Statistical Analysis of a certain Chutes-and-Ladders Game By Shalosh B. Ekhad Consider the following game with a 1-dimensional board with, 100, squares numbered from 1 to, 100 You start at square 1 and proceed by rolling a die (or spinner) with the fol\ lowing possible outcomes. It lands on, 1, with probability , 1/6 It lands on, 2, with probability , 1/6 It lands on, 3, with probability , 1/6 It lands on, 4, with probability , 1/6 It lands on, 5, with probability , 1/6 It lands on, 6, with probability , 1/6 At each turn, you roll the die (or spin the spinner), and advance forward th\ e number of squares indicated. Your goal is to reach the last square, square, 100, it does not matter if in\ the last round you went beyond it, either way the game ended. However there are a certain number of Chutes (snakes) and Ladders If your new position is at the bottom of a ladder, you immediately move to t\ he top, and if it landed at the top of the chute (snake), your slide down to its bottom The Chutes (Snakes) are as follows From , 16, down to , 5 From , 50, down to , 8 From , 63, down to , 20 From , 57, down to , 25 From , 98, down to , 46 From , 98, down to , 76 From , 95, down to , 90 The Ladders are as follows From , 2, up to , 44 From , 6, up to , 13 From , 9, up to , 31 From , 28, up to , 84 From , 59, up to , 61 From , 67, up to , 93 From , 70, up to , 73 From , 79, up to , 100 The probabilty generating function for the random variable, "number of turns\ to get to the end" is 7 76 75 74 73 72 - t (7 t + 3359 t + 790569 t + 121865337 t + 13851178200 t 71 70 69 + 1238470637112 t + 90737333711640 t + 5601948512747184 t 68 67 + 297432798366624480 t + 13792988834956880640 t 66 65 + 565504052931395058816 t + 20701254794058416325888 t 64 63 + 682148174796464941094400 t + 20373476142406180973626368 t 62 61 + 554768981940324846954332160 t + 13843276983143523283048722432 t 60 59 + 317974058896122036598479888384 t + 6749800298270638132181036728320 t 58 + 132881765379280297317636696834048 t 57 + 2433759146682397564563254064513024 t 56 + 41585571248150125274455478505701376 t 55 + 664572789822708340161362739982761984 t 54 + 9954897957143779694560143635191431168 t 53 + 140046085310514129163150626573105758208 t 52 + 1853459442506757421438506702955408785408 t 51 + 23110479048188318951397993863396431233024 t 50 + 271823076113704907475355092170979076472832 t 49 + 3018967261582495811540661193248254623481856 t 48 + 31686672635560498143445363558901387762860032 t 47 + 314487264734215935125303160156656225237336064 t 46 + 2952651950018419516731504730552177160066433024 t 45 + 26229678694613631098301730322826417667760455680 t 44 + 220470031172573800965778484668198496332544999424 t 43 + 1753086173082855120628740861776265441009444323328 t 42 + 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3742042951225759540014535187298779136 t + 22452257707354557240087211123792674816)) and in Maple notation it is -1/39182082048*t^7*(7*t^76+3359*t^75+790569*t^74+121865337*t^73+13851178200*t^ 72+1238470637112*t^71+90737333711640*t^70+5601948512747184*t^69+ 297432798366624480*t^68+13792988834956880640*t^67+565504052931395058816*t^66+ 20701254794058416325888*t^65+682148174796464941094400*t^64+ 20373476142406180973626368*t^63+554768981940324846954332160*t^62+ 13843276983143523283048722432*t^61+317974058896122036598479888384*t^60+ 6749800298270638132181036728320*t^59+132881765379280297317636696834048*t^58+ 2433759146682397564563254064513024*t^57+41585571248150125274455478505701376*t^ 56+664572789822708340161362739982761984*t^55+ 9954897957143779694560143635191431168*t^54+ 140046085310514129163150626573105758208*t^53+ 1853459442506757421438506702955408785408*t^52+ 