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Thread: World record formula for nxnxn cubes

  1. 04-28-2012 #11
    JK
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    Oh,that's interesting,how did you find that?
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  2. 04-28-2012 #12
    Odder
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    Without too much hack etc I got this:
    2.10(x-1)^1.892*floor(x/2)

    Kinda close for 1-5:
    1 | 0 (spot on yay! )
    2 | 2.1 (0.02 off)
    3 | 7.79149 (0.15 off)
    4 | 33.5709 (spot on)
    5 | 57.8558 (0.08 off)
    6 | 132.371 (11.94 off)
    7 | 186.897 (13.58 off in the other direction than 6x6x6 )
    I'm too lazy to contribute :/ Weekly Comp - Pyraminx Methods - UWR list - Megaminx LL img generator - Mega alg manipulator - Meffert's Speed Pyraminx (under pic 26.)
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  3. 04-28-2012 #13
    cmowla
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    I think this thread is kind of silly, but to play along, I just came up with a formula. I am not sure what the lowest averages for computer cubes are, so I not sure how values for n greater than 7 compare with human records (computer or by hand).

    The Potential time (in seconds) for world record averages of 5 for a cube of size n is:

    P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)}
    , where

    {f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n.

    {f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}}
    \text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}

    Which I first showed in this post.

    \left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor is power of ten of the rate of change for the number of positions for the nxnxn.

    So for n = 1 to 1000 (ignore the value for n = 1), if we use the actual derivative (rate of change formula), with Mathematica input:

