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The beginnings of probability and \ statistics derive from questions asked by gamblers from their experiences at \ the gaming tables. Here we consider games where reasoning, not chance, is \ paramount. In particular, we consider the class of so-called ", StyleBox["impartial games", FontSlant->"Italic"], ", defined below. The basic mathematical theory of such games is easily \ explained and there is an algorithm, known as the Sprague-Grundy algorithm, \ that enables a player with sufficient computational resources to win, when \ confronted with a winnable position. Many interesting questions can be asked \ and answered by students, at all levels, about particular impartial games. We \ will show how ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to play various games and to investigate which player will \ win (assuming proper play). In this way, we can make and test conjectures, \ and can discover and prove new theorems. All this while playing games!" }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is an ideal environment in which to implement the Sprague-Grundy \ algorithm for playing two-player impartial games. This is largely because the \ algorithm is recursively defined, and ", StyleBox["Mathematica", FontSlant->"Italic"], " allows the user to program recursions on many types of objects. In this \ paper, we illustrate a particular game called \[OpenCurlyDoubleQuote]Cram\ \[CloseCurlyDoubleQuote], which is played on an ", Cell[BoxData[ \(TraditionalForm\`m\[Times]n\)]], " checkerboard, optionally with some squares removed. The two players \ alternate placing dominoes on adjacent pairs of uncovered squares, and the \ last player able to place a domino wins. (The game has been called various \ names, including \[OpenCurlyDoubleQuote]Plugg\[CloseCurlyDoubleQuote] and \ \[OpenCurlyDoubleQuote]Dots and Pairs\[CloseCurlyDoubleQuote]; and \ discussions of the game can be found in [1] and [2].)" }], "Text"], Cell[TextData[{ "As described, Cram is an example of an impartial two-player game. In such \ a game, there is a finite set of positions, each of which has associated with \ it a finite set of possible moves\[LongDash]positions that can be obtained \ from the given position by a single legal move. If the set of possible moves \ from a position is empty, the position is called an ", StyleBox["end position", FontSlant->"Italic"], ". The game is \[OpenCurlyDoubleQuote]impartial\[CloseCurlyDoubleQuote] \ because the set of possible moves for a position is the same for each \ player." }], "Text"], Cell[TextData[{ "The Sprague-Grundy algorithm is based on the following recursive numbering \ scheme: positions are assigned the smallest nonnegative integer not already \ assigned to one of its possible moves. (The name Mex, or Minimal Excludant, \ is often used for the smallest nonnegative integer not a member of a set of \ nonnegative integers.) It should be noted that, under this definition, end \ positions are numbered 0, since the Mex of the empty set is 0.", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "The function that assigns these numbers to positions is called the Grundy \ function." }], "Text"], Cell["There are two important properties of this numbering scheme:", "Text"], Cell["\<\ \t1. Any nonzero position must have a zero position as one of its possible \ moves.\ \>", "Text"], Cell["\<\ \t2. A zero position cannot have a zero position as a possible move.