Thank you for doing this analysis. For the #of positions in column#1 - Is this NOT counting transpositions to the same position like U2 d2 R' = d2 U2 R', and equivalent positions reached by different turns +cube rotation like U D' = Ey?Code:
block turns (half-turns allowed) moves positions cumulative positions cumulative pos. positions mod M mod M 0 1 1 1 1 1 54 55 8 9 2 2070 2125 87 96 3 78649 80774 2052 2148 4 2973289 3054063 66299 68447 5 111963451 115017514 2376654 2445101 6 4212573974 4327591488 88205742 90650843 7 158458718464 162786309952 3305688035 3396338878
In each of your examples, the two sequences you give produce the same position, so are only counted as one position. I used a hash table to avoid counting duplicates such as these more than once. Whole cube rotations are not counted as moves, so Ey (or U D') is counted as a position 1 move from solved, since E is a single block turn. The cube can be rotated so DLB corner is at DLB to prevent counting "positions" differing by only the orientation of the cube as different positions.Thank you for doing this analysis. For the #of positions in column#1 - Is this NOT counting transpositions to the same position like U2 d2 R' = d2 U2 R', and equivalent positions reached by different turns +cube rotation like U D' = Ey?
Can you do another table (same metric), that also shows the total number of positions that are possible (including duplicates) before ANY equivalency elimination? Showing the various branching factors would be nice too.In each of your examples, the two sequences you give produce the same position, so are only counted as one position. I used a hash table to avoid counting duplicates such as these more than once. Whole cube rotations are not counted as moves, so Ey (or U D') is counted as a position 1 move from solved, since E is a single block turn. The cube can be rotated so DLB corner is at DLB to prevent counting "positions" differing by only the orientation of the cube as different positions.
I have done basically the same thing in six different metrics. See this thread on the Domain of the Cube forum.
Did he say it was? He's asking pretty good questions that I think shows that he knows math pretty well. Nevertheless, his questions pertain to the scrambles which make the permutations, but this will not help very much considering what qqwref pointed out about the amount of hard-drive space needed for information on the permutations alone. At least RtC is contributing to this idea. Not all math problems have an obvious real world application, and, sometimes, somethings previously overlooked become foundational.hey guys, learn math. symmetry is NOT going to help reduce ANYTHING by a factor of a hundred million million million.
If you're asking for the number of unique possible sequences, that should be easy to calculate by hand. If you're asking for something else, I'm not sure what it is exactly what you want, or what the point of calculating it is. The "positions" column in my table gives positions from the set of 7.4*10^45 positions of the 4x4x4. The "positions mod M" is the number of symmetry-reduced positions. (M is the symbol Dan Hoey used for the symmetry group of the cube in the old cube-lovers archives.)Can you do another table (same metric), that also shows the total number of positions that are possible (including duplicates) before ANY equivalency elimination? Showing the various branching factors would be nice too.
My take on that, was that the number of combinations generated, would initially be much larger than the actual number of "distinct" 4x4x4 positions. Therefore, the reductions in positions from this larger set would also be larger- if all the equivalencies were eliminated. My intention was that "symmetrically equivalent" would be positions mod M, but also include any isomorphic positions reached by different move orders, or whole cube rotations.Instead, why don't we make scrambles using all combinations of slices.
