(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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[ 106106, 2707]*) (*NotebookOutlinePosition[ 106749, 2729]*) (* CellTagsIndexPosition[ 106705, 2725]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Chutes & Ladders", "Title"], Cell["\<\ The game is played on a board with 100 squares, numbered 1 through 100. \ Players take turns rolling a die and moving forward the resulting number of \ squares. Players begin off the board, so that an initial roll of 2 lands the \ player on the square labeled \"2\". Some pairs of squares are connected by \ ladders: when you land on the lower square you must advance to the higher \ square. Others are connected by chutes: when you land on the higher square \ you must retreat to the lower square. If you move beyond the end (if you are \ at 99 and roll a 6) then you move to the end. While there are 101 places (100 squares and \"off the board\"), you cannot \ end your turn at the base of a ladder or at the top of a chute. Thus, play \ happens on a Markov chain with 82 states. There is one sink (100) and four \ sources (off, 2, 3, 4, 5).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Squares = Range[0, 100];\)\), "\[IndentingNewLine]", \(\(Chutes = {16 \[Rule] 6, 48 \[Rule] 26, 49 \[Rule] 11, 56 \[Rule] 53, 62 \[Rule] 19, 64 \[Rule] 60, 87 \[Rule] 24, 93 \[Rule] 73, 95 \[Rule] 75, 98 \[Rule] 78};\)\), "\[IndentingNewLine]", \(\(Ladders = {1 \[Rule] 38, 4 \[Rule] 14, 9 \[Rule] 31, 21 \[Rule] 42, 28 \[Rule] 84, 36 \[Rule] 44, 51 \[Rule] 67, 80 \[Rule] 100, 71 \[Rule] 91};\)\), "\[IndentingNewLine]", \(\(Enders = {101 \[Rule] 100, 102 \[Rule] 100, 103 \[Rule] 100, 104 \[Rule] 100, 105 \[Rule] 100, 106 \[Rule] 100};\)\), "\[IndentingNewLine]", \(\(Terrain = Join[Chutes, Ladders, Enders];\)\), "\[IndentingNewLine]", \(States = Union[Squares /. 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o`030000003oool0oooo00/0oooo0`00003o0?oooj`0oooo000S0?ooo`030000003oool0oooo00`0 oooo00<000000?ooo`3oool0o`3ooon[0?ooo`008`3oool00`000000oooo0?ooo`3o0?oookX0oooo 000S0?ooo`030000003oool0oooo0140oooo00<000000?ooo`3oool0o`3ooonV0?ooo`008`3oool0 0`000000oooo0?ooo`0@0?ooo`<00000o`3ooonW0?ooo`008`3oool00`000000oooo0?ooo`0A0?oo o`030000003oool0oooo0?l0ooooYP3oool002<0oooo0`00003o0?oookX0oooo000S0?ooo`030000 003oool0oooo00`0oooo00D000000?ooo`3oool0oooo0000003o0?ooojT0oooo000S0?ooo`030000 003oool0oooo00/0oooo0`000000103oool0000000000000003o0?ooojP0oooo000S0?ooo`030000 003oool0oooo00`0oooo00D000000?ooo`3oool0oooo0000003o0?ooojT0oooo000S0?ooo`030000 003oool0oooo00d0oooo0`00003o0?ooojX0oooo000S0?ooo`030000003oool0oooo00`0oooo1@00 003o0?ooojT0oooo000S0?ooo`030000003oool0oooo00d0oooo0`00003o0?ooojX0oooo000S0?oo o`030000003oool0oooo0?l0oooo^P3oool002<0oooo00<000000?ooo`3oool0o`3ooonj0?ooo`00 8`3oool00`000000oooo0?ooo`3o0?oookX0oooo000S0?ooo`<00000o`3ooonj0?ooo`008`3oool0 0`000000oooo0?ooo`3o0?oookX0oooo000S0?ooo`030000003oool0oooo0?l0oooo^P3oool00?l0 ooooh03oool00?l0ooooh03oool00?l0ooooh03oool00001\ \>"], ImageRangeCache->{{{0, 478.438}, {295.313, 0}} -> {-41.275, -0.00190009, \ 1.15742, 0.000103111}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[N[ExactFinish[k]], {k, 15, 22}]\)], "Input"], Cell[BoxData[ \({0.02394950143963995`, 0.025097381684376748`, 0.02629435590703123`, 0.027100283380821245`, 0.027427514773673955`, 0.02747427038772581`, 0.02739781519541385`, 0.02717754129331068`}\)], "Output"] }, Open ]], Cell["\<\ The answer would seem to be 20. How about the average time to finish? This is \ touchy because we haven't computed the exact time to finish for all k. But \ here's a lower bound that's in the right ballpark.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[Sum[k*ExactFinish[k], {k, 1, 500}] + 501*\((1 - Sum[ExactFinish[k], {k, 1, 500}])\)]\)], "Input"], Cell[BoxData[ \(36.19307020160661`\)], "Output"] }, Open ]], Cell["\<\ This lower bound is based on the game ending in 501 moves or less. And what \ is the probability that the game takes at most 500 moves?