# World record formula for nxnxn cubes

#### JensRenders

##### Member
Not really accurate, but here it is

y=7.5(x-2)^2

where x is the n of the nxnxn cube and y is the estimate average world record

the 7.5 is the value of skill that humans have, the lower this number, the more skill we have, if this number goes down a bit then the time for a 3x3x3 cube would also change a bit and the time of a 7x7x7 cube would change alot more

if the value of 7.5 becomes 0 then we can solve all cubes in no time

I just discovered that 7.5 is what it gives for 3x3x3 haha if you change that to the exact wr 7.64 than thats what the formula gives ofcourse

Last edited:

#### JensRenders

##### Member
2x2 solves can get lower but they are almost zero (and yes, 2.12=0 thats about the accuracy of this formula)

#### ben1996123

##### Banned
Single WR estimate:

Code:
x	y	WR - y
2	0.43571	0.254287
3	6.63572	-0.97572
4	23.8381	2.9319
5	57.2429	-2.3829
6	112.05	2.76
7	193.46	-0.35
8	306.672
9	456.886
10	649.303
11	889.122
$$y=0.866667x^3-2.29881x^2+1.22738x+0.242857$$

Last edited:

#### Sa967St

##### Not A Moderator
Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.

#### ben1996123

##### Banned
Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.
Wolframalpha gives $$y=2.92575e^{0.606276x}$$ with R² = 0.996411 with 3-7 averages. Doesn't really seem that good though

Code:
x	y	WR - y
2	9.83653	-7.71653
3	18.0362	-10.3962
4	33.0709	0.4991
5	60.6386	-2.6986
6	111.186	9.244
7	203.87	-3.4
8	373.814
9	685.422
10	1256.78
11	2304.43

#### Mike Hughey

##### Super Moderator
Staff member
Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.
If you use this formula for 11x11x11, you get 92 minutes!

#### JensRenders

##### Member
If you use this formula for 11x11x11, you get 92 minutes!
if you use mine, you get 10 minutes!

#### The Bloody Talon

##### Member
Not really accurate, but here it is

y=7.5(x-2)^2

the 7.5 is the value of skill that humans have, the lower this number, the more skill we have, if this number goes down a bit then the time for a 3x3x3 cube would also change a bit and the time of a 7x7x7 cube would change alot more
the value of my skill is approximately 13..

#### Carrot

##### Member
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 )

#### Christopher Mowla

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:

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
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954    20771653766    346194229.4    5769904    240412.6593    658.6648201
955    20855707126    347595118.8    5793252    241385.4991    661.3301346
956    20940037717    349000628.6    5816677    242361.5477    664.0042401
957    21024592848    350409880.8    5840165    243340.195    666.6854658
958    21109438156    351823969.3    5863733    244322.2009    669.3758928
959    21194497019    353241617    5887360    245306.6785    672.0730917
960    21279847181    354664119.7    5911069    246294.5276    674.7795276
961    21365417692    356090294.9    5934838    247284.927    677.4929507
962    21451274663    357521244.4    5958687    248278.6419    680.2154574
963    21537358825    358955980.4    5982600    249274.9864    682.9451682
964    21623724566    360395409.4    6006590    250274.5899    685.6838079
965    21710324388    361838739.8    6030646    251276.9026    688.4298702
966    21797206893    363286781.6    6054780    252282.4872    691.1848964
967    21884312315    364738538.6    6078976    253290.6518    693.9469912
968    21971713584    366195226.4    6103254    254302.2406    696.7184673
969    22059338647    367655644.1    6127594    255316.4195    699.4970398
970    22147254634    369120910.6    6152015    256333.9657    702.2848375
971    22235401355    370590022.6    6176500    257354.1823    705.0799516
972    22323840134    372064002.2    6201067    258377.7793    707.8843269
973    22412504438    373541740.6    6225696    259403.9866    710.6958536
974    22501461910    375024365.2    6250406    260433.5869    713.5166765
975    22590651898    376510865    6275181    261465.8784    716.3448725
976    22680136192    378002269.9    6300038    262501.5763    719.1824008
977    22769847747    379497462.5    6324958    263539.9045    722.0271356
978    22859854725    380997578.8    6349960    264581.6519    724.8812381
979    22950096004    382501600.1    6375027    265626.1112    727.7427703
980    23040633847    384010564.1    6400176    266674.0029    730.6137065
981    23131400695    385523344.9    6425389    267724.5451    733.4919043
982    23222465226    387041087.1    6450685    268778.5327    736.3795417
983    23313765853    388562764.2    6476046    269835.2529    739.2746656
984    23405365307    390089421.8    6501490    270895.4318    742.1792652
985    23497195517    391619925.3    6526999    271958.2814    745.091182
986    23589325678    393155428    6552590    273024.6028    748.0126103
987    23681693735    394694895.6    6578248    274093.6775    750.9415822
988    23774362893    396239381.6    6603990    275166.2372    753.8801019
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:

#### ben1996123

##### Banned
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.

$$y=-0.510403x^6+11.5106x^5-102.079x^4+453.215x^3-1047.51x^2+1177.21x-491.84$$

#### Godmil

$$P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)}$$
$${f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n.$$
$${f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}}$$
$$\text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}$$