\ \>", "Text"], Cell["\<\ Both properties are easy to prove. In the case of property 1, if a \ position does not have a zero position as a possible move, the Mex of its \ possible moves is 0, that is, the number of the position would be zero. For \ property 2, if a position does have a zero position as a possible move, the \ Mex of its possible moves would be nonzero. The winning strategy is to move, whenever possible, from a nonzero position \ to a zero position. If a player can achieve a nonzero position this strategy \ will win, since one can always move from a nonzero position to a zero \ position, and the opponent must then move to a nonzero position. Since this \ process cannot continue forever, it must terminate in an end position, and \ since end positions are numbered zero, the player using this strategy will \ have won.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Programming the Sprague-Grundy Algorithm", "Section"], Cell[TextData[{ "The ", StyleBox["Mex", "Input"], " function, which accepts a list of nonnegative integers and returns the \ smallest nonnegative integer not in the list, can be rendered as follows." }], "Text"], Cell[BoxData[ \(Mex[L_List] := Min[Complement[Range[0, Length[L]], L]]\)], "Input"], Cell[TextData[{ "Assuming for the moment we have a ", StyleBox["PossibleMoves", "Input"], " function\[LongDash]that is, a function that returns for a given position, \ all positions attainable with a single legal move\[LongDash]we can \ recursively compute Grundy numbers for possible board states in the following \ way." }], "Text"], Cell[BoxData[ \(Grundy[boardstate_] := \(Grundy[boardstate] = Mex[Map[Grundy, PossibleMoves[boardstate]]]\)\)], "Input"], Cell[TextData[{ "(We cache intermediate values of the Grundy function using the form ", StyleBox[ "Grundy[boardstate_] := Grundy[boardstate] = \[CenterEllipsis]", "Input"], ", which saves time in future computations, but which greatly increases the \ memory used while playing a game. If memory becomes tight, it may be \ necessary to execute the command ", StyleBox["Clear[Grundy]", "Input"], " from time to time.)" }], "Text"], Cell[TextData[{ "This recursive definition instructs ", StyleBox["Mathematica", FontSlant->"Italic"], " to construct a tree of positions with ", StyleBox["boardstate", "Input"], " as the root and with each position on the tree branching to all its \ possible moves, until, finally, all positions at the end of branches have no \ possibles moves (i.e., are end positions). The function then numbers the \ positions on the tree, using the ", StyleBox["Mex", "Input"], " function, starting from the end positions, which are numbered 0, and \ working backwards to the root." }], "Text"], Cell[TextData[{ "We can now define a function ", StyleBox["PossibleMoves", "Input"], ", which, if not given an end position, moves to a zero-numbered position \ (the optimal move); and otherwise selects a nonzero move. This definition \ will depend, of course, on how we choose to \[OpenCurlyDoubleQuote]represent\ \[CloseCurlyDoubleQuote] the positions. For Cram it is natural to represent \ the positions as matrices with entries: 0s, 1s and ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\)]], "s; \[OpenCurlyDoubleQuote]1\[CloseCurlyDoubleQuote] represents an \ uncovered square, \[OpenCurlyDoubleQuote]0\[CloseCurlyDoubleQuote] a covered \ square, and \[OpenCurlyDoubleQuote]", Cell[BoxData[ \(TraditionalForm\`\(-1\)\)]], "\[CloseCurlyDoubleQuote] a square that has been (optionally) removed at \ the outset. The ", StyleBox["PossibleMoves", "Input", FontWeight->"Bold"], " function, then, searches for adjacent pairs of 1s and replaces them by \ 0s. It is convenient to split the search into a horizontal and a vertical \ component." }], "Text"], Cell[TextData[{ "The function that tests for possible horizontal moves, called ", StyleBox["PossibleMovesHorizontal", "Input"], ", searches for horizontal adjacent pairs of empty squares in the current \ board state, returning a list of all possible board states after a single \ legal play; ", StyleBox["PossibleMovesVertical", "Input"], " finds all possible vertical moves by finding the horizontal moves for the \ transpose of the current board state; and ", StyleBox["PossibleMoves", "Input"], " simply joins the two lists." }], "Text"], Cell[BoxData[ \(PossibleMovesHorizontal[boardstate_] := \n\t ReplaceList[ boardstate, {L1___, {bef___, 1, 1, aft___}, L2___} \[Rule] {L1, {bef, 0, 0, aft}, L2}]\)], "Input"], Cell[BoxData[ \(PossibleMovesVertical[boardstate_] := Map[Transpose, PossibleMovesHorizontal[Transpose[boardstate]]]\)], "Input"], Cell[BoxData[ \(PossibleMoves[boardstate_] := \n\t Join[PossibleMovesHorizontal[boardstate], PossibleMovesVertical[boardstate]]\)], "Input"], Cell["\<\ There are several ways we can select a particular move from a list of all \ possible moves. The simplest is to take the first possible move, using the \ following function.\ \>", "Text"], Cell[BoxData[ \(ChooseFirstMove[boardstate_] := \n\t\t If[Grundy[boardstate] == 0, \n\t\tFirst[PossibleMoves[boardstate]], \n \t\tFirst[Select[PossibleMoves[boardstate], Grundy[#] == 0&, 1]]]\)], "Input"], Cell[TextData[{ "To make repeated playings of a game more interesting, we can select one \ move at random from the complete list of possible moves, using our functions \ ", StyleBox["Choose", "Input"], StyleBox["RandomMove", "Input"], " and ", StyleBox["ChooseMove", "Input"], "." }], "Text"], Cell[BoxData[ \(ChooseRandomMove[choices_] := \(Part[#, Random[Integer, {1, Length[#]}]]&\)[choices]\)], "Input"], Cell[BoxData[ \(ChooseMove[boardstate_] := \n\t\t ChooseRandomMove[ If[Grundy[boardstate] == 0, PossibleMoves[boardstate], Select[PossibleMoves[boardstate], Grundy[#] == 0&]]]\)], "Input"], Cell[TextData[{ "Next we construct the initial position in our game, starting with a ", Cell[BoxData[ \(TraditionalForm\`4\[Times]4\)]], " version of Cram. We start with a list of sixteen 1s (where a 1 indicates \ an empty square on the checkerboard), and then replace three of the 1s with \ \[Dash]1s (again, numbers other than 1 indicate missing or played squares), \ the replacements being chosen randomly; finally partitioning the list to make \ a ", Cell[BoxData[ \(TraditionalForm\`4\[Times]4\)]], " matrix." }], "Text"], Cell[BoxData[ \(r1 := Random[Integer, {1, 5}]; \nr2 := Random[Integer, {6, 10}]; \n r3 := Random[Integer, {11, 16}]; \n pos\ = \ Partition[ ReplacePart[Table[1, {16}], \(-1\), {{r1}, {r2}, {r3}}], 4]; \)], "Input"], Cell[TextData[{ "To avoid retyping the same graphics options for each plot, we use ", StyleBox["SetOptions", "Input"], "." }], "Text"], Cell[BoxData[ \(\(SetOptions[ListDensityPlot, Frame \[Rule] None, AspectRatio \[Rule] Automatic]; \)\)], "Input"], Cell["Here is the initial state of the checkerboard.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ListDensityPlot[pos]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% DensityGraphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0192308 0.240385 0.0192308 0.240385 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath % Start of gray image p .01923 .01923 translate .96154 .96154 scale 4 string 4 4 8 [4 0 0 4 0 0] { \tcurrentfile \t1 index \treadhexstring \tpop } Mimage 00FFFFFF FFFF00FF FFFFFF00 FFFFFFFF pop P % End of image 0 g .25 Mabswid [ ] 0 setdash .01923 .01923 m .01923 .98077 L s .25962 .01923 m .25962 .98077 L s .5 .01923 m .5 .98077 L s .74038 .01923 m .74038 .98077 L s .98077 .01923 m .98077 .98077 L s .01923 .01923 m .98077 .01923 L s .01923 .25962 m .98077 .25962 L s .01923 .5 m .98077 .5 L s .01923 .74038 m .98077 .74038 L s .01923 .98077 m .98077 .98077 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{150, 150}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLc"], ImageRangeCache->{{{0, 149}, {149, 0}} -> {-0.0800208, -0.0800208, 0.0279197, 0.0279197}}] }, Open ]], Cell["\<\ Here is the Grundy function applied to the initial state of the board.