23110479048188318951397993863396431233024*t^51+ 271823076113704907475355092170979076472832*t^50+ 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27505449835430375825092379853900931637146012492459605214887936*t^4+ 25207048467629160380102803947621884335417281125444155934244864*t^3+ 17438217969583684872544335094274508638935263635929311058329600*t^2+ 7314237313364434797055109171457129526030280242408456124891136*t+ 1143238558994257775583689010435896407131843776598467942547456)/(t^20+105*t^19+ 5220*t^18+163404*t^17+3613248*t^16+59980176*t^15+774652896*t^14+7949902464*t^13 +65540295936*t^12+435064214016*t^11+2310170720256*t^10+9638792183808*t^9+ 30521753524224*t^8+68895160934400*t^7+98817210925056*t^6+71703210147840*t^5+ 19747769352192*t^4+50779978334208*t^3+203119913336832*t^2+304679870005248*t-\ 1828079220031488)/(t^48+240*t^47+28332*t^46+2186568*t^45+123615072*t^44+ 5443122240*t^43+194327465472*t^42+5804751216384*t^41+149282415783936*t^40+ 3398244360210432*t^39+70230855799397376*t^38+1342457422075054080*t^37+ 23919771781118103552*t^36+396146117339038457856*t^35+6043206267730834587648*t^ 34+84070164896360739864576*t^33+1058124662122160717365248*t^32+ 11987995308247387185610752*t^31+121927269651307258128629760*t^30+ 1112049013708509333255880704*t^29+9093221101123050594471247872*t^28+ 66665101981642742219705155584*t^27+438096709133589649922633760768*t^26+ 2578184045301117524692105691136*t^25+13557786333158799768420712710144*t^24+ 63463977152715476392127672352768*t^23+262847598342859256266858954752000*t^22+ 954521572389116638746566832685056*t^21+2998044924240819663815125921431552*t^20+ 7966673947394268394906225962123264*t^19+17188195301820880976401896296153088*t^ 18+27238488628860263125604976172204032*t^17+19964000214688162916848503411965952 *t^16-47386820884438409229962646889955328*t^15-\ 230504172751388667806513014448848896*t^14-540974912570354228691276114117525504* t^13-877752981061358843671985910492168192*t^12-\ 1075945117713470416265779489340915712*t^11-\ 1105375946743040161715017958931038208*t^10-\ 1085077406285805951533004671315607552*t^9-970587043476365843197642928667230208* t^8-665941842936117143791595628861063168*t^7-\ 394608438786667029682602733228523520*t^6-265638851475902683396093547246518272*t ^5-571701006437268818613331764726202368*t^4-\ 1871021475612879770007267593649389568*t^3-1247347650408586513338178395766259712 *t^2-3742042951225759540014535187298779136*t+ 22452257707354557240087211123792674816) The expected duration of the (solitaire) game is 29.45521780 The variance of the duration of the (solitaire) game is 259.8428632 The scaled , 3, -th moment about the mean is 1.589362184 The scaled , 4, -th moment about the mean is 7.155880106 The scaled , 5, -th moment about the mean is 30.87977994 The scaled , 6, -th moment about the mean is 170.5829208 Summarizing the, expectation, variance and the scaled moments up to the, 6, -th , are [29.45521780, 259.8428632, 1.589362184, 7.155880106, 30.87977994, 170.5829208] Let's compare it to simulating , 1000, random games , the corresponding list is [30.48100000, 312.1036390, 1.901616092, 8.101476074, 32.44846263, 147.0937690] [Of course this changes (hopefully only slightly, by the law of large number\ s) each time] So far the game was Solitaire. Suppose that there are two players, and they \ keep taken turns. Assuming that the game did not last beyond, 2000, moves The probability, that the first player is going to win is 0.5112590928 Just for fun, let's compare it to the result of simulating, 1000, such games, and see the fraction of times the first player won 0.5230000000 ------------------------- This ends this article, that took , 53.090, seconds to generate.