the annual International Olympiad in Informatics (IOI). He also \ writes a column, \"", StyleBox["Cowculations,\" ", FontSlant->"Italic"], "in", StyleBox[" ", FontSlant->"Italic"], StyleBox["Quantum Magazine", FontVariations->{"Underline"->True}], StyleBox[" ", FontSlant->"Italic"], ", devoted to computer algorithms using ", StyleBox["Mathematica", FontSlant->"Italic"], ". The web address for the USACO is: www.usaco.org" }], "Text"] }, Open ]], Cell["\<\ Don Piele Mathematics Department University of Wisconsin-Parkside Kenosha, WI 53141 don@piele.com \ \>", "Subsubsection"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 1024}, {0, 704}}, WindowSize->{916, 577}, WindowMargins->{{-5, Automatic}, {Automatic, -4}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1, StyleDefinitions -> "Default.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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