# Random Turn Sequence vs De Novo Random Cube

#### Bruce MacKenzie

##### Member
I've been playing around with my cube simulator and wrote a new routine for generating a random cube by placing the cubies one by one into the cube at random positions and orientations (as constrained by parity). To test the routine I used it to generate a million configurations and counted the number of times each cubie was given each of the 24 possible states. One would expect that a cubie should have an equal probability to end up in each of the possible states. I then went ahead and repeated the calculation using a random q-turn cube scrambler. It is a dumb scrambler which does nothing to filter out short identities such as R R' or R R R = R' , etc.

Following is a table of my results. The table lists the raw counts for the possible states followed by the ave deviation for that set of data. Listed are the data for the de novo random cube generator, a 30 q-turn scramble, a 50 q-turn scramble and a 100 q-turn scramble. From a quick look at the data I would say that there is significant improvement going from 30 turns to 50, but very little going from 50 to 100 turns.

De Novo Random Cube State
Cubie: UF 41994 41685 41696 41848 41794 41684 41502 41916 41687 41420 41428 41626 41668 41348 41580 41843 41848 41520 41725 41592 41418 41598 41929 41651
0.33%
Cubie: UR 41801 41719 41509 41894 41669 41621 41848 41531 41853 41622 41484 41643 41524 41559 42009 41612 41498 41631 41520 41866 41482 41395 41736 41974
0.34%
Cubie: UB 41745 41996 41340 41557 41708 41801 42077 41878 41715 41538 41728 41460 41613 41417 41747 41666 41911 41821 41337 41582 41649 41769 41562 41383
0.38%
Cubie: UL 41697 41545 41928 41472 41679 41733 41478 41635 41791 41581 41971 42070 41546 41854 41216 41608 41689 41973 41447 41884 41195 41372 41876 41760
0.45%
Cubie: DF 41591 41729 41848 41780 41737 41683 41427 41590 40940 41790 41660 41622 41560 41515 41695 41549 42057 41861 41382 41539 41473 41953 42139 41880
0.43%
Cubie: DR 41550 41770 41771 41346 41885 42058 41616 41301 41572 41587 42100 41673 41590 41611 41443 41758 41875 41228 41558 41694 41545 41799 41610 42060
0.42%
Cubie: DB 41744 41840 41567 41689 41563 41894 41627 41469 41327 41863 41906 41681 41474 41731 41541 41984 41523 41674 41756 41464 41765 41657 41750 41511
0.32%
Cubie: DL 41624 41424 41851 41856 41775 41760 41835 41617 41487 41712 41806 41390 41730 41779 41539 41907 41350 41446 41526 41349 41863 42032 41697 41645
0.39%
Cubie: FR 41807 41693 41353 41948 41355 41323 41902 41656 41749 41722 41500 41754 41724 41507 41802 41705 41389 41766 41779 41223 41787 41866 41934 41756
0.41%
Cubie: FL 41535 41808 41520 41610 41778 41812 41564 41641 42025 42090 41593 41876 41565 41578 41515 41796 41563 42131 41842 40925 41521 41351 41692 41669
0.44%
Cubie: BR 41644 41539 41876 41990 41569 41727 41795 41804 41871 41742 41474 41586 41583 41948 41380 41692 41852 41565 41762 41344 41307 41847 41593 41510
0.38%
Cubie: BL 41836 41586 41562 41946 41750 41471 41587 41251 42118 41138 41433 41801 41739 41676 41622 41645 41884 41775 41852 41550 41566 41842 41638 41732
0.39%
Cubie: UFR 41653 41593 42047 41424 41981 41225 41492 41770 41492 42016 41459 41606 41557 41747 41895 41975 41524 41500 41295 41938 41899 41212 41940 41760
0.53%
Cubie: URB 41726 41947 41555 41798 41437 41367 41588 41975 41442 41400 41672 41671 41593 41574 41742 41494 41673 41895 41506 41557 41636 41778 42254 41720
0.37%
Cubie: UBL 41839 41708 41873 41451 41504 42180 41635 41643 41671 41945 41459 41444 41339 41784 41372 41902 41608 41447 41766 41714 41718 41928 41469 41601
0.41%
Cubie: ULF 41797 41719 41298 41473 42050 41536 41701 41621 41944 41680 41788 41622 41377 41324 41636 41645 41642 41778 41869 41840 41751 41802 41572 41535
0.34%
Cubie: DRF 41818 41808 41692 41537 41887 41402 41739 41596 41366 41764 41854 41570 41500 41603 41660 41615 41648 41830 41706 41451 41870 41591 41786 41707
0.29%
Cubie: DFL 41453 41473 41577 41893 41752 41658 41393 41786 41859 41823 41546 41557 41662 41733 41603 41339 41825 41914 41873 41917 41462 41543 41816 41543
0.37%
Cubie: DLB 41701 41940 41798 41934 41390 41590 41559 41625 42116 41563 41537 41582 41626 41331 41858 41754 41509 41704 41805 41575 41582 41615 41748 41558
0.34%
Cubie: DBR 41474 41558 42015 41839 41677 41630 41984 41585 41598 41851 41794 41290 41653 41612 41782 41497 41384 41825 41533 41846 41514 41687 41415 41957
0.38%
AVE: 0.39%

