A table of the difference between the Expected Duration of the Dym-Luks Ball\ and Cell game with n cells and r balls and r / n \(j-1) ----- |-----| \ \n - 1/ ) -------- / j ----- j = 1 for n from 2 to, 10, and r from 1 to, 150 By Shalosh B. Ekhad Let's call the difference E_n(r) when there are, 2, cells, the first , 150, terms of the seqence E_n(r) are [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0. , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0\ ., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] when there are, 3, cells, the first , 150, terms of the seqence E_n(r) are [0., -.2500000000, -.2500000000, -.1937500000, -.1562500000, -.1163194444, -.\ 8304255610e-1, -.5554972888e-1, -.3347162452e-1, -.1603342521e-1, -.2419005672e\ -2, .8123277604e-2, .1623830806e-1, .2245744000e-1, .2720763762e-1, .3082639037\ e-1, .3357747934e-1, .3566543411e-1, .3724791452e-1, .3844591399e-1, .\ 3935196897e-1, .4003666024e-1, .4055370479e-1, .4094391216e-1, .4123824013e-1, .4146014392e-1, .4162737578e-1, .4175335960e-1, .4184823846e-1, .4191967130e-1, .4197343795e-1, .4201389795e-1, .4204433808e-1, .4206723533e-1, .4208445574e-1, .4209740469e-1, .4210714026e-1, .4211445889e-1, .4211995993e-1, .4212409430e-1, .4212720122e-1, .4212953578e-1, .4213128984e-1, .4213260762e-1, .4213359757e-1, .4213434118e-1, .4213489971e-1, .4213531921e-1, .4213563426e-1, .4213587086e-1, .4213604853e-1, .4213618194e-1, .4213628212e-1, .4213635733e-1, .4213641380e-1, .4213645620e-1, .4213648803e-1, .4213651192e-1, .4213652986e-1, .4213654333e-1, .4213655343e-1, .4213656102e-1, .4213656672e-1, .4213657099e-1, .4213657420e-1, .4213657661e-1, .4213657841e-1, .4213657977e-1, .4213658079e-1, .4213658155e-1, .4213658213e-1, .4213658256e-1, .4213658288e-1, .4213658312e-1, .4213658330e-1, .4213658344e-1, .4213658354e-1, .4213658362e-1, .4213658368e-1, .4213658372e-1, .4213658375e-1, .4213658378e-1, .4213658379e-1, .4213658381e-1, .4213658382e-1, .4213658383e-1, .4213658383e-1, .4213658384e-1, .4213658384e-1, .4213658384e-1, .4213658384e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1, .4213658385e-1] when there are, 4, cells, the first , 150, terms of the seqence E_n(r) are [0., -.3333333333, -.3925925926, -.3629629630, -.3313580247, -.2985436511, -.25\ 70083645, -.2167431911, -.1764986328, -.1380024878, -.1016263468, -.6768994504e\ -1, -.3630679553e-1, -.7480796787e-2, .1885805275e-1, .4282534758e-1, .\ 6456305531e-1, .8422634194e-1, .1019746733, .1179659209, .1323525714, .14527940\ 23, .1568821705, .1672869966, .1766102174, .1849585510, .1924294601, .199111638\ 7, .2050855663, .2104240930, .2151930299, .2194517286, .2232536376, .2266468290 , .2296744938, .2323754001, .2347843181, .2369324098, .2388475852, .2405548276, .2420764893, .2434325591, .2446409054, .2457174948, .2466765897, .2475309267, .\ 2482918761, .2489695865, .2495731134, .2501105355, .2505890585, .2510151085, .2\ 513944150, .2517320860, .2520326747, .2523002388, .2525383945, .2527503639, .25\ 29390178, .2531069133, .2532563286, .2533892925, .2535076120, .2536128963, .253\ 7065783, .2537899340, .2538640992, .2539300855, .2539887930, .2540410234, .2540\ 874900, .2541288279, .2541656024, .2541983164, .2542274176, .2542533046, .25427\ 63318, .2542968148, .2543150344, .2543312404, .2543456551, .2543584763, .254369\ 8799, .2543800226, .2543890436, .2543970669, .2544042027, .2544105491, .2544161\ 934, .2544212131, .2544256774, .2544296477, .2544331785, .2544363185, .25443911\ 09, .2544415942, .2544438026, .2544457664, .2544475128, .2544490658, .254450446\ 9, .2544516749, .2544527670, .2544537381, .2544546016, .2544553694, .2544560522 , .2544566594, .2544571992, .2544576793, .2544581062, .2544584857, .2544588232, .2544591233, .2544593901, .2544596274, .2544598383, .2544600259, .2544601927, .