Here it is folks, the Ultimate Expression of Cubism! Welcome to... ----------------------------------------------------------------- God's Algorithm Calculations for Rubik's Cube, Rubik's Subgroups, and Related Puzzles ----------------------------------------------------------------- This is the work of Dan Hoey, Jerry Bryan, Dik Winter, Micheal Reid, Martin Schoenert, Tom Rokicki, Jaap Scherphuis and Mark Longridge. Not all results are known but with computing power continuing to increase we will undoubtably see more calculations of this sort in the future. Size of Cube Space = (8! * 12!/2) * (2^12 /2) * (3^8 /3) = 43,252,003,274,489,856,000 (approx 43 quintillion) Real size of Cube Space = 901,083,404,981,813,616 (approx 901 quadrillion) The Real Size of Cube Space was first calculated on Fri, 4 Nov 94 by Dan Hoey. Analysis of Analysis of ------------------------ ---------------------------- Level Number of Level Number of Positions Positions 0 1 0 1 1 3 1 4 2 5 2 10 3 8 3 24 4 13 4 53 5 21 5 64 6 23 6 31 7 13 7 4 8 5 8 1 9 3 --- 10 1 192 -- 96 Antipode 2 X order 2 = R2 T2 L2 R2 T2 R2 D2 T2 Antipode 2 H order 2 = (R2 D2)^2 (T2 R2)^2 D2 T2 Analysis of Level Number of Positions 0 1 1 3 2 6 3 12 4 24 5 48 6 93 7 180 8 315 9 489 10 604 11 522 12 250 13 42 14 3 ----- 2,592 Analysis of < U, R2 > (using both U and -U) --------------------- Level Positions Total Elapsed Time 0) 1 1) 3 4 0 2) 5 9 0 3) 8 17 0 4) 13 30 0 5) 21 51 0 6) 34 85 0 7) 55 140 0 8) 85 225 0 9) 125 350 0 10) 188 538 0 11) 286 824 0 12) 432 1256 0 13) 646 1902 0 14) 952 2854 1 15) 1404 4258 1 16) 1794 6052 1 17) 2170 8222 2 18) 2306 10528 3 19) 1964 12492 4 20) 1312 13804 5 21) 512 14316 5 22) 75 14391 5 23) 5 14396 5 24) 3 14399 5 25) 1 14400 5 Antipode of < U, R2 > TOP U U U U U U U U U LEFT FRONT RIGHT BACK F L B L B R B R F R F L L L L F F F R R R B B B L L L F F R B R F R B B DOWN D D D D D D D D D Analysis of < U, R2 > (counting U2 as 1 move) --------------------- Level Positions Total Elapsed Time 0) 1 1) 4 5 0 2) 6 11 0 3) 12 23 0 4) 18 41 0 5) 36 77 0 6) 53 130 0 7) 100 230 0 8) 144 374 0 9) 252 626 0 10) 364 990 0 11) 644 1634 0 12) 898 2532 1 13) 1504 4036 1 14) 1934 5970 1 15) 2544 8514 2 16) 2662 11176 3 17) 1988 13164 4 18) 1111 14275 5 19) 116 14391 5 20) 9 14400 5 Analysis of < U, R2 > (using clockwise U only) --------------------- Level Positions Total Elapsed Time 0) 1 1) 2 3 0 2) 3 6 0 3) 5 11 0 4) 7 18 0 5) 10 28 0 6) 15 43 0 7) 22 65 0 8) 32 97 0 9) 46 143 0 10) 66 209 0 11) 95 304 0 12) 133 437 0 13) 188 625 0 14) 266 891 0 15) 373 1264 0 16) 515 1779 0 17) 700 2479 0 18) 923 3402 0 19) 1194 4596 1 20) 1487 6083 1 21) 1752 7835 1 22) 1899 9734 2 23) 1817 11551 2 24) 1472 13023 3 25) 895 13918 4 26) 368 14286 4 27) 100 14386 4 28) 13 14399 4 29) 1 14400 4 Antipode of < U, R2 > (same as previous) TOP U U U U U U U U U LEFT FRONT RIGHT BACK F L B L B R B R F R F L L L L F F F R R R B B B L L L F F R B R F R B B DOWN D D D D D D D D D