    Spoiler:
    AccountingForm[(7.64/(3^2/Log[3]*
    Floor[Log[
    10, ((8! 3^7) (12! 2^10) (Exp[(Log[4008235231]/4 +
    Log[6075]/2 + (11/2) Log[2]) ((3 + 1) (3 - 3) +
    3 Cos[3 (Pi/2)]^2)] Log[
    24!^(4 (3 - 1))/4!^(24 (3 - 2))]))/(8 (24 12! 2^10)^
    Cos[3 (Pi/2)]^2 4!^((3/2) ((3 - 1)*(3 - 3) +
    Cos[3 (Pi/2)]^2)))]])) (Range[1000]^2/
    Log[Range[1000]])*
    Floor[Log[
    10, ((8! 3^7) (12! 2^10) (Exp[(Log[4008235231]/4 +
    Log[6075]/
    2 + (11/2) Log[
    2]) ((Range[1000] + 1) (Range[1000] - 3) +
    3 Cos[Range[1000] (Pi/2)]^2)] Log[
    24!^(4 (Range[1000] - 1))/
    4!^(24 (Range[1000] - 2))]))/(8 (24 12! 2^10)^
    Cos[Range[1000] (Pi/2)]^2 4!^((3/
    2) ((Range[1000] - 1)*(Range[1000] - 3) +
    Cos[Range[1000] (Pi/2)]^2)))]] // N]
    Yields (in seconds, but I included minutes, hours, days, and years)
    Code:
    n     Seconds      Minutes       Hours        Days          Years
    Code:
    2    2.05022    0.034170333    0.00057    2.37294E-05    6.5012E-08       
3    7.64    0.127333333    0.002122    8.84259E-05    2.42263E-07       
4    24.0901    0.401501667    0.006692    0.000278821    7.63892E-07       
5    52.427    0.873783333    0.014563    0.000606794    1.66245E-06       
6    105.288    1.7548    0.029247    0.001218611    3.33866E-06       
7    181.161    3.01935    0.050323    0.002096771    5.74458E-06       
8    299.332    4.988866667    0.083148    0.003464491    9.49176E-06       
9    456.763    7.612716667    0.126879    0.005286609    1.44839E-05       
10    678.896    11.31493333    0.188582    0.007857593    2.15277E-05       
11    956.883    15.94805    0.265801    0.011075035    3.03426E-05       
12    1325.37    22.0895    0.368158    0.015339931    4.20272E-05       
13    1773.2    29.55333333    0.492556    0.020523148    5.62278E-05       
14    2341.76    39.02933333    0.650489    0.027103704    7.42567E-05       
15    3010.87    50.18116667    0.836353    0.034848032    9.54741E-05       
16    3833.91    63.8985    1.064975    0.044373958    0.000121572       
17    4788.17    79.80283333    1.330047    0.055418634    0.000151832       
18    5933.95    98.89916667    1.648319    0.068679977    0.000188164       
19    7236.12    120.602    2.010033    0.083751389    0.000229456       
20    8770.03    146.1671667    2.436119    0.101504977    0.000278096       
21    10498.2    174.97    2.916167    0.121506944    0.000332896       
22    12502.8    208.38    3.473    0.144708333    0.000396461       
23    14730.2    245.5033333    4.091722    0.170488426    0.000467092       
24    17281    288.0166667    4.800278    0.200011574    0.000547977       
25    20099.9    334.9983333    5.583306    0.232637731    0.000637364       
26    23293.6    388.2266667    6.470444    0.269601852    0.000738635       
27    26786.9    446.4483333    7.440806    0.310033565    0.000849407       
28    30708.6    511.81    8.530167    0.355423611    0.000973763       
29    34982.6    583.0433333    9.717389    0.404891204    0.001109291       
30    39743    662.3833333    11.03972    0.459988426    0.001260242       
31    44889.9    748.165    12.46942    0.519559028    0.001423449       
32    50596.2    843.27    14.0545    0.585604167    0.001604395       
33    56736.8    945.6133333    15.76022    0.656675926    0.001799112       
34    63473.7    1057.895    17.63158    0.734649306    0.002012738       
35    70722.9    1178.715    19.64525    0.818552083    0.002242608       
36    78634.4    1310.573333    21.84289    0.91012037    0.00249348       
37    87097.2    1451.62    24.19367    1.008069444    0.002761834       
38    96307.7    1605.128333    26.75214    1.114672454    0.003053897       
39    106108    1768.466667    29.47444    1.228101852    0.003364663       
40    116747    1945.783333    32.42972    1.351238426    0.003702023       
41    128013    2133.55    35.55917    1.481631944    0.004059266       
42    140196    2336.6    38.94333    1.622638889    0.004445586       
43    153105    2551.75    42.52917    1.772048611    0.004854928       
44    166969    2782.816667    46.38028    1.932511574    0.005294552       
45    181623    3027.05    50.45083    2.102118056    0.005759228       
46    197335    3288.916667    54.81528    2.283969907    0.006257452       
47    213877    3564.616667    59.41028    2.475428241    0.006781995       
48    231588    3859.8    64.33    2.680416667    0.007343607       
49    250197    4169.95    69.49917    2.895798611    0.007933695       
50    270037    4500.616667    75.01028    3.125428241    0.008562817       
51    290842    4847.366667    80.78944    3.366226852    0.009222539       
52    312969    5216.15    86.93583    3.622326389    0.009924182       
53    336162    5602.7    93.37833    3.890763889    0.010659627       
54    360772    6012.866667    100.2144    4.175601852    0.011440005       
55    386523    6442.05    107.3675    4.473645833    0.012256564       
56    413755    6895.916667    114.9319    4.788831019    0.013120085       
57    442204    7370.066667    122.8344    5.118101852    0.014022197       
58    472268    7871.133333    131.1856    5.466064815    0.01497552       
59    503629    8393.816667    139.8969    5.829039352    0.015969971       
60    536670    8944.5    149.075    6.211458333    0.017017694       
61    571089    9518.15    158.6358    6.609826389    0.018109113       
62    607331    10122.18333    168.7031    7.029293981    0.01925834       
63    645037    10750.61667    179.1769    7.465706019    0.020453989       
64    684631    11410.51667    190.1753    7.923969907    0.021709507       
65    725775    12096.25    201.6042    8.400173611    0.023014174       
66    768961    12816.01667    213.6003    8.900011574    0.024383593       
67    813785    13563.08333    226.0514    9.41880787    0.025804953       
68    860720    14345.33333    239.0889    9.962037037    0.027293252       