Random q-Turns(30)
Cubie: UF 43167 40905 41797 42045 41142 41239 41376 41550 41175 41492 41736 41599 41343 41533 42590 42706 42148 42354 41257 41733 41387 41116 41400 41210
1.06%
Cubie: UR 43439 42281 41813 41034 41209 41762 41684 41477 41664 41479 41609 41387 42279 42622 42450 42451 41164 41333 40922 41144 40884 41222 41319 41372
1.16%
Cubie: UB 43250 40975 42124 42167 41802 41547 41509 41300 41585 41080 41415 41476 41252 41377 42310 42390 42550 42633 40986 41103 41164 41206 41475 41324
1.18%
Cubie: UL 43370 41931 41968 40799 41438 41482 41664 41324 41261 41335 41699 42096 42398 42008 42537 42621 41329 41134 41445 41098 41215 41332 41472 41044
1.13%
Cubie: DF 43272 40683 42019 41908 41482 41807 41522 41566 41259 41031 41538 41334 41112 41404 42616 42805 42575 42555 41166 41382 41259 41370 41313 41022
1.24%
Cubie: DR 43161 41789 42031 40855 41494 41561 41694 41476 41160 41169 41343 41313 42583 42702 42536 42532 41217 41456 41215 41576 41301 41458 41064 41314
1.14%
Cubie: DB 42844 40480 41930 42323 41358 41234 41297 41707 41341 41696 41202 41561 41455 41506 42665 42706 42650 42573 40957 41072 41307 41791 41173 41172
1.24%
Cubie: DL 43539 41854 42088 40584 41593 41417 41751 41379 41622 41328 41532 41727 42298 42413 42499 42551 41108 41349 41344 41066 41329 41322 40838 41469
1.14%
Cubie: FR 43045 42371 40859 42003 41624 41597 41386 41095 41169 41299 41693 41362 42310 42661 41039 41753 42334 42484 41502 41616 41488 41177 41073 41060
1.13%
Cubie: FL 43124 42126 40541 41974 41306 41279 41420 41394 41716 41201 41556 41371 42765 42488 40881 41443 42657 42564 40833 41502 41423 41656 41112 41668
1.22%
Cubie: BR 43619 42383 40867 41968 41573 41407 41260 41274 41510 41580 41219 41132 42697 42092 41241 41445 42117 42483 41269 41434 41407 41450 41271 41302
1.14%
Cubie: BL 43389 41789 41173 42067 41238 41290 41327 41430 41722 41505 41368 41541 42320 42296 41046 41439 42960 42217 41289 41365 41290 41184 41237 41518
1.09%
Cubie: UFR 42118 41327 41663 41742 41788 41553 41379 41748 41646 41668 41550 41597 42132 41774 41545 41833 41739 41692 41562 41563 41626 41535 41786 41434
0.34%
Cubie: URB 41894 41699 41596 41281 41553 41625 41693 41897 41666 41842 41816 41388 42006 41706 41654 41945 41452 41523 41554 41650 41843 41681 41797 41239
0.36%
Cubie: UBL 41664 41263 41430 41598 41887 41213 42040 41576 41790 41630 41487 41642 41846 41657 41882 41599 41709 41616 41549 41728 41863 41759 41691 41881
0.35%
Cubie: ULF 41249 41827 41536 41558 41762 41946 41303 41464 41768 41628 41680 41424 41782 42243 41464 41666 41784 41058 41816 41668 42129 41733 41568 41944
0.48%
Cubie: DRF 41635 41492 41697 41566 41601 42137 41550 41797 41545 41497 41900 41658 41789 41812 41924 41559 41883 41746 41585 41622 41693 41281 41433 41598
0.34%
Cubie: DFL 41646 41842 41695 41224 41790 41810 41786 41534 41573 41652 41792 41585 41732 41588 41426 41977 41677 41457 41748 41698 41395 41887 41961 41525
0.35%
Cubie: DLB 41643 41701 41883 41650 41691 41375 41498 41770 41541 41906 41696 41479 41660 41626 41981 41647 41870 41818 41262 41609 41461 41605 41715 41913
0.32%
Cubie: DBR 41590 41758 41662 41691 41917 41754 41563 41476 41685 41734 41496 41749 41631 41693 41540 41592 41444 41962 41815 41851 41534 41602 41687 41574
0.26%
AVE: 0.83%