\ 2544603410, .2544604729, .2544605901, .2544606943, .2544607870, .2544608694, .2\ 544609427, .2544610078, .2544610658, .2544611173, .2544611631, .2544612038, .25\ 44612400, .2544612721, .2544613008, .2544613262, .2544613488, .2544613689, .254\ 4613868, .2544614027, .2544614168, .2544614294, .2544614406, .2544614505, .2544\ 614593, .2544614672, .2544614742, .2544614804, .2544614859, .2544614908, .25446\ 14951] when there are, 5, cells, the first , 150, terms of the seqence E_n(r) are [0., -.3750000000, -.4791666667, -.4763764881, -.4570745678, -.4394788789, -.41\ 26003026, -.3785812717, -.3437841737, -.3067182134, -.2691304464, -.2314285973, -.1941114801, -.1575058000, -.1218548734, -.8732953358e-1, -.5404617147e-1, -.\ 2207888692e-1, .8530799478e-2, .3776637420e-1, .6563037763e-1, .9213972387e-1, .1173220236, .1412126995, .1638527275, .1852868734, .2055623193, .2247275979, .\ 2428317703, .2599237942, .2760520442, .2912639499, .3056057289, .3191221931, .3\ 318566139, .3438506327, .3551442080, .3657755907, .3757813213, .3851962441, .39\ 40535347, .4023847375, .4102198102, .4175871743, .4245137686, .4310251057, .437\ 1453300, .4428972755, .4483025244, .4533814645, .4581533463, .4626363381, .4668\ 475798, .4708032353, .4745185431, .4780078643, .4812847296, .4843618840, .48725\ 13292, .4899643644, .4925116256, .4949031220, .4971482717, .4992559347, .501234\ 4449, .5030916403, .5048348910, .5064711268, .5080068623, .5094482211, .5108009\ 591, .5120704853, .5132618828, .5143799278, .5154291077, .5164136383, .51733747\ 99, .5182043525, .5190177502, .5197809545, .5204970475, .5211689233, .521799299\ 7, .5223907285, .5229456059, .5234661815, .5239545674, .5244127464, .5248425800 , .5252458154, .5256240931, .5259789525, .5263118388, .5266241083, .5269170341, .5271918109, .5274495602, .5276913341, .5279181202, .5281308453, .5283303790, .\ 5285175373, .5286930861, .5288577438, .5290121846, .5291570412, .5292929071, .5\ 294203392, .5295398600, .5296519597, .5297570983, .5298557072, .5299481913, .53\ 00349303, .5301162806, .5301925764, .5302641312, .5303312392, .5303941764, .530\ 4532016, .5305085575, .5305604721, .5306091589, .5306548184, .5306976386, .5307\ 377957, .5307754553, .5308107725, .5308438928, .5308749527, .5309040802, .53093\ 13955, .5309570112, .5309810329, .5310035597, .5310246845, .5310444946, .531063\ 0716, .5310804923, .5310968285, .5311121477, .5311265132, .5311399843, .5311526\ 167, .5311644624, .5311755705, .5311859869, .5311957545, .5312049139, .53121350\ 27] when there are, 6, cells, the first , 150, terms of the seqence E_n(r) are [0., -.4000000000, -.5371428571, -.5577142857, -.5494394492, -.5422099603, -.52\ 98246096, -.5072245463, -.4801906673, -.4514799024, -.4200312904, -.3871647703, -.3533543139, -.3189157887, -.2841991713, -.2494421280, -.2148473908, -.1805754\ 622, -.1467552390, -.1134891592, -.8085783690e-1, -.4892376642e-1, -.1773438493\ e-1, .1267538357e-1, .4228100412e-1, .7106655222e-1, .9902322262e-1, .126148078\ 6, .1524430027, .1779138174, .2025695476, .2264218032, .2494842619, .2717722365 , .2933023127, .3140920464, .3341597102, .3535240828, .3722042716, .3902195668, .4075893183, .4243328350, .4404693003, .4560177033, .4709967822, .4854249782, .