Statistics of the Number of Rounds for Reaching an integer with property, "n -> IsDP(n,3)", 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,3)" 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=, 116 [Ave,SD,ThirdScaledMoment,Etc]= , [24.698287233745604388, 8.4587330154859495477, 2.0682354230009892701, 7.4183711141066600758] --------------------- Number of faces=, 3, Number of rounds=, 119 [Ave,SD,ThirdScaledMoment,Etc]= , [21.901700445479070042, 9.4083475643194419867, 1.5048408994550812801, 5.3764964839073292159] --------------------- Number of faces=, 4, Number of rounds=, 119 [Ave,SD,ThirdScaledMoment,Etc]= , [20.269167408080197223, 9.9273547831752105726, 1.3544820666304623286, 4.9909010532068063976] --------------------- Number of faces=, 5, Number of rounds=, 122 [Ave,SD,ThirdScaledMoment,Etc]= , [18.760973989873190675, 9.6447604986914498451, 1.3097945684034885793, 5.1232318855533095600] --------------------- Number of faces=, 6, Number of rounds=, 117 [Ave,SD,ThirdScaledMoment,Etc]= , [17.616887668092340735, 9.3260517770208134777, 1.2566850326127203801, 5.0405163509552011060] --------------------- Number of faces=, 7, Number of rounds=, 111 [Ave,SD,ThirdScaledMoment,Etc]= , [16.674219028146247040, 9.0500840703190467451, 1.2418686260999851859, 5.0692371751267422689] --------------------- Number of faces=, 8, Number of rounds=, 113 [Ave,SD,ThirdScaledMoment,Etc]= , [16.080049989532709736, 9.1036187097463752620, 1.2740719179562951879, 5.2165447170024369155] --------------------- Number of faces=, 9, Number of rounds=, 116 [Ave,SD,ThirdScaledMoment,Etc]= , [15.434941677016227530, 8.9625568456030982030, 1.3079940390217612781, 5.3905177983272443350] --------------------- Number of faces=, 10, Number of rounds=, 112 [Ave,SD,ThirdScaledMoment,Etc]= , [14.838344167382847319, 8.7139742830360702964, 1.3017351314049092855, 5.3913445151455475772] --------------------- Number of faces=, 11, Number of rounds=, 108 [Ave,SD,ThirdScaledMoment,Etc]= , [14.327377438326924472, 8.5462296240214527798, 1.3110655665399549837, 5.4546305131991564270] --------------------- Number of faces=, 12, Number of rounds=, 108 [Ave,SD,ThirdScaledMoment,Etc]= , [14.153539097464377213, 8.5250693036610289870, 1.3015225696081365638, 5.4590101444899949742] --------------------- Number of faces=, 13, Number of rounds=, 108 [Ave,SD,ThirdScaledMoment,Etc]= , [13.738937618496698643, 8.3695640339766352767, 1.3225911997722039180, 5.5623337094632690036] --------------------- Number of faces=, 14, Number of rounds=, 107 [Ave,SD,ThirdScaledMoment,Etc]= , [13.389172657879405615, 8.2162672558200132868, 1.3324401953340882554, 5.6091631049206772251] --------------------- Number of faces=, 15, Number of rounds=, 106 [Ave,SD,ThirdScaledMoment,Etc]= , [13.036217328516446046, 8.1068010978124471516, 1.3451671117016813192, 5.6798725304678866415] --------------------- Number of faces=, 16, Number of rounds=, 107 [Ave,SD,ThirdScaledMoment,Etc]= , [12.780715901670173169, 8.1229866475686013385, 1.3633230598935133647, 5.7165496150099895831] --------------------- Number of faces=, 17, Number of rounds=, 106 [Ave,SD,ThirdScaledMoment,Etc]= , [12.513847765721866743, 8.0319696337873868016, 1.3763705628668398724, 5.7784531175981213938] --------------------- Number of faces=, 18, Number of rounds=, 105 [Ave,SD,ThirdScaledMoment,Etc]= , [12.283265448192502948, 7.9255042775587431948, 1.3821628900474483540, 5.8048260761686484246] --------------------- Number of faces=, 19, Number of rounds=, 103 [Ave,SD,ThirdScaledMoment,Etc]= , [12.080623274672759580, 7.8295626723724033883, 1.3949339164904039083, 5.8703668347144841630] --------------------- Number of faces=, 20, Number of rounds=, 103 [Ave,SD,ThirdScaledMoment,Etc]= , [11.946791072358925453, 7.8060932910576316555, 1.4115974826484674797, 5.9277251415893483504] --------------------- Number of faces=, 21, Number