69    909382    15156.36667    252.6061    10.52525463    0.028836314       
70    960318    16005.3    266.755    11.11479167    0.030451484       
71    1013076    16884.6    281.41    11.72541667    0.032124429       
72    1068176    17802.93333    296.7156    12.36314815    0.033871639       
73    1125192    18753.2    312.5533    13.02305556    0.035679604       
74    1184782    19746.36667    329.1061    13.71275463    0.037569191       
75    1246273    20771.21667    346.1869    14.42445602    0.039519058       
76    1310464    21841.06667    364.0178    15.16740741    0.041554541       
77    1376710    22945.16667    382.4194    15.93414352    0.043655188       
78    1445727    24095.45    401.5908    16.73295139    0.045843702       
79    1516896    25281.6    421.36    17.55666667    0.048100457       
80    1591031    26517.18333    441.9531    18.41471065    0.050451262       
81    1667420    27790.33333    463.1722    19.29884259    0.052873541       
82    1746845    29114.08333    485.2347    20.21811343    0.055392092       
83    1828696    30478.26667    507.9711    21.16546296    0.05798757       
84    1913650    31894.16667    531.5694    22.14872685    0.060681443       
85    2001209    33353.48333    555.8914    23.1621412    0.063457921       
86    2092009    34866.81667    581.1136    24.21306713    0.06633717       
87    2185378    36422.96667    607.0494    25.29372685    0.069297882       
88    2282276    38037.93333    633.9656    26.41523148    0.072370497       
89    2381854    39697.56667    661.6261    27.56775463    0.075528095       
90    2485031    41417.18333    690.2864    28.76193287    0.078799816       
91    2591079    43184.65    719.7442    29.98934028    0.082162576       
92    2700875    45014.58333    750.2431    31.26012731    0.085644184       
93    2813577    46892.95    781.5492    32.56454861    0.089217941       
94    2930174    48836.23333    813.9372    33.91405093    0.092915208       
95    3049880    50831.33333    847.1889    35.29953704    0.09671106       
96    3173546    52892.43333    881.5406    36.73085648    0.100632484       
97    3300440    55007.33333    916.7889    38.19953704    0.104656266       
98    3431631    57193.85    953.2308    39.71795139    0.108816305       
99    3565988    59433.13333    990.5522    41.27300926    0.113076738       
100    3704803    61746.71667    1029.112    42.87966435    0.117478532       
101    3846999    64116.65    1068.611    44.52545139    0.121987538       
102    3993818    66563.63333    1109.394    46.22474537    0.126643138       
103    4144043    69067.38333    1151.123    47.96346065    0.131406742       
104    4299055    71650.91667    1194.182    49.75758102    0.13632214       
105    4457701    74295.01667    1238.25    51.59376157    0.141352771       
106    4621198    77019.96667    1283.666    53.48608796    0.146537227       
107    4788563    79809.38333    1330.156    55.42318287    0.151844337       
108    4960951    82682.51667    1378.042    57.41841435    0.157310724       
109    5137229    85620.48333    1427.008    59.45866898    0.162900463       
110    5318815    88646.91667    1477.449    61.5603588    0.168658517       
111    5504309    91738.48333    1528.975    63.70728009    0.174540493       
112    5695285    94921.41667    1582.024    65.91765046    0.180596303       
113    5890419    98173.65    1636.228    68.17614583    0.186783961       
114    6091219    101520.3167    1692.005    70.50021991    0.193151287       
115    6296190    104936.5    1748.942    72.87256944    0.199650875       
116    6507007    108450.1167    1807.502    75.31258102    0.206335838       
117    6722257    112037.6167    1867.294    77.80390046    0.213161371       
118    6943540    115725.6667    1928.761    80.3650463    0.220178209       
119    7169398    119489.9667    1991.499    82.97914352    0.227340119       
120    7401214    123353.5667    2055.893    85.66219907    0.234690956       
121    7638011    127300.1833    2121.67    88.40290509    0.24219974       
122    7880951    131349.1833    2189.153    91.21471065    0.249903317       
123    8128887    135481.45    2258.024    94.08434028    0.257765316       
124    8383292    139721.5333    2328.692    97.02884259    0.265832445       
125    8642702    144045.0333    2400.751    100.0312731    0.274058283       
126    8908916    148481.9333    2474.699    103.1124537    0.282499873       
127    9180138    153002.3    2550.038    106.2515972    0.291100266       
128    9458362    157639.3667    2627.323    109.4717824    0.299922692       
129    9742041    162367.35    2706.123    112.7551042    0.308918094       
130    10032622    167210.3667    2786.839    116.1183102    0.318132357       
131    10328813    172146.8833    2869.115    119.5464468    0.327524512       
132    10632258    177204.3    2953.405    123.0585417    0.337146689       
133    10941313    182355.2167    3039.254    126.6355671    0.346946759       
134    11257985    187633.0833    3127.218    130.3007523    0.356988363       
135    11580260    193004.3333    3216.739    134.030787    0.367207636       
136    11910360    198506    3308.433    137.8513889    0.377675038       
137    12246555    204109.25    3401.821    141.7425347    0.388335712       
138    12590449    209840.8167    3497.347    145.7227894    0.399240519       
139    12940600    215676.6667    3594.611    149.775463    0.410343734       
140    13298831    221647.1833    3694.12    153.9216551    0.421703165       
141    13663309    227721.8167    3795.364    158.1401505    0.433260686       
142    14036259    233937.65    3898.961    162.4567014    0.445086853       
143    14415622    240260.3667    4004.339    166.8474769    0.457116375       
144    14803493    246724.8833    4112.081    171.3367245    0.469415684       
145    15197942    253299.0333    4221.651    175.9021065    0.481923579       