\ 4993203981, .5127007854, .5255834963, .5379854825, .5499232782, .5614129902, .5\ 724702928, .5831104240, .5933481854, .6031979426, .6126736288, .6217887491, .63\ 05563852, .6389892029, .6470994584, .6548990067, .6623993099, .6696114458, .676\ 5461170, .6832136605, .6896240567, .6957869396, .7017116056, .7074070238, .7128\ 818452, .7181444123, .7232027684, .7280646671, .7327375810, .7372287112, .74154\ 49957, .7456931182, .7496795165, .7535103907, .7571917113, .7607292270, .764128\ 4722, .7673947748, .7705332630, .7735488725, .7764463534, .7792302764, .7819050\ 397, .7844748749, .7869438532, .7893158910, .7915947555, .7937840705, .79588732\ 14, .7979078601, .7998489104, .8017135725, .8035048272, .8052255412, .806878470\ 6, .8084662654, .8099914733, .8114565437, .8128638314, .8142156001, .8155140257 , .8167611999, .8179591332, .8191097583, .8202149324, .8212764409, .8222959996, .8232752573, .8242157988, .8251191472, .8259867659, .8268200615, .8276203853, .\ 8283890361, .8291272619, .8298362617, .8305171878, .8311711471, .8317992035, .8\ 324023788, .8329816552, .8335379759, .8340722473, .8345853404, .8350780917, .83\ 55513050, .8360057524, .8364421757, .8368612875, .8372637725, .8376502882, .838\ 0214665, .8383779142, .8387202142, .8390489266, .8393645892, .8396677188, .8399\ 588118, .8402383448, .8405067758, .8407645448, .8410120741, .8412497696, .84147\ 80212] when there are, 7, cells, the first , 150, terms of the seqence E_n(r) are [0., -.4166666667, -.5787037037, -.6192129630, -.6209579772, -.6204958931, -.61\ 85256241, -.6073717920, -.5884433191, -.5668125514, -.5431005935, -.5166933662, -.4886035956, -.4593283830, -.4290223514, -.3979979683, -.3664632120, -.3345907\ 433, -.3025319646, -.2704122057, -.2383382747, -.2064005968, -.1746757656, -.14\ 32285094, -.1121133377, -.8137593188e-1, -.5105433020e-1, -.2117994590e-1, .\ 8221553124e-2, .3712948175e-1, .6552747483e-1, .9340291021e-1, .1207464063, .14\ 75513839, .1738136850, .1995312395, .2247037761, .2493325694, .2734202198, .296\ 9704624, .3199879993, .3424783545, .3644477472, .3859029808, .4068513466, .4273\ 005402, .4472585873, .4667337803, .4857346220, .5042697770, .5223480290, .53997\ 82436, .5571693360, .5739302426, .5902698962, .6061972047, .6217210325, .636850\ 1842, .6515933908, .6659592973, .6799564525, .6935933001, .7068781713, .7198192\ 781, .7324247081, .7447024201, .7566602404, .7683058602, .7796468326, .79069057\ 19, .8014443517, .8119153044, .8221104208, .8320365503, .8417004011, .851108540\ 5, .8602673964, .8691832575, .8778622754, .8863104652, .8945337076, .9025377502 , .9103282096, .9179105728, .9252901996, .9324723240, .9394620570, .9462643880, .9528841872, .9593262078, .9655950880, .9716953534, .9776314189, .9834075911, .\ 9890280703, .9944969529, .9998182334, 1.004995807, 1.010033469, 1.014934924, 1.