ALL SLICES ALL SLICES SYMM SYMM SAVINGS #of positions Commulative #of positions Commulative #of positions 0 1 1 1 1 0 1 72 73 8 9 64 2 4968 5041 87 96 4945 3 342792 347833 2052 2148 345685 4 23652648 24000481 66299 68477 23932004 5 1632032712 1656033193 2376654 2445101 1653588092 6 1.1261E+11 1.1427E+11 8.8206E+07 9.0651E+07 1.1418E+11 7 7.7701E+12 7.8844E+12 3.3057E+09 3.3963E+09 7.8810E+12 8 5.3614E+14 5.4402E+14 1.2396E+11 1.2736E+11 5.4389E+14 9 3.6993E+16 3.7538E+16 4.6486E+12 4.7760E+12 3.7533E+16 10 2.5526E+18 2.5901E+18 1.7432E+14 1.7910E+14 2.5899E+18 11 1.7613E+20 1.7872E+20 6.5371E+15 6.7162E+15 1.7871E+20 12 1.2153E+22 1.2331E+22 2.4514E+17 2.5186E+17 1.2331E+22 13 8.3854E+23 8.5087E+23 9.1928E+18 9.4447E+18 8.5086E+23 14 5.7859E+25 5.8710E+25 3.4473E+20 3.5418E+20 5.8709E+25 15 3.9923E+27 4.0510E+27 1.2927E+22 1.3282E+22 4.0510E+27 16 2.7547E+29 2.7952E+29 4.8478E+23 4.9806E+23 2.7952E+29 17 1.9007E+31 1.9287E+31 1.8179E+25 1.8677E+25 1.9287E+31 18 1.3115E+33 1.3308E+33 6.8172E+26 7.0040E+26 1.3308E+33 19 9.0493E+34 9.1824E+34 2.5564E+28 2.6265E+28 9.1824E+34 20 6.2440E+36 6.3359E+36 9.5867E+29 9.8493E+29 6.3359E+36 21 4.3084E+38 4.3717E+38 3.5950E+31 3.6935E+31 4.3717E+38 22 2.9728E+40 3.0165E+40 1.3481E+33 1.3851E+33 3.0165E+40 23 2.0512E+42 2.0814E+42 5.0555E+34 5.1940E+34 2.0814E+42 24 1.4153E+44 1.4362E+44 1.8958E+36 1.9477E+36 1.4362E+44 25 9.7659E+45 9.9095E+45 7.1093E+37 7.3040E+37 9.9095E+45 26 6.7384E+47 6.8375E+47 2.6660E+39 2.7390E+39 6.8375E+47 27 4.6495E+49 4.7179E+49 9.9974E+40 1.0271E+41 4.7179E+49 28 3.2082E+51 3.2554E+51 3.7490E+42 3.8517E+42 3.2554E+51 29 2.2136E+53 2.2462E+53 1.4059E+44 1.4444E+44 2.2462E+53 30 1.5274E+55 1.5499E+55 5.2721E+45 5.4165E+45 1.5499E+55
pfffh. As the table shows (lol), there would be a bulk savings of approx. 1.5*10^55 positions by turn 30 - if all of the symmetrically equivalent positions are eliminated. This is (lol) not surprisingly > than a billion times more than the total number of distinct positions for 4x4x4 ~ 7.4*10^45. But viewing symmetry as limited to mod M, within the already determined to be "distinct positions", then those additional reductions will only be ~/48.How do you think I overestimated? If you think you can get a factor of 10^20 or more using symmetry, please, let me know how.
If I have to teach you a lesson a hundred million million million times, will you become my exponent?hey guys, learn math. symmetry is NOT going to help reduce ANYTHING by a factor of a hundred million million million.
He just needs a lesson on exponents.
Uh, what? The concept of "bulk savings" is useless here, because (a) I think you're only saving moves because you massively overcount in the "all" column, (b) subtracting two very large numbers tells us nothing because what's important here is the quotient, and (c) my point was that, at the very least, you still need to store a yes/no for each possible position to see which ones have been seen so far, and that would be technologically impossible to store, even with a factor-of-48ish reduction. Canceling moves in scrambles does not fix this problem.As the table shows (lol), there would be a bulk savings of approx. 1.5*10^55 positions by turn 30 - if all of the symmetrically equivalent positions are eliminated.
Just because we know the maximum number of moves necessary to solve any scrambled cube doesn't mean we can find the optimal solutions by hand.i'm a new member hi everyone, my question is really simple, if you did work out how to do every scrambled cube in twenty moves or less and could show everyone how this is done and why you dont need more than 20 moves everytime, if you could show scrambled positions that by sight indicate you could pure solve it in 19, 18,17,16 and below would you not just kill the enigma of the rubiks cube, and i'm talking 4x 4's upwards to infinitum sized cubes, what would happen to the mystery of the cube? is it worth having knowing gods algorithym?