146    15601305    260021.75    4333.696    180.5706597    0.494714136       
147    16011416    266856.9333    4447.616    185.3173148    0.507718671       
148    16430474    273841.2333    4564.021    190.1675231    0.521006913       
149    16856448    280940.8    4682.347    195.0977778    0.53451446       
150    17291789    288196.4833    4803.275    200.1364468    0.548319032       
151    17734220    295570.3333    4926.172    205.2571759    0.562348427       
152    18186046    303100.7667    5051.679    210.4866435    0.576675736       
153    18645135    310752.25    5179.204    215.8001736    0.591233352       
154    19114053    318567.55    5309.459    221.2274653    0.606102645       
155    19590412    326506.8667    5441.781    226.7408796    0.621207889       
156    20076625    334610.4167    5576.84    232.3683449    0.636625602       
157    20570454    342840.9    5714.015    238.0839583    0.652284817       
158    21074587    351243.1167    5854.052    243.918831    0.66827077       
159    21586517    359775.2833    5996.255    249.8439468    0.684503964       
160    22108771    368479.5167    6141.325    255.8885532    0.701064529       
161    22639228    377320.4667    6288.674    262.0281019    0.717885211       
162    23180021    386333.6833    6438.895    268.2872801    0.735033644       
163    23729202    395486.7    6591.445    274.6435417    0.752448059       
164    24289187    404819.7833    6746.996    281.1248495    0.770205067       
165    24857754    414295.9    6904.932    287.7054861    0.788234209       
166    25437129    423952.15    7065.869    294.4112153    0.806606069       
167    26025278    433754.6333    7229.244    301.2184954    0.825256152       
168    26624963    443749.3833    7395.823    308.159294    0.844272038       
169    27233128    453885.4667    7564.758    315.1982407    0.863556824       
170    27853080    464218    7736.967    322.3736111    0.883215373       
171    28481949    474699.15    7911.653    329.6521875    0.903156678       
172    29122608    485376.8    8089.613    337.0672222    0.923471842       
173    29772633    496210.55    8270.176    344.5906597    0.944083999       
174    30434443    507240.7167    8454.012    352.2504977    0.965069857       
175    31105816    518430.2667    8640.504    360.0210185    0.986358955       
176    31789754    529829.2333    8830.487    367.9369676    1.008046487       
177    32482929    541382.15    9023.036    375.9598264    1.030026922       
178    33188929    553148.8167    9219.147    384.1311227    1.052414035       
179    33904631    565077.1833    9417.953    392.4147106    1.075108796       
180    34633150    577219.1667    9620.319    400.8466435    1.098209982       
181    35371849    589530.8167    9825.514    409.3964005    1.121633974       
182    36123631    602060.5167    10034.34    418.097581    1.145472825       
183    36885519    614758.65    10245.98    426.9157292    1.169632135       
184    37660753    627679.2167    10461.32    435.8883449    1.194214644       
185    38446584    640776.4    10679.61    444.9836111    1.219133181       
186    39245739    654095.65    10901.59    454.2330903    1.24447422       
187    40055702    667595.0333    11126.58    463.607662    1.270157978       
188    40879849    681330.8167    11355.51    473.1464005    1.296291508       
189    41714421    695240.35    11587.34    482.8057986    1.322755613       
190    42563453    709390.8833    11823.18    492.6325579    1.349678241       
191    43423421    723723.6833    12062.06    502.5858912    1.376947647       
192    44298133    738302.2167    12305.04    512.7098727    1.404684583       
193    45183681    753061.35    12551.02    522.9592708    1.432765126       
194    46084254    768070.9    12801.18    533.3825694    1.461322108       
195    46996189    783269.8167    13054.5    543.9373727    1.490239377       
196    47923439    798723.9833    13312.07    554.6694329    1.519642282       
197    48861942    814365.7    13572.76    565.5317361    1.549402017       
198    49816046    830267.4333    13837.79    576.5746065    1.579656456       
199    50781945    846365.75    14106.1    587.7539931    1.610284912       
200    51763739    862728.9833    14378.82    599.1173495    1.641417396       
201    52757212    879286.8667    14654.78    610.6158796    1.672920218       
202    53766871    896114.5167    14935.24    622.3017477    1.704936295       
203    54788766    913146.1    15219.1    634.1292361    1.737340373       
204    55827146    930452.4333    15507.54    646.1475231    1.770267187       
205    56877989    947966.4833    15799.44    658.3100579    1.8035892       
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944    19944754521    332412575.4    5540210    230842.0662    632.444017       
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989    23867264563    397787742.7    6629796    276241.488    756.8259945       
990    23960468460    399341141    6655686    277320.2368    759.7814707       
991    24053912061    400898534.4    6681642    278401.76    762.7445479       
992    24147659042    402460984    6707683    279486.7945    765.7172451       
993    24241640298    404027338.3    6733789    280574.5405    768.6973712       
994    24335926067    405598767.8    6759979    281665.811    771.6871533       
995    24430453352    407174222.5    6786237    282759.8768    774.6845939       
996    24525286307    408754771.8    6812580    283857.4804    777.6917271       
997    24620355305    410339255.1    6838988    284957.816    780.7063453       
998    24715737515    411928958.6    6865483    286061.7768    783.7308953       
999    24811356666    413522611.1    6892044    287168.4799    786.7629587       
1000    24907283784    415121396.4    6918690    288278.7475    789.8047877
    For \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor, I made the approximation formula (which is accurate up to n = 31).