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Grundy[pos]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell["\<\ If the initial position has a nonzero Grundy number, the Sprague-Grundy \ algorithm indicates we should place the domino in a location that puts the \ board in a zero position. Here are the Grundy values for all possible \ moves.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Grundy, PossibleMoves[pos]]\)], "Input"], Cell[BoxData[ \({3, 0, 3, 1, 3, 3, 2, 3, 3, 0, 0, 1, 0, 3, 3}\)], "Output"] }, Open ]], Cell["Here we see all possible first moves.", "Text"], Cell[BoxData[ \(\(ShowOpts[boardstate_] := \n\ \ Show[GraphicsArray[\n\ \ \ \ \ \ \(Partition[#, Floor[Length[#]/2]]&\)@\n\ \ \ \ \ \ \ \ Append[Map[ ListDensityPlot[#, DisplayFunction \\[Rule] Identity]&, \n \ \ \ \ \ \ \ \ \ \ \ \ PossibleMoves[boardstate]], Graphics[{}]]]]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ShowOpts[pos]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .24138 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 0.0238095 0.119048 0.00574713 0.119048 [ [ 0 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ImageRangeCache->{{{0, 332.625}, {76.5625, 0}} -> {-0.121766, -0.0232607, 0.0127577, 0.0127577}, {{9.5, 82.4375}, {74.6875, 1.8125}} -> {-0.624108, -0.187054, 0.0570845, 0.0570845}, {{89.75, 162.625}, {74.6875, 1.8125}} -> {-5.20336, -0.187054, 0.0570845, 0.0570845}, {{169.938, 242.875}, {74.6875, 1.8125}} -> {-9.78261, -0.187054, 0.0570845, 0.0570845}, {{250.125, 323.062}, {74.6875, 1.8125}} -> {-14.3601, -0.187054, 0.0570845, 0.0570845}}] }, Open ]], Cell[TextData[{ "An entire game is encapsulated in a single list by accumulating the \ successive states of the playing board, using ", StyleBox["FixedPointList", "Input"], "." }], "Text"], Cell[BoxData[ \(PlayTheGame[initialstate_] := \n\t FixedPointList[ChooseMove, initialstate, SameTest \[Rule] \((PossibleMoves[#2] == {}&)\)]\)], "Input"], Cell["Here is one complete sample game.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(thegame = PlayTheGame[pos]\)], "Input"], Cell[BoxData[ \({{{\(-1\), 1, 1, 1}, {1, 1, \(-1\), 1}, {1, 1, 1, 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Cell["\<\ It is possible to display each move more clearly by using different colors \ for each player, as well as graphics that emphasize the placement of each \ domino.\ \>", "Text"], Cell["\<\ The following variables store colors in which to display dominoes placed by \ the respective players.\ \>", "Text"], Cell[BoxData[ \(color[0] = RGBColor[1, 0, 0]; \ncolor[1] = RGBColor[0, 0, 1]; \n$Player = 0; \)], "Input"], Cell[TextData[{ "The following definition of ", StyleBox["domino", "Input"], " generates graphics primitives for a domino placed on the checkerboard. " }], "Text"], Cell[BoxData[ \(domino[M_?MatrixQ] := Module[{L}, L = Reverse/@Sort[M]; $Player = 1 - $Player; {color[$Player], Rectangle[L[\([1]\)] - {3/4, 3/4}, L[\([2]\)] - {1/4, 1/4}]}]\)], "Input"], Cell["Here we begin a new game.", "Text"], Cell[BoxData[ \($Player = 0; start\ = \ Partition[ReplacePart[Table[1, {16}], \(-1\), {{r1}, {r2}, {r3}}], 4]; \)], "Input"], Cell[TextData[{ "Here we tell ", StyleBox["Mathematica", FontSlant->"Italic"], " to play the game, and display its progress using ", StyleBox["ListDensityPlot", "Input"], "." }], "Text"], Cell[BoxData[ \(\(thegame = PlayTheGame[start]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[ GraphicsArray[ \(ListDensityPlot[#, DisplayFunction \[Rule] Identity]&\)/@thegame]]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .15385 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% 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However, for \ a rectangular board with no squares removed, simple symmetry arguments tell \ us that the first player will win on a ", Cell[BoxData[ \(TraditionalForm\`\((2 n + 1)\)\[Times]2 m\)]], " board by playing in the \[OpenCurlyDoubleQuote]center\ \[CloseCurlyDoubleQuote] and then answering symmetrically about the center; \ while the second player wins on a ", Cell[BoxData[ \(TraditionalForm\`2 n\[Times]2 m\)]], " board by answering symmetrically the first player\[CloseCurlyQuote]s \ move. However, the ", Cell[BoxData[ \(TraditionalForm\`\((2 n + 1)\)\[Times]\((2 m + 1)\)\)]], " case is largely unknown (except when ", Cell[BoxData[ \(TraditionalForm\`n = 0\)]], ", in which case the Grundy numbers are \[OpenCurlyDoubleQuote]periodic\ \[CloseCurlyDoubleQuote]\[LongDash]see [3]). In [2], even the ", Cell[BoxData[ \(TraditionalForm\`5\[Times]5\)]], " board is listed as unsolved." }], "Text"], Cell[TextData[{ "We shall use ", StyleBox["Mathematica", FontSlant->"Italic"], " to show that the ", Cell[BoxData[ \(TraditionalForm\`5\[Times]5\)]], " board is a win for the second player. Simply using our program to compute \ the Grundy number of the board proves inadequate, and so we must modify our \ approach." }], "Text"], Cell["\<\ First, we need to compute the Grundy numbers of only six \ representative first moves, because all other first moves made on the board \ are identical to one of these six by symmetry :\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"firstmoves", "=", RowBox[{"With", "[", RowBox[{ \({w = {{{1}, {2}}, {{2}, {3}}, {{6}, {7}}, {{7}, {8}}, {{11}, { 12}}, {{12}, {13}}}}\), ",", "\n", "\t", RowBox[{"Table", "[", "\n", "\t\t\t\t\t\t", RowBox[{ \(Partition[ReplacePart[Table[1, {25}], 0, w[\([i]\)]], 5]\), ",", "\n", "\t\t\t", StyleBox[\({i, Length[w]}\), FontColor->GrayLevel[0]]}], StyleBox["]", FontColor->GrayLevel[0]]}]}], StyleBox["]", FontColor->GrayLevel[0]]}]}], StyleBox[";", FontColor->GrayLevel[0]]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[ GraphicsArray[ \(ListDensityPlot[Reverse[#], Frame \[Rule] None, DisplayFunction 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\[OpenCurlyDoubleQuote]primitive\ \[CloseCurlyDoubleQuote] Grundy numbering which replaces all nonzero numbers \ by the number 1. Thus a position will now have the number 1 if and only if \ one of its possible moves is numbered 0. This numbering has the two \ properties listed above for the ordinary Grundy numbering, and hence the \ first player wins if and only if the board has number 1. Moreover, it often \ allows us to compute the number of a position without knowing the numbers \ assigned to all its possible moves! For as soon as one of the possible moves \ for a position has Grundy number 0, we can assign the number 1 to the \ position. Of course, it is possible that all the possible moves have number 1 \ (in which case it is necessary to compute them all), which means the position \ has Grundy number 0."], "Text"], Cell[TextData[{ "An added benefit to the primitive Grundy numbering is that we don\ \[CloseCurlyQuote]t have to use the ", StyleBox["Mex", "Input"], " function\[LongDash]we can simply play ", StyleBox["Throw", "Input", FontWeight->"Bold"], " and ", StyleBox["Catch", "Input", FontWeight->"Bold"], ": as