Maybe it's a coincidence, but when I looked at my derivative formula a second time (from this post), I noticed something interesting.Of course I cannot prove this, but I am pretty sure that God's number for the nxnxn cube is
C(n)->0.25Log(24!)-1.5Log(4!) for n->Infinity
n=2: 11 n=3: 20 n=4: 31.2659 n=5: 62.70665485 n=6: 69.24544149 n=7: 117.7741065 n=8: 117.4613029 n=9: 186.0953306 n=10: 176.0280599 n=11: 267.5914284 n=12: 244.8247329 n=13: 362.0839241 n=14: 323.7094834 n=15: 469.3931065 n=16: 412.547601 n=17: 589.3598688 n=18: 511.2207599 n=19: 721.8427516 n=20: 619.6252563 n=21: 866.7141865 n=22: 737.6687839 n=23: 1 023.861951 n=24: 865.2680173 n=25: 1 193.190987 n=26: 1 002.345821 n=27: 1 374.595445 n=28: 1 148.839832 n=29: 1 567.999255 n=30: 1 304.681743 n=31: 1 773.336688 n=32: 1 469.816769 n=33: 1 990.520222 n=34: 1 644.193294 n=35: 2 219.501815 n=36: 1 827.75701 n=37: 2 460.212149 n=38: 2 020.468033 n=39: 2 712.594266 n=40: 2 222.285904 n=41: 2 976.592701 n=42: 2 433.164604 n=43: 3 252.167974 n=44: 2 653.067145 n=45: 3 539.265871 n=46: 2 881.953595 n=47: 3 837.848056 n=48: 3 119.795215 n=49: 4 147.868404 n=50: 3 366.572571 n=51: 4 469.29323 n=52: 3 622.228207 n=53: 4 802.071821 n=54: 3 886.75986 n=55: 5 146.163056 n=56: 4 160.128579 n=57: 5 501.560725 n=58: 4 442.303052 n=59: 5 868.217385 n=60: 4 733.260235 n=61: 6 246.09646 n=62: 5 032.839342 n=63: 6 635.359829 n=64: 5 341.292848 n=65: 7 035.600007 n=66: 5 658.465512 n=67: 7 446.96595 n=68: 5 984.343194 n=69: 7 869.451746 n=70: 6 318.8773 n=71: 8 303.025603 n=72: 6 662.077494 n=73: 8 747.646669 n=74: 7 013.909263 n=75: 9 203.312673 n=76: 7 374.355562 n=77: 9 669.983959 n=78: 7 743.394437 n=79: 10 147.65561 n=80: 8 121.018605 n=81: 10 636.28889 n=82: 8 507.196532 n=83: 11 135.86593 n=84: 8 901.930092 n=85: 11 646.35073 n=86: 9 305.232389 n=87: 12 167.67032 n=88: 9 717.06143 n=89: 12 699.93 n=90: 10 137.37871 n=91: 13 243.06844 n=92: 10 566.13131 n=93: 13 797.00818 n=94: 11 003.40002 n=95: 14 361.83359 n=96: 11 449.0952 n=97: 14 937.42493 n=98: 11 903.25648 n=99: 15 523.81908 n=100: 12 365.83595
Not really - why would God's number be higher for, say, a 95x95x95 cube than a 96x96x96 cube? Especially considering the lower one can be simulated on the higher one?Do these results seem to be reasonable?
The two numbers are unrelated; FMC scrambles rarely have an optimal solution of 20 moves. (And being Christian has nothing to do with people not being perfect. Simply put, nobody could expect someone to find an optimal solution to a random scramble in an hour when a computer requires so much processing power to do it!)God's Number is 20...but the FMC WR is 22, proving that humans aren't perfect.
In fact, István Kocza is unlucky, his WR solution should have been less than 20 moves.God's Number is 20...but the FMC WR is 22, proving that humans aren't perfect.
Yes, I am a Christian.
The two numbers are unrelated; FMC scrambles rarely have an optimal solution of 20 moves.
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