\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sum[ExactFinish[k], {k, 1, 500}]\)], "Input"], Cell[BoxData[ \(595107187869493019973483154245712068762573722586088965267822807764291642\ 852114745423387300171599697203740529816195972938660044080545272172987372726784\ 543632278253648766210683253755298909715201431311276599513068007439649579742661\ 315172059032573963347980096744705406718049403749820632208605540749780355183082\ 570614377248866702028871642199124209550374244016338638357840131012452277324282\ 64829/595107188324818962688616487187155081693773040604900526287205908427604833\ 504717435294682686150200950855074910367713640480413016514200339681002702322652\ 069845433037712903501106210282764365797377406942688000507934996476421296710022\ 414402708501416579588473045758534687076892348584957872736709857877527291792734\ 584875623884208056979300933166299296156124881551312404618685547546875871363994\ 74688\)], "Output"] }, Open ]], Cell["\<\ I acknowledge that we haven't computed the expected length of a game, because \ we are just considering one player. If playing with several players (and who \ would play with themselves), the game will typically be much shorter, but how \ much shorter?\ \>", "Text"], Cell[TextData[{ "My daughter and I play cooperative Chutes & Ladders sometimes: we have all \ four pieces on the board, and after rolling the die, the roller decides which \ piece to move. When some piece gets to the end, then \"we win!\" \ Unfortunately, my daughter has realized that I won't put her to bed if we're \ in the middle of a game, so she tries to keep the pieces ", StyleBox["away", FontSlant->"Italic"], " from 100. It takes awhile, but eventually it seems that the dice win out \ and some piece will get to 100. Is this a theorem?" }], "Text"], Cell["\<\ Of course, Chutes & Ladders is not as interesting as Markov chains in \ general. Have Markov games been studied?\ \>", "Text"] }, Open ]] }, Open ]], Cell["Blackjack", "Subtitle"], Cell[TextData[{ "In blackjack, the dealer has no choices. Number-cards count as the shown \ number, face cards count as 10, and aces count as either 1 or 11. The dealer \ must take a card if the sum of her cards is less than 16, and must stop if \ the sum of her cards is 17, 18, 19, 20, or 21. More than 21, and dealer has \ \"busted\": all players who haven't busted win. Let's call a total that \ includes an ace counting as 11 soft, and other totals hard. The dealer's hand \ has a number of states:\n\thard: \ 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21\n\tsoft: \ 11,12,13,14,15,16,17,18,19,20,21\n\tadditional: busted.\nThat's a total of 32 \ states, and if we assume (incorrectly) that the likelihood of the next card \ being a 2 is always ", Cell[BoxData[ \(TraditionalForm\`1\/13\)]], ", being a face card is always ", Cell[BoxData[ \(TraditionalForm\`3\/13\)]], ", and so on, then we can compute the probability of the dealer starting \ with an ace and ending up busting by taking powers of a 32\[Cross]32 matrix. \ Although the constant probability assumption is not correct (if the dealer \ has a 2, then there are fewer 2's left in the deck so it is less likely to \ draw another 2 than to draw a 7), casinos frequently shuffle together 6 decks \ with the purpose of making this assumption very close to the truth." }], "Text"], Cell["\<\ If the dealer is showing an ace, and you have a total of 16, should you take \ another card?\ \>", "Text"], Cell["Monopoly", "Subtitle"], Cell["\<\ A Monopoly board has 40 squares: although you can never be on \"Go To Jail\", \ you could be on \"Just Visiting\". But there is that peculiar doubles rule: \ if you roll three doubles in a row, then you go to jail, and you get out of \ jail only if you roll doubles or pay $50 (which you must do after three \ tries). Suppose that your opponents stay in jail for as long as possible (as is good \ strategy after several of your properties have been developed). Are they more \ likely to land on Tennessee Avenue or on Illinois Avenue?\ \>", "Text"], Cell["\<\ This is a Markov chain: there are 120 states (40 squares and you can be on \ each after having rolled 0, 1, or 2 doubles). You also need to figure out \ what happens Chance and Community Chest (Go To Jail, Take a Walk on the \ Boardwalk, et cetera), but there is a fixed probability of moving from any \ particular state to any other particular state. You can answer the question \ above by taking a large power of a 120\[Cross]120 matrix. \ \>", "Text"] }, FrontEndVersion->"5.1 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowSize->{955, 909}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (******************************************************************* 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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