soon as one of the possible moves of a position is found to have \ number 0, we immediately ", StyleBox["Throw", "Input"], " a 1 to the enclosing ", StyleBox["Catch", "Input"], "; while if all of the possible moves have number 1, we ", StyleBox["Throw", "Input"], " a 0. (Note that there are some drawbacks to this simplified numbering. In \ particular, it lacks the \[OpenCurlyDoubleQuote]bogus nim property\ \[CloseCurlyDoubleQuote] which makes it easy to compute the sum of games\ \[LongDash]see [3].) We thus define the ", StyleBox["QuickGrundy", "Input", FontWeight->"Bold"], " function." }], "Text"], Cell[BoxData[ \(QuickGrundy[u_] := \(QuickGrundy[u] = \n\t\t\t With[{p = PossibleMoves[u]}, \n\t\t\t\t Catch[\n\t\t\t\t Do[\n\t\t\t\t\t If[QuickGrundy[p\[LeftDoubleBracket]j\[RightDoubleBracket]] == 0, Throw[1]], \n\t\t\t\t\t\t{j, 1, Length[p]}\n\t\t\t\t\t]; \n\t\t\t\t\tThrow[0]\n\t\t\t]]\)\)], "Input"], Cell[TextData[{ "An additional complication is that each first move in the ", Cell[BoxData[ \(TraditionalForm\`5\[Times]5\)]], " board has itself over 30 possible moves; caching values for all of these \ and their possible moves, etc., is too large a strain on the computer\ \[CloseCurlyQuote]s memory, even when extended with virtual memory. Therefore \ we break up the calculation by testing each of the possible moves separately, \ reintroducing the ", StyleBox["QuickGrundy", "Input"], " function between calls (which clears out the stored function values each \ time!). This is our function ", StyleBox["BreakUpQuickGrundy", "Input"], ", which is thus a tradeoff: decreased speed for lower memory \ requirements." }], "Text"], Cell[TextData[{ "We handle the re-definition of ", StyleBox["QuickGrundy", "Input"], " inside ", StyleBox["BreakUpQuickGrundy", "Input"], " by saving the original definition in the variable ", StyleBox["QuickGrundyOriginalDefn", "Input"], ". The complete definition of any function, including all of its cached \ values, is stored in the list ", StyleBox["DownValues[", "Input"], StyleBox["FunctionName", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", "Input"], "; to restore the original definition, we simply set ", StyleBox["DownValues[QuickGrundy]", "Input"], " equal to ", StyleBox["QuickGrundyOriginalDefn", "Input"], " before testing each separate possible move." }], "Text"], Cell[BoxData[ \(\(QuickGrundyOriginalDefn = DownValues[QuickGrundy]; \)\)], "Input"], Cell[BoxData[ \(BreakUpQuickGrundy[boardstate_] := \n\t With[{p = PossibleMoves[boardstate]}, \n\t Catch[\n\t\t Scan[\((DownValues[QuickGrundy] = QuickGrundyOriginalDefn; \n\t\t\t If[QuickGrundy[#] == 0, Throw[1]])\)&, \n\t\t\tp]; \n\t\t Throw[0]\n\t\t]]\)], "Input"], Cell[TextData[{ "Finally, ", StyleBox["Compute5by5", "Input"], " uses each of the six representative first moves for the ", Cell[BoxData[ \(TraditionalForm\`5\[Times]5\)]], " board, and computes their respective Grundy numbers using ", StyleBox["BreakUpQuickGrundy", "Input"], "." }], "Text"], Cell[BoxData[ \(Compute5by5[] := Map[BreakUpQuickGrundy, firstmoves]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Compute5by5[]\)], "Input"], Cell[BoxData[ \({1, 1, 1, 1, 1, 1}\)], "Output"] }, Open ]], Cell[TextData[{ "It works. All of the six representative first moves have Grundy number 1 \ (which means ", StyleBox["all", FontSlant->"Italic"], " first moves have Grundy number 1) and thus the ", Cell[BoxData[ \(TraditionalForm\`5\[Times]5\)]], " board has Grundy number 0: hence it is a win for the second player. The \ computation takes under 2 hours on a PowerMac clone (PowerCenter Pro 210)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Final Remarks", "Section", FontSize->16], Cell[TextData[{ "Other \[OpenCurlyDoubleQuote]odd-by-odd\[CloseCurlyDoubleQuote] cases can \ also be investigated with ", StyleBox["Mathematica", FontSlant->"Italic"], " for the game of Cram. The ", Cell[BoxData[ \(TraditionalForm\`3\[Times]5\)]], " and ", Cell[BoxData[ \(TraditionalForm\`3\[Times]7\)]], " boards can be computed without difficulty and are wins for the first \ player; we have not tried the ", Cell[BoxData[ \(TraditionalForm\`3\[Times]9\)]], " and ", Cell[BoxData[ \(TraditionalForm\`3\[Times]11\)]], " boards, but they should be readily computable as well. The ", Cell[BoxData[ \(TraditionalForm\`5\[Times]7\)]], " should prove more difficult. We feel the ", Cell[BoxData[ \(TraditionalForm\`7\[Times]7\)]], " will need some further breakthrough before it can be done, even on a much \ faster computer. Perhaps sharing the computation among several computers \ (distributive processing) will be necessary. As we pointed out earlier, much \ can be learned from these games\[LongDash]not the least of which is learning \ which ", StyleBox["Mathematica", FontSlant->"Italic"], " programming techniques are most effective in carrying out these \ calculations." }], "Text"], Cell[TextData[{ "Furthermore, because the Grundy algorithm applies to any impartial \ two-player game, we can play variations of Cram with very little change to \ the original code. 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After all, playing games is not only instructive, it\ \[CloseCurlyQuote]s fun!" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "[1] E.\[NonBreakingSpace]Berlekamp, J.\[NonBreakingSpace]Conway, R.\ \[NonBreakingSpace]Guy, ", StyleBox["Winning Ways for Your Mathematical Plays", FontSlant->"Italic"], ", Academic Press, London, 1985." }], "Text"], Cell[TextData[{ "[2] M. Gardner, \[OpenCurlyDoubleQuote]Mathematical Games\ \[CloseCurlyDoubleQuote], ", StyleBox["Scientific American", FontSlant->"Italic"], ", February 1974, pp.\[NonBreakingSpace]106\[Dash]108." }], "Text"], Cell[TextData[{ "[3] R.\[NonBreakingSpace]McEliece, R.\[NonBreakingSpace]Ash, C.\ \[NonBreakingSpace]Ash, ", StyleBox["Introduction to Discrete Mathematics", FontSlant->"Italic"], ", Random House, New York, 1989." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["About the Authors", "Section"], Cell[TextData[{ "Robert Cowen teaches math at Queens College of The City University of New \ York where he gives a course in mathematical algorithms that meets in the \ Math Department\[CloseCurlyQuote]s ", StyleBox["Mathematica", FontSlant->"Italic"], " lab. It was for this course that he started programming games in ", StyleBox["Mathematica", FontSlant->"Italic"], ". When not playing games, he does research in logic, set theory, and graph \ theory." }], "Text"], Cell["\tcowen@qcvaxa.acc.qc.edu", "Text", FontFamily->"Courier New"], Cell["\<\ Robert Dickau is a former member of technical staff of Wolfram Research, and \ is currently senior technical trainer at InstallShield Software Corporation \ in Schaumburg, Illinois. He holds degrees in mathematics and actuarial \ science from California Polytechnic University, Pomona, and University of \ Illinois at Champaign-Urbana. His mathematical interests include fractals and \ combinatorial enumeration.\ \>", "Text"], Cell[TextData[StyleBox["\tpops@prairienet.org", FontFamily->"Courier New"]], "Text", FontFamily->"Courier New"] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{512, 438}, WindowMargins->{{83, Automatic}, {Automatic, 63}}, Magnification->1.25 ] (*********************************************************************** Cached data follows. 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