(*********************************************************************** 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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At the 1997 ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " Developers Conference in Champaign, Illinois, a programming contest was \ held for the participants. The problem, created by Matthew Cook of Wolfram \ Research, was to write a program to create a graphical object with certain \ properties. The problem, and the winning solution by Richard Mercer of Wright \ State University, is presented here along with Matthew's example solution.\n", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "THE PROBLEM\n", StyleBox["by Matthew Cook (mcook@wolfram.com)", FontWeight->"Plain", FontSlant->"Italic"] }], "Subsection"], Cell[TextData[StyleBox["", FontSlant->"Italic"]], "Text", FontWeight->"Plain"], Cell[BoxData[ \(TextForm \`Create\ the\ function\ TreeGraphic[d_Integer]\ \ that\ produces\ a\ Graphics\ object\ for\ a\ tree\ with\ 3\[CenterDot]2\^d\ leaves, \ arranged\ and\ connected\ as\ shown\ below . \ In\ particular, \ the\ 3\[CenterDot]2\^d\ leaves\ should\ be\ arranged\ on\ a\ circle, \ their\ 3\[CenterDot]2\^\(d - 1\)\ neighbors\ should\ be\ arranged\ between \ them\ on\ a\ smaller\ concentric\ circle\ and\ so\ on, \ until\ one\ vertex\ is\ placed\ at\ the\ center\ of\ all\ the\ circles . \ As\ you\ can\ see, \ every\ non\ leaf\ vertex\ is\ connected\ to\ exactly\ three\ other\ vertices . \ If\ your\ output\ differs\ from\ that\ shown\ below\ by\ only\ a\ rotation, \ that' s\ \(fine . \)\)], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[GraphicsArray[{\ TreeGraphic[3], \ TreeGraphic[5]}]]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .47619 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.47619 0.011338 0.47619 [ [ 0 0 0 0 ] [ 1 .47619 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .47619 L 0 .47619 L closepath clip newpath % Start of sub-graphic p 0.02381 0.011338 0.477324 0.464853 MathSubStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.120075 0.5 0.120075 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] 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05aoo`00G7oo00<007ooOol06goo00<007ooOol09goo00<007ooOol06goo00<007ooOol0Fgoo001K Ool00`00Oomoo`0LOol00`00Oomoo`0WOol00`00Oomoo`0LOol00`00Oomoo`1JOol005Yoo`03001o ogoo01ioo`03001oogoo02Eoo`03001oogoo01ioo`03001oogoo05Uoo`00Ngoo00<007ooOol09Goo 00<007ooOol0NWoo001kOol00`00Oomoo`0UOol00`00Oomoo`1jOol007]oo`03001oogoo02Eoo`03 001oogoo07Yoo`00O7oo00<007ooOol08goo00<007ooOol0Ngoo001lOol00`00Oomoo`0SOol00`00 Oomoo`1kOol007aoo`03001oogoo02=oo`03001oogoo07]oo`00O7oo00<007ooOol08goo00<007oo Ool0Ngoo001mOol00`00Oomoo`0QOol00`00Oomoo`1lOol007eoo`03001oogoo025oo`03001oogoo 07aoo`00OGoo00<007ooOol08Goo00<007ooOol0O7oo003oOolQOol00?moob5oo`00ogoo8Goo003o OolQOol00?moob5oo`00ogoo8Goo0000\ \>"], ImageRangeCache->{{{0, 287}, {287, 0}} -> {-4.16411, -4.16411, 0.0290182, 0.0290182}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["REPORT ", "Section"], Cell["\<\ What is given above under SOLUTION and EXAMPLES is exactly what I \ submitted during the contest with two minor modifications : (1) \"Pi\" has replaced the actual Greek letter to improve readability. (2) A semicolon was added to the line beginning \"Lines[0] =\" to suppress \ output.\ \>", "Text"], Cell["\<\ We had 90 minutes to work on this problem and submit a solution. I \ thought that would be plenty of time, as I had often done harder things in \ 90 minutes, as had many others in the room. Boy was I wrong! I felt like I \ was not thinking clearly, and the time went by quickly. One solution was \ submitted in a little over an hour, and one began to hear numerous muttered \ curses (including a few of my own) as the deadline approached. I was \ fortunate to find something that worked, and submitted my solution with a few \ minutes to spare. The deadline was extended to 10:20, and apparently two \ other solutions were submitted by that time. Reportedly solutions continued \ to come until after midnight; I left the room after the 10:20 deadline.\ \>", "Text"], Cell["\<\ We had been told the solutions would be based on elegance. Elegance \ is subjective, of course. I spent a lot of time trying to decide whether to \ write a constructive program like Matthew Cook's sample solution (below), or \ to write a recursive solution like the one I submitted. I would have \ preferred to use a constructive solution as they tend to be shorter and \ faster, but they are also more subtle to construct. As time grew shorter, I \ decided the first condition of elegance is that the program should work, and \ that was the only thing I could accomplish within the time limit! That turned \ out to be the right strategy. My solution won not because it was elegant (I \ don't claim that it is) but because it was the only one of the four submitted \ \"on time\" that actually did what it was supposed to do.\ \>", "Text"], Cell["\<\ Incidentally, I just have to mention that my solution actually \ produces the picture given to us during the contest, without rotation. \ Matthew's solution produces an image rotated 90 degrees. But then he said it \ was OK to do so! I want to express my thanks to: Wolfram Research for the generous prize of my \ choice of application package. (I chose Wavelet Explorer.) ; Matthew Cook, who came up with a problem of appropriate difficulty, not too \ easy but not impossible , and the other employees of Wolfram Research who \ helped administer the contest that evening; and finally all the contestants \ in the competition, probably 25-30 altogether, who joined in the spirit of \ competition for the fun of it. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Matthew Cook's Example Solution", "Section"], Cell[BoxData[ \(TreeGraphic[d_Integer] := Show[\ Graphics[ Table[\(Line[{k + 1, k, k + 1} #]&\)\ /@\ Partition[ Table[\({Cos[#], Sin[#]}&\)[2 Pi\ t/\((6*2^k)\)], \n{t, 1 + 6*2^k}], 3, 4], {k, 0, d}], AspectRatio -> Automatic]]\)], "Input"] }, Open ]], Cell[BoxData[ \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)], "Input"], Cell["A PREVIOUS PROBLEM", "Section"], Cell["KEITH NUMBERS", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ In Vol 6 No 3, a problem involving Keith numbers was presented. A Keith \ number is an n-digit integer N with the following property: If a \ Fibonacci-like sequence (in which each term in the sequence is the sum of the \ n previous terms) is formed, with the first n terms being the decimal digits \ of the number N, then N itself occurs as a term in the sequence. For example, \ 197 is a 3-digit Keith number since starting with the first three digits \ 1,9,7 the Fibonacci-like sequence of summing the previous 3 terms eventually \ includes the number 197. 1, 9, 7, 17, 33, 57, 107, 197, ... \ \>", "Text"], Cell[TextData[{ StyleBox["The problem was to write a function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Keith[n]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" which returns the list of all n digit Keith numbers. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "SOLUTION \n", StyleBox["by Colin Rose (crose@c2.telstra-mm.net.au) ", FontWeight->"Plain", FontSlant->"Italic"] }], "Subsection"], Cell[BoxData[ \(\(\nEgbert\ = \ Compile[{{num, \ _Integer}, \ {lis, \ _Integer, \ 1}}, \ Block[{s = Plus@@lis, \ r = lis}, \n\ \ While[s\ < \ num, \ r = Append[Rest[r], \ s]; \ s = Plus@@r]; \ If[s\ === \ num, \ Print[num]]]]\ ; \n\n Keith[n_]\ := \ Module[{y, \ u}, \ u\ = \ Table[Unique[y], \ {n}]; \n\ \ \ \(Do[\ Egbert[\ FromDigits[u], \ u], \ ##]&\)\ @@\ ReplacePart[Map[{#, 0, 9}&, \ u], \ 1, \ {1, 2}]]\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Keith[4] // Timing\)], "Input"], Cell[BoxData[ \(1104\)], "Print"], Cell[BoxData[ \(1537\)], "Print"], Cell[BoxData[ \(2208\)], "Print"], Cell[BoxData[ \(2580\)], "Print"], Cell[BoxData[ \(3684\)], "Print"], Cell[BoxData[ \(4788\)], "Print"], Cell[BoxData[ \(7385\)], "Print"], Cell[BoxData[ \(7647\)], "Print"], Cell[BoxData[ \(7909\)], "Print"], Cell[BoxData[ \({2.40999999999985448`\ Second, Null}\)], "Output"] }, Open ]], Cell["NEW PROBLEM", "Section"], Cell["\<\ The following problem comes from the National Championship Round \ of the 1997 USA Computing Olympiad \ (http://usaco.uwp.edu/usaco97/NationalChampProb.html#P1).\ \>", "Text"], Cell["Stamps", "Subsection"], Cell[TextData[{ "Given a set of stamp values (e.g., {1 cent, 3 cents}) and an upper limit K \ to the number of stamps that can fit on an envelope, find the largest \ unbroken list of postages that can be created. \n\nFor example, consider \ stamps whose values are limited to 1 cent and 3 cents; you can use at most 5 \ stamps. It's easy to see how to assemble postage of 1 through 5 cents (just \ use that many 1 cent stamps), and successive values aren't much harder: 6 = \ 3+3; 7 = 3+3+1; 8 = 3+3+1+1; 9=3+3+3; 10=3+3+3+1; 11=3+3+3+1+1; 12=3+3+3+3; \ 13=3+3+3+3+1 .\n\nHowever, there is no way to make 14 cents of postage with 5 \ or fewer stamps of value 1 and 3 cents. Thus, for this set of two stamp \ values and a limit of K=5, the answer is {1,2,3,4,5,6,7,8,9,10,11,13}.\n\n\ Write a function ", StyleBox["Stamps[mylist, K] ", FontWeight->"Bold"], "which returns the largest unbroken list of postages with values in ", StyleBox["mylist", FontWeight->"Bold"], " which can fit on an envelope with at most ", StyleBox["K", FontWeight->"Bold"], " stamps. It is sufficient to output the lowest and highest values of the \ unbroken string. For example, \n\nIn []:= Stamps[{1,3},5] \n\nOut[]:= {1,13}\n\ " }], "Text"], Cell[" If there is more than one answer, find them all.", "Text"], Cell["Show me the pearls!", "Subsection"], Cell["\<\ Have you got a Mathematica Pearl burried in your files? I invite \ you to share them in this column and add them to our string. If you have the \ goods, then send them along to piele@uwp.edu. \ \>", "Text"], 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, which selects \ the top four American high school computer programmers to represent the USA \ at the annual International Computing Olympiad. He also writes the column, ", StyleBox["Cowculations", FontSlant->"Italic"], ", devoted to computer algorithms using Mathematica in ", StyleBox["Quantum Magazine", FontVariations->{"Underline"->True}], ". The Web address for the USACO is: http:// usaco.uwp.edu." }], "Text"], Cell["\<\ Don Piele Mathematics Department University of Wisconsin-Parkside Kenosha, WI 53141 piele@cs.uwp.edu \ \>", "Subsubsection"] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 800}, {0, 544}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{761, 508}, WindowMargins->{{3, Automatic}, {Automatic, 6}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (*********************************************************************** Cached data follows. 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