\ 019703778, 1.024343549, 1.028857664, 1.033249465, 1.037522205, 1.041679058, 1.0\ 45723112, 1.049657381, 1.053484796, 1.057208216, 1.060830424, 1.064354131, 1.06\ 7781979, 1.071116539, 1.074360317, 1.077515752, 1.080585219, 1.083571034, 1.086\ 475447, 1.089300653, 1.092048787, 1.094721929, 1.097322103, 1.099851280, 1.1023\ 11379, 1.104704268, 1.107031766, 1.109295642, 1.111497619, 1.113639376, 1.11572\ 2544, 1.117748712, 1.119719427, 1.121636194, 1.123500479, 1.125313707, 1.127077\ 266, 1.128792506, 1.130460741, 1.132083251, 1.133661280, 1.135196040, 1.1366887\ 08, 1.138140432, 1.139552327, 1.140925480, 1.142260948, 1.143559757, 1.14482291\ 0, 1.146051378] when there are, 8, cells, the first , 150, terms of the seqence E_n(r) are [0., -.4285714286, -.6099773243, -.6675089083, -.6785728974, -.6830120460, -.68\ 78684519, -.6862483798, -.6755610244, -.6598709954, -.6422934587, -.6224842520, -.6001479710, -.5761351972, -.5509097654, -.5245421057, -.4972664139, -.4692880\ 224, -.4407382944, -.4117477477, -.3824275285, -.3528725546, -.3231664100, -.29\ 33819650, -.2635830749, -.2338257282, -.2041590196, -.1746259767, -.1452642650, -.1161067967, -.8718226211e-1, -.5851559546e-1, -.3012838634e-1, -.2039244068e-\ 2, .2573587836e-1, .5318339591e-1, .8029183425e-1, .1070516002, .1334547765, .1\ 594949389, .1851669922, .2104670243, .2353921751, .2599405202, .2841109663, .30\ 79031580, .3313173946, .3543545558, .3770160347, .3993036789, .4212197367, .442\ 7668100, .4639478114, .4847659259, .5052245770, .5253273960, .5450781942, .5644\ 809386, .5835397296, .6022587810, .6206424023, .6386949825, .6564209757, .67382\ 48883, .6909112667, .7076846876, .7241497479, .7403110563, .7561732261, .771740\ 8675, .7870185822, .8020109572, .8167225604, .8311579355, .8453215985, .8592180\ 338, .8728516913, .8862269832, .8993482816, .9122199165, .9248461734, .93723129\ 19, .9493794642, .9612948334, .9729814928, .9844434846, .9956847995, 1.00670937\ 5, 1.017521098, 1.028123798, 1.038521253, 1.048717189, 1.058715274, 1.068519124 , 1.078132301, 1.087558311, 1.096800609, 1.105862593, 1.114747610, 1.123458951, 1.131999858, 1.140373517, 1.148583063, 1.156631579, 1.164522098, 1.172257599, 1\ .179841015, 1.187275225, 1.194563062, 1.201707308, 1.208710698, 1.215575920, 1.\ 222305613, 1.228902372, 1.235368743, 1.241707231, 1.247920293, 1.254010343, 1.2\ 59979753, 1.265830850, 1.271565920, 1.277187209, 1.282696919, 1.288097213, 1.29\ 3390217, 1.298578013, 1.303662649, 1.308646133, 1.313530436, 1.318317493, 1.323\ 009202, 1.327607427, 1.332113995, 1.336530701, 1.340859304, 1.345101533, 1.3492\ 59082, 1.353333613, 1.357326758, 1.361240116, 1.365075257, 1.368833721, 1.37251\ 7020, 1.376126633, 1.379664016, 1.383130593, 1.386527762, 1.389856896, 1.393119\ 339, 1.396316411] when there are, 9, cells, the first , 150, terms of the seqence E_n(r) are [0., -.4375000000, -.6343750000, -.7065252131, -.7263183594, -.7348062939, -.74\ 40452183, -.7496180730, -.7466675575, -.7367321513, -.7237978216, -.7092742838, -.6924887504, -.6733766170, -.6526998589, -.6308687234, -.6079078414, -.5839707\ 459, -.5592618351, -.5338927449, -.5079624123, -.4815709317, -.4548001633, -.42\ 77225565, -.4004031938, -.3728996887, -.3452635493, -.3175408792, -.2897729935, -.2619969587, -.2342460438, -.2065501117, -.1789359598, -.1514276201, -.1240466\ 244, -.9681224032e-1, -.6974168367e-1, -.4285030841e-1, -.1615177858e-1, .