of rounds=, 102 [Ave,SD,ThirdScaledMoment,Etc]= , [11.772616052526099579, 7.7400029334044001466, 1.4167172987297954627, 5.9369435970957026162] --------------------- Number of faces=, 22, Number of rounds=, 101 [Ave,SD,ThirdScaledMoment,Etc]= , [11.583196511406778573, 7.6491887571817942993, 1.4171146332674403327, 5.9344495530019041711] --------------------- Number of faces=, 23, Number of rounds=, 101 [Ave,SD,ThirdScaledMoment,Etc]= , [11.416049319369706213, 7.5709976604727479112, 1.4252981689037511710, 5.9778457399200888182] --------------------- Number of faces=, 24, Number of rounds=, 101 [Ave,SD,ThirdScaledMoment,Etc]= , [11.347073339197820093, 7.5605672922543132795, 1.4317269530614536940, 6.0102733545323831676] --------------------- Number of faces=, 25, Number of rounds=, 101 [Ave,SD,ThirdScaledMoment,Etc]= , [11.236646544010547444, 7.5303881791214288072, 1.4386284660271688753, 6.0262907972129375226] --------------------- Number of faces=, 26, Number of rounds=, 100 [Ave,SD,ThirdScaledMoment,Etc]= , [11.095624775635875384, 7.4494237418098028866, 1.4429964070944982465, 6.0416041239783247879] --------------------- Number of faces=, 27, Number of rounds=, 100 [Ave,SD,ThirdScaledMoment,Etc]= , [11.008338935116088731, 7.4266210302843765370, 1.4511527430341407231, 6.0663677107491088764] --------------------- Number of faces=, 28, Number of rounds=, 100 [Ave,SD,ThirdScaledMoment,Etc]= , [10.952870501279541022, 7.4174335870014355202, 1.4606187366264510426, 6.1086871743125208337] --------------------- Number of faces=, 29, Number of rounds=, 99 [Ave,SD,ThirdScaledMoment,Etc]= , [10.832138306634991976, 7.3522536468268766720, 1.4717492515415350429, 6.1616203842708350737] --------------------- Number of faces=, 30, Number of rounds=, 98 [Ave,SD,ThirdScaledMoment,Etc]= , [10.432948926747325723, 7.3870229648926666647, 1.4303687082471250592, 6.0219654859489437565] --------------------- Number of faces=, 31, Number of rounds=, 97 [Ave,SD,ThirdScaledMoment,Etc]= , [10.361786096740690828, 7.3260501365218506625, 1.4412477948738075953, 6.0697103633911501445] --------------------- Number of faces=, 32, Number of rounds=, 97 [Ave,SD,ThirdScaledMoment,Etc]= , [10.344955462867106720, 7.3144976196910468569, 1.4509622415122522174, 6.1133393323030720981] --------------------- Number of faces=, 33, Number of rounds=, 97 [Ave,SD,ThirdScaledMoment,Etc]= , [10.282532962798401440, 7.2696083909804115046, 1.4587182704061309426, 6.1442272638307966088] --------------------- Number of faces=, 34, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [10.200294522901590569, 7.2054887232846617293, 1.4625866496375410897, 6.1582504084466918809] --------------------- Number of faces=, 35, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [10.152581009319083699, 7.1847135368601937141, 1.4669772828868358683, 6.1662841087674919797] --------------------- Number of faces=, 36, Number of rounds=, 97 [Ave,SD,ThirdScaledMoment,Etc]= , [10.163536929234736620, 7.2024492286281877331, 1.4731802597828907059, 6.1959260369272778956] --------------------- Number of faces=, 37, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [10.082457951170420824, 7.1532322300983085580, 1.4808341155730401687, 6.2299175307767618092] --------------------- Number of faces=, 38, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [10.000906767351029677, 7.0943407664900171315, 1.4842123513103986430, 6.2445480034679101804] --------------------- Number of faces=, 39, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [9.9354359076972314094, 7.0562043696372630651, 1.4902836806587830991, 6.2698907868425330506] --------------------- Number of faces=, 40, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [9.9039168381808373899, 7.0454928153129493969, 1.4964051981626302788, 6.3024632390068437148] --------------------- This ends this paper that took, 332.781, seconds to produce.