Statistics of the Number of Rounds for Reaching an integer with property, "n -> IsDP(n,2)", rolling a fair die with n faces for n from 2 to, 40 By Shalsoh B. Ekhad Suppose you start at, 0, and roll a fair die with n faces, with 1 through n d\ ots, and walk forward according to the number of dots, and stop as soon \ as you hit an integer with property, "n -> IsDP(n,2)" You are allowed up to, 400, die-rolls, but you are interested in the smallest\ number of rolls that would guarantee that you would reach your goal wit\ h prob at least, 0.99999990000000000000 The table below tells you that number of rolls, followed by, conditioned on \ finishing by that number of rolls, the expected duration of such a game,\ followed by the standard-deviation and the scaled moments up to the, 4 with Fair Dice with number of faces from 2 to, 40 Number of faces=, 2, Number of rounds=, 12 [Ave,SD,ThirdScaledMoment,Etc]= , [5.2338867187500000000, 1.9068861978477788539, 1.0121851413917371646, 3.1721338580638416791] --------------------- Number of faces=, 3, Number of rounds=, 22 [Ave,SD,ThirdScaledMoment,Etc]= , [4.5776909279538779182, 2.0704121733646061249, 1.3008542759159493547, 5.7847408735709106520] --------------------- Number of faces=, 4, Number of rounds=, 36 [Ave,SD,ThirdScaledMoment,Etc]= , [4.6140243587780702211, 2.4197799411444355590, 1.5101974964660311828, 6.5290894412012876379] --------------------- Number of faces=, 5, Number of rounds=, 39 [Ave,SD,ThirdScaledMoment,Etc]= , [4.4091532322889596501, 2.4919010078694252834, 1.7429189643763109352, 7.7053859622887276153] --------------------- Number of faces=, 6, Number of rounds=, 38 [Ave,SD,ThirdScaledMoment,Etc]= , [3.7889180372935410569, 2.5739136938282850950, 1.6179645617510462604, 7.1829742959043674369] --------------------- Number of faces=, 7, Number of rounds=, 39 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8658193771235218454, 2.6263423641444356096, 1.6895010282409483123, 7.4986786676223303272] --------------------- Number of faces=, 8, Number of rounds=, 42 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8829800542724138052, 2.6170604978137128517, 1.7112297362181246042, 7.6843971818806147509] --------------------- Number of faces=, 9, Number of rounds=, 42 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8797306490203022309, 2.5792763716988456338, 1.7717709600924457707, 8.0850619942082399028] --------------------- Number of faces=, 10, Number of rounds=, 40 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5457217074218403579, 2.5365667587748450060, 1.7935622909889370604, 8.2142625171027275882] --------------------- Number of faces=, 11, Number of rounds=, 41 [Ave,SD,ThirdScaledMoment,Etc]= , [3.6229006864975795237, 2.5867643820501739656, 1.8219095298326713692, 8.2908946837255797357] --------------------- Number of faces=, 12, Number of rounds=, 43 [Ave,SD,ThirdScaledMoment,Etc]= , [3.7266423656801871126, 2.6905750945647852990, 1.8318611218821876988, 8.1699400539325726363] --------------------- Number of faces=, 13, Number of rounds=, 43 [Ave,SD,ThirdScaledMoment,Etc]= , [3.7448325225484955531, 2.7053494384450241268, 1.8450587160993395640, 8.1721734270802270826] --------------------- Number of faces=, 14, Number of rounds=, 43 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5115492698720402196, 2.6590893452080935073, 1.8805784929701354427, 8.3666600719548588580] --------------------- Number of faces=, 15, Number of rounds=, 43 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3308358962315302392, 2.6039690190978793308, 1.9087798340992788738, 8.5535997393504499808] --------------------- Number of faces=, 16, Number of rounds=, 44 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4393437998115955394, 2.6509523603817129318, 1.8761888057038841076, 8.4333689142049894055] --------------------- Number of faces=, 17, Number of rounds=, 44 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4841146815868846003, 2.6702829471087248980, 1.8921559902417271614, 8.5195013897302164338] --------------------- Number of faces=, 18, Number of rounds=, 44 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5121839382657612636, 2.6738556914113208668, 1.9015379484460193112, 8.5842087175823710410] --------------------- Number of