\ 1034177641e-1, .3661962115e-1, .6267228655e-1, .8849145398e-1, .1140698501, .13\ 94011509, .1644798953, .1893014054, .2138617151, .2381575044, .2621860403, .285\ 9451223, .3094330337, .3326484966, .3555906317, .3782589207, .4006531732, .4227\ 734959, .4446202649, .4661941000, .4874958423, .5085265322, .5292873913, .54977\ 98041, .5700053022, .5899655504, .6096623323, .6290975393, .6482731584, .667191\ 2632, .6858540034, .7042635972, .7224223232, .7403325132, .7579965456, .7754168\ 396, .7925958494, .8095360594, .8262399795, .8427101405, .8589490907, .87495939\ 20, .8907436163, .9063043432, .9216441562, .9367656413, .9516713834, .966363965\ 0, .9808459639, .9951199512, 1.009188490, 1.023054134, 1.036719424, 1.050186890 , 1.063459049, 1.076538400, 1.089427430, 1.102128606, 1.114644380, 1.126977185, 1.139129435, 1.151103523, 1.162901824, 1.174526691, 1.185980457, 1.197265433, 1\ .208383908, 1.219338149, 1.230130401, 1.240762885, 1.251237801, 1.261557325, 1.\ 271723610, 1.281738786, 1.291604959, 1.301324213, 1.310898606, 1.320330176, 1.3\ 29620934, 1.338772870, 1.347787950, 1.356668116, 1.365415287, 1.374031359, 1.38\ 2518205, 1.390877675, 1.399111595, 1.407221770, 1.415209980, 1.423077985, 1.430\ 827520, 1.438460301, 1.445978018, 1.453382342, 1.460674922, 1.467857384, 1.4749\ 31333, 1.481898354, 1.488760009, 1.495517841, 1.502173372, 1.508728102, 1.51518\ 3513, 1.521541064, 1.527802199, 1.533968336, 1.540040879, 1.546021210, 1.551910\ 692, 1.557710671, 1.563422471] when there are, 10, cells, the first , 150, terms of the seqence E_n(r) are [0., -.4444444444, -.6539468762, -.7387454795, -.7667141734, -.7788988719, -.79\ 10278477, -.8017060706, -.8054467672, -.8012732294, -.7924332185, -.7817715968, -.7695726521, -.7551405230, -.7385895938, -.7206216886, -.7015928634, -.6814954\ 511, -.6604153939, -.6385455103, -.6160004670, -.5928487739, -.5691743456, -.54\ 50530072, -.5205457877, -.4957091257, -.4705942026, -.4452466687, -.4197079246, -.3940155227, -.3682035661, -.3423030956, -.3163423947, -.2903472573, -.2643412\ 206, -.2383457694, -.2123805166, -.1864633640, -.1606106463, -.1348372615, -.10\ 91567882, -.8358159272e-1, -.5812292634e-1, -.3279101468e-1, -.7595139215e-2, .\ 1745628783e-1, .4235565469e-1, .6709608684e-1, .9167138880e-1, .1160759906, .14\ 03048985, .1643536497, .1882182702, .2118952367, .2353814411, .2586741575, .281\ 7710124, .3046699572, .3273692421, .3498673932, .3721631906, .3942556483, .4161\ 439962, .4378276627, .4593062598, .4805795678, .5016475228, .5225102041, .54316\ 78230, .5636207123, .5838693168, .6039141844, .6237559576, .6433953664, .662833\ 2209, .6820704048, .7011078695, .7199466286, .7385877525, .7570323638, .7752816\ 327, .7933367731, .8111990384, .8288697182, .8463501352, .8636416414, .88074561\ 60, .8976634625, .9143966057, .9309464904, .9473145781, .9635023461, .979511284\ 6, .9953428958, 1.010998692, 1.026480193, 1.041788927, 1.056926427, 1.071894231 , 1.086693880, 1.101326918, 1.115794889, 1.130099338, 1.144241809, 1.158223846, 1.172046990, 1.185712777, 1.199222744, 1.212578419, 1.225781328, 1.238832992, 1\ .251734925, 1.264488635, 1.277095622, 1.289557382, 1.301875402, 1.314051159, 1.\ 326086125, 1.337981761, 1.349739523, 1.361360852, 1.372847186, 1.384199950, 1.3\ 95420559, 1.406510420, 1.417470928, 1.428303471, 1.439009423, 1.449590150, 1.46\ 0047007, 1.470381336, 1.480594472, 1.490687737, 1.500662442, 1.510519888, 1.520\ 261364, 1.529888149, 1.539401510, 1.548802703, 1.558092974, 1.567273556, 1.5763\ 45671, 1.585310532, 1.594169339, 1.602923281, 1.611573537, 1.620121272, 1.62856\ 7644, 1.636913797, 1.645160864] This ends this paper that took, 20107.881, seconds to generate .