Analysis of the Group ------------------------------ Level Number of Time Branching Positions Factor 0 1 0 s -- 1 4 0 s 4 2 10 0 s 2.5 3 24 0 s 2.4 4 58 0 s 2.416 5 140 2 s 2.414 6 338 11 s 2.414 7 816 67 s 2.414 8 1,909 433 s 2.339 9 4,296 2793 s 2.250 10 8,893 17355 s 2.070 Calculation later completed by B MacKenzie confirmed by Brac37 Depth whole cube total unique total 0 1 1 1 1 1 4 5 4 5 2 10 15 10 15 3 24 39 24 39 4 58 97 58 97 5 140 237 140 237 6 338 575 338 575 7 816 1391 816 1391 8 1909 3300 1909 3300 9 4296 7596 4296 7596 10 8893 16489 8893 16489 11 17160 33649 17160 33649 12 28891 62540 28891 62540 13 37996 100536 37996 100536 14 37678 138214 37678 138214 15 27186 165400 27186 165400 16 13051 178451 13051 178451 17 4128 182579 4128 182579 18 1199 183778 1199 183778 19 372 184150 372 184150 20 122 184272 122 184272 21 36 184308 36 184308 22 10 184318 10 184318 23 2 184320 2 184320 Analysis of the 3x3x3 5 Generator Group --------------------------------------- Level Number of Local Branching Positions Max Factor 0 1 0 1 10 0 10.000 2 77 0 7.700 3 584 0 7.584 4 4,434 0 7.592 5 33,664 0 7.592 6 255,320 0 7.584 7 1,933,936 7.575 8 14,635,503 7.568 Analysis of the 3x3x3 Slice & Anti-Slice Groups ----------------------------------------------- arrangements M arrangements M Moves Deep (2q or slice) conjugates (4q or double slice) conjugates 0 1 1 1 1 1 6 1 9 2 2 27 2 51 4 3 120 6 247 15 4 287 16 428 25 5 258 15 32 3 6 69 9 --- -- --- -- 768 50 768 50 arrangements arrangements Moves Deep (2q or anti-slice moves) (4q or double anti-slice moves) 0 1 1 1 1 1 6 1 9 2 2 27 3 51 5 3 120 10 265 25 4 423 37 864 75 5 1,098 93 1,785 152 6 1,770 166 2,017 184 7 1,650 147 1,008 108 8 851 89 144 16 9 198 21 ----- --- ----- --- 6,144 568 6,144 568 Analysis of the 2x2x2 cube group -------------------------------- Originally computed on a DEC VAX 11/780 in over 51 hours of CPU time on Sept. 9, 1981 Moves Deep arrangements (q+h) arrangements (q) loc max (q+h) loc max (q) 0 1 1 0 0 1 9 6 0 0 2 54 27 0 0 3 321 120 0 0 4 1,847 534 11 0 5 9,992 2,256 8 0 6 50,136 8,969 96 0 7 227,536 33,058 904 16 8 870,072 114,149 13,212 53 9 1,887,748 360,508 413,392 260 10 623,800 930,588 604,516 1,460 11 2,644 1,350,852 2,644 34,088 12 782,536 402,260 13 90,280 88,636 14 276 276 --------- --------- --------- ------- 3,674,160 3,674,160 1,034,783 527,049 Analysis of the full 3x3x3 cube group ------------------------------------- Moves Deep arrangements (q+h) bf arrangements (q only) * 0 1 -- 1 1 18 18 12 2 243 13.5 114 3 3,240 13.33 1,068 4 43,239 13.34 10,011 5 574,908 13.29 93,840 (March 22, 1981) 6 7,618,438 13.25 878,880 (August 14, 1981) 7 100,803,036 13.23 8,221,632 (December 7, 1981) 8 1,332,343,288 13.217 76,843,595 (July 18, 1994) 9 17,596,479,795 13.207 717,789,576 10 6,701,836,858 11 62,549,615,248 (February 4, 1995) PH[0] = 1 PH[1] <= 6*3*PH[0] PH[2] <= 6*2*PH[1] + 9*3*PH[0] PH[n] <= 6*2*PH[n-1] + 9*2*PH[n-2] for n > 2 Solving yields the