faces=, 19, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5463431800266693816, 2.6844090383753742700, 1.9069542320937349771, 8.6080440060856655981] --------------------- Number of faces=, 20, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5795498373939516326, 2.7026027999508189013, 1.9153001202661873674, 8.6434888446988882944] --------------------- Number of faces=, 21, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4383435805706763373, 2.6748288864125434536, 1.9481304032472628420, 8.8074154814488976416] --------------------- Number of faces=, 22, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3181422164047594298, 2.6564247899252652581, 1.9865797024768566615, 8.9796240519576479089] --------------------- Number of faces=, 23, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3747611898917918097, 2.6791858761296255803, 1.9671241396709375010, 8.8890296611378566146] --------------------- Number of faces=, 24, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4447367363714605464, 2.7187416220396070705, 1.9495579806738816777, 8.7988658627982153688] --------------------- Number of faces=, 25, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4922824975594537762, 2.7316293646261644077, 1.9348017491128887476, 8.7384225816328650071] --------------------- Number of faces=, 26, Number of rounds=, 45 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3804312132682257246, 2.7095029099875309774, 1.9760385383581376612, 8.9336805249248092208] --------------------- Number of faces=, 27, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4113515856862999133, 2.7103523375774048470, 1.9717449515458767289, 8.9355847335537545772] --------------------- Number of faces=, 28, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4458033942548755385, 2.7245343214431544958, 1.9762756913549527813, 8.9741491331068956357] --------------------- Number of faces=, 29, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4805058172798642439, 2.7359186857149178258, 1.9599000970128184801, 8.8591540195278939186] --------------------- Number of faces=, 30, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4955994245112247613, 2.7343446894062097350, 1.9641793245655660007, 8.8964033525644192520] --------------------- Number of faces=, 31, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5179355447279696730, 2.7380769862045305062, 1.9583107769846397716, 8.8754235971582567514] --------------------- Number of faces=, 32, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5368165081762767683, 2.7452399993762468159, 1.9764875886000220027, 9.0145629447478446830] --------------------- Number of faces=, 33, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4482205322969352646, 2.7267702775209142119, 1.9995551943609346410, 9.1303143816806038117] --------------------- Number of faces=, 34, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3691354942491224019, 2.7117088891896871746, 2.0255251519932161602, 9.2767425131915352692] --------------------- Number of faces=, 35, Number of rounds=, 46 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3008516499021353371, 2.7023041869833715646, 2.0493961216672296151, 9.3887150918365334259] --------------------- Number of faces=, 36, Number of rounds=, 47 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3548028759306572318, 2.7374522616371172443, 2.0359915524960886082, 9.3366212143703213692] --------------------- Number of faces=, 37, Number of rounds=, 47 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3802525897993289814, 2.7443740547026369168, 2.0294628490619344484, 9.2892007208118904265] --------------------- Number of faces=, 38, Number of rounds=, 47 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3101289003261002887, 2.7275223692986620217, 2.0534831233323182522, 9.4151625699606611345] --------------------- Number of faces=, 39, Number of rounds=, 47 [Ave,SD,ThirdScaledMoment,Etc]= , [3.2535231478831835725, 2.7179132874271213272, 2.0727946511466951227, 9.5111512849840675605] --------------------- Number of faces=, 40, Number of rounds=, 47 [Ave,SD,ThirdScaledMoment,Etc]= , [3.2907213334961597313, 2.7339936740980192815, 2.0622593889407368877, 9.4718042274603672281] --------------------- This ends this paper that took, 66.407, seconds to produce.