    \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor \approx \frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right)

    which I first showed in this post.

    By using this approximation, we can have a formula at wolframalpha.

    \frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left( 21 \right)}\frac{n^{2}}{\ln \left( n \right)}\frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right)

    Note that \left\lfloor \log _{10}\left( {f}'\left( 3 \right) \right) \right\rfloor =21 with the approximation formula as well as the actual derivative formula (that's where the 21 comes from).

    To get values for a certain n, just change the 2 (located on the far right of the input bar) to a different number in the input bar.
    Last edited by cmowla; 04-28-2012 at 06:18 PM.
    Methods for Forming Parity Algorithms Supercube Centers and Odd Parity # of K4 OLLs
    # of Permutations on the nxnxn Cube F3L OLL 4x4x4 Parity Algorithms
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  4. 04-28-2012 #14
    ben1996123
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    Default

    Code:
    x	y
2	0.669408
3	5.59801
4	26.6197
5	54.5381
6	154.179
7	191.958
8	-804.047
9	-5651.51
    Pretty much exact for 2-7, but lol8,9.

    y=-0.510403x^6+11.5106x^5-102.079x^4+453.215x^3-1047.51x^2+1177.21x-491.84
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  5. 04-29-2012 #15
    RNewms27
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    I'm curious to see a BLD formula
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  6. 04-30-2012 #16
    Godmil
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    Quote Originally Posted by JensRenders View Post
    y=7.5(x-2)^2
    Quote Originally Posted by cmowla View Post
    P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)}
    , where

    {f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n.

    {f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}}
    \text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}
    You know what, I'm going to give JensRenders the point for the elegance of his formula.
    3X3 Warp 2: (Snl/5/12) 15.26, 18.77, 20.78.
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