following upper bounds: htw new total htw new total 0 1 1 10 2.447*10^11 2.646*10^11 1 18 19 11 3.267*10^12 3.531*10^12 2 243 262 12 4.360*10^13 4.713*10^13 3 3240 3502 13 5.820*10^14 6.292*10^14 4 43254 46756 14 7.769*10^15 8.398*10^15 5 577368 624124 15 1.037*10^17 1.121*10^17 6 7706988 8331112 16 1.385*10^18 1.497*10^18 7 102876480 111207592 17 1.848*10^19 1.998*10^19 8 1373243544 1484451136 18 2.467*10^20 2.667*10^20 9 18330699168 19815150304 Analysis of < U2, F2, L2, R2 > ------------------------------ Level Positions 0 1 1 4 2 11 3 30 4 82 5 224 6 589 7 1,484 8 3,649 9 8,488 10 18,424 11 34,890 12 47,802 13 36,757 14 12,360 15 1,067 16 26 ------- 165,888 Analysis of the 3x3x3 squares group ----------------------------------- branching Moves Deep arrangements (h only) factor loc max (h only) 0 1 -- 0 1 6 6 0 2 27 4.5 0 3 120 4.444 0 4 519 4.325 0 5 1,932 3.722 0 6 6,484 3.356 1 (6 X pattern) 7 20,310 3.132 0 8 55,034 2.709 65 9 113,892 2.069 1,482 10 178,495 1.567 7,379 11 179,196 1.004 25,980 12 89,728 0.501 50,320 13 16,176 0.180 11,328 14 1,488 0.092 912 15 144 0.096 144 ------- ------ 663,552 97,611 Analysis of the 3x3x3 squares group (5 generators) -------------------------------------------------- Moves Deep arrangements (h only) 0 1 1 5 2 18 3 64 4 223 5 726 6 2,360 7 7,315 8 21,619 9 57,283 10 130,243 11 207,350 12 171,907 13 58,469 14 5,353 15 564 16 52 ------- 663,552 Analysis of the 3x3x3 group ---------------------------------- ML's Conjecture: The < U, R > group is >=20 turns deep in qt metric Now confirmed, Sept 1, 1994 branching Moves Deep arrangements (q only) factor 0 1 -- 1 4 4 2 10 2.5 3 24 2.4 4 58 2.416 5 140 2.413 6 338 2.414 7 816 2.414 8 1,970 2.414 9 4,756 2.414 10 11,448 2.407 11 27,448 2.401 12 65,260 2.378 13 154,192 2.363 14 360,692 2.339 15 827,540 2.294 16 1,851,345 2.237 17 3,968,840 2.144 18 7,891,990 1.988 19 13,659,821 1.755 20 18,471,682 1.352 21 16,586,822 0.898 22 8,039,455 0.485 23 1,511,110 0.188 24 47,351 0.031 25 87 0.002 ---------- 73,483,200 Moves Deep arrangements (q+h) total 0 1 1 1 6 7 2 18 25 3 54 79 4 162 241 5 486 727 6 1,457 2,184 7 4,360 6,544 8 13,016 19,560 9 38,482 58,042 10 113,094 171,136 11 328,920 500,056 12 942,351 1,442,407 13 2,616,973 4,059,380 14 6,774,848 10,834,228 15 15,105,592 25,939,820 16 24,231,019 50,170,839 17 19,421,274 69,592,113 18 3,843,568 73,435,681 19 47,465 73,483,146 20 54 73,483,200 Analysis of 3x3x3 corners only ------------------------------ Moves Deep arrangements (q+h) arrangements (q only) * loc max (q only) 0 1 1 0 1 18 12 0 2 243 114 0 3 2,874 924 0 4 28,000 6,539 0 5 205,416 39,528 0 6 1,168,516 199,926 114 7 5,402,628 806,136 600 8 20,776,176 2,761,740 17,916 9 45,391,616 8,656,152 10,200 10 15,139,616 22,334,112 35,040 11 64,736 32,420,448 818,112 12 18,780,864 9,654,240 13 2,166,720 2,127,264 14 6,624 6,624 ---------- ---------- ---------- 88,179,840 88,179,840 11,870,110 Analysis of 3x3x3 edges only using q turns ------------------------------------------ Distance Number of Branching Number of Branching from M-Conjugate Factor M-Conjugate Factor Start Classes Classes Without With Centers Centers 0 1 1 1 1 1.00 1 1.00 2 5 5.00 5 5.00 3 25 5.00 25 5.00 4 215 8.60 215 8.60 5 1,860 8.65 1,886 8.77 6 16,481 8.86 16,902 8.96 7 144,334 8.76 150,442 8.90 8 1,242,992 8.61 1,326,326 8.81 9 10,324,847 8.31 11,505,339 8.67 10 76,993,295 7.46 96,755,918 8.40 11 371,975,385 4.83 750,089,528 7.75 12 382,690,120 1.03 .... 13 8,235,392 0.02 work 14 54 0.00 in 15 1 0.02 progress ----------- Total 851,625,008 Posted to Yahoo by Tom Rokicki on Jan 2, 2004 Dist Positions Unique wrt M Unique wrt M+inv 0 1 1 1 1 12 1 1 2 114 5 5 3 1,068 25 17 4 9,819 215 128 5 89,392 1,886 986 6 807,000 16,902 8,652 7 7,209,384 150,442 75,740 8 63,624,107 1,326,326 665,398 9 552,158,812 11,505,339 5,759,523 10 4,643,963,023 96,755,918 48,408,203 11 36,003,343,336 750,089,528 375,164,394 12 208,075,583,101 4,334,978,635 2,167,999,621 13 441,790,281,226 9,204,132,452 4,603,365,303 14 277,713,627,518 5,785,844,935 2,894,003,596 15 12,144,555,140 253,044,012 126,739,897 16 23,716 750 677 17 30 3 3 18 1 1 1 --------------- ------------- -------------- 980,995,276,800 20,437,847,376 10,222,192,146 Analysis of 3x3x3 edges only using q+h turns -------------------------------------------- Dist Positions Unique mod M Unique mod M+inv ---- --------------- ------------- ---------------- 0 1 1 1 1 18 2 2 2 243 9 8 3 3,240 75 48 4 42,807 925 505 5 555,866 11,684 6,018 6 7,070,103 147,680 74,618 7 87,801,812 1,830,601 918,432 8 1,050,559,626 21,890,847 10,960,057 9 11,588,911,021 241,449,652 120,788,522 10 110,409,721,989 2,300,251,615 1,150,428,080 11 552,734,197,682 11,515,452,614 5,759,027,817 12 304,786,076,626 6,349,914,756 3,176,487,580 13 330,335,518 6,896,891 3,500,434 14 248 24 24 Total number of positions on edges-only 3x3x3: (2 ^ 12 / 2 ) * 12! = 980,995,276,800 Total number of positions on edges-only 3x3x3 without centres: (2 ^ 11 / 2 ) * 11! = 40,874,803,200 *Note* that normally there would be only half the number of positions since on a real 3x3x3 cube you can't exchange one pair of edges alone. Analysis of Pyraminx -------------------- Moves Deep arrangements branching factor 0 1 -- 1 8 8 2 48 6 3 288 6 4 1,728 6 5 9,896 5.726 6 51,808 5.235 7 220,111 4.248 8 480,467 2.183 9 166,276 0.346 10 2,457 0.015 11 32 0.013 ------- 933,120 (If tips are included: 933,120 * 3^4 = 75,582,720) Analysis of the Skewb --------------------- The correct numbers for H = < RUF, LUB, RDB, LDF > are as follows Moves Deep Arrangements ---------- ------------ 0 1 1 8 2 48 3 288 4 1,728 5 10,248 6 59,304 7 315,198 8 1,225,483 9 1,455,856 10 81,028 11 90 --------- 3,149,280 Analysis of the Dino Cube (calculated by Jaap on January 29 2002) Moves Deep Arrangements ---------- ------------ 1 16 2 160 3 1,408 4 11,712 5 90,912 6 640,192 7 3,740,838 8 11,138,597 9 4,313,963 10 20,577 11 24 -------- 19,958,400