Random q-Turns(50)
Cubie: UF 41716 41748 41562 41400 41521 41918 41726 41924 41941 41670 41334 41253 41513 41191 41957 41699 41476 41759 41852 41842 41914 41978 41377 41729
0.47%
Cubie: UR 41820 41538 41601 41440 41898 41858 41519 41651 41244 41772 41696 41699 41490 41657 41761 41762 42051 41659 41901 41711 41740 41664 41538 41330
0.33%
Cubie: UB 41241 41787 42001 41828 41813 41904 41188 41467 41745 41677 41073 41415 41878 41785 41678 41719 41914 41745 41909 41720 41402 41687 41605 41819
0.46%
Cubie: UL 41519 41532 41457 41677 41857 41699 41364 41640 41711 41798 41692 41753 41761 41689 41639 41815 41981 41441 41931 41472 41536 41555 41681 41800
0.30%
Cubie: DF 41915 41707 41521 41602 41622 41722 41749 41660 41222 41741 41440 41502 41701 41570 42104 41974 41613 41769 41610 41817 41542 41723 41771 41403
0.34%
Cubie: DR 41557 41413 41467 41435 41543 41377 41946 41716 41279 41828 41666 41795 41831 41985 41898 41764 41835 41712 41504 41629 41708 41643 41768 41701
0.36%
Cubie: DB 41962 41936 41171 41406 41765 41495 41886 41592 41578 41560 41919 41700 42188 41508 41682 41449 41522 41469 41980 41625 41731 41766 41377 41733
0.45%
Cubie: DL 41980 41464 41900 41787 41381 41639 41878 41581 41829 41280 41782 41568 41473 41618 41865 41719 41401 41743 41685 41646 41576 41708 41772 41725
0.34%
Cubie: FR 41503 41547 41805 42139 41667 41533 41654 41806 41956 41801 41524 41197 41501 41727 41663 41905 41218 41891 41509 41579 41878 41711 41909 41377
0.44%
Cubie: FL 41681 41699 41507 41992 41205 41638 41516 41701 41529 41582 41588 41985 41536 41929 41456 41703 41739 41762 41730 41900 41631 41478 41539 41974
0.36%
Cubie: BR 41413 41884 41369 41468 41632 41814 41662 42227 42185 41811 41591 41502 41681 41670 41957 41645 41609 41388 41442 41867 41546 41725 41604 41308
0.43%
Cubie: BL 41699 41517 41467 41837 41598 41876 41963 41705 41982 41205 41641 41526 41557 41780 41888 41765 41690 42323 41535 41620 41387 41518 41455 41466
0.43%
Cubie: UFR 41893 41517 41600 41632 41418 41821 41622 41493 41706 41692 41857 41600 41755 41904 41868 41592 41663 41721 41901 41505 41709 41559 41676 41296
0.30%
Cubie: URB 41600 41918 41803 41641 41391 41456 41705 41682 41348 41600 41478 42061 41841 41752 41974 41717 41517 41753 41465 41717 41745 41690 41831 41315
0.37%
Cubie: UBL 41967 41410 41734 41603 41347 41524 41668 41786 41747 41665 41815 41704 41693 41297 41754 41916 41630 41821 41888 41474 41918 41619 41712 41308
0.36%
Cubie: ULF 41747 41868 41362 41230 41737 41543 41814 41904 41670 41362 41503 42024 41603 41347 41876 41325 41624 41785 41470 42186 42183 41525 41571 41741
0.51%
Cubie: DRF 41699 41663 41484 41632 41593 41761 41336 41809 41808 41628 41377 41658 41953 41546 41817 41722 41434 41834 42027 41501 41986 41742 41604 41386
0.36%
Cubie: DFL 41871 41568 41318 41485 41485 41818 41419 41683 41589 41403 41265 42056 42077 41859 41276 41812 41901 41740 42149 41884 41441 41556 41749 41596
0.52%
Cubie: DLB 42027 41830 41824 41702 41707 41818 41767 41497 41696 41657 41471 41912 41657 41765 41626 41359 41692 41557 41649 41637 41584 41566 41380 41620
0.28%
Cubie: DBR 42084 41793 41810 41587 41548 41485 42078 41576 41997 41543 41625 41896 41424 41642 41844 41465 41479 41864 41419 41719 41810 41400 41615 41297
0.45%
AVE: 0.39%

Random q-Turns(100)
Cubie: UF 41876 41562 41705 41431 41788 41793 41698 41733 41326 41436 41554 41508 41598 41750 41688 42102 41815 41688 41928 41414 41541 41680 41992 41394
0.38%
Cubie: UR 41743 41944 41437 41495 41309 41995 41748 41572 41760 41375 41777 41659 41633 41717 41635 41919 41341 41500 41825 41714 41710 41786 42035 41371
0.40%
Cubie: UB 41748 41868 41453 41747 41551 41535 41917 41708 41647 41523 41922 41804 41748 41637 41613 41704 41767 41543 41309 41695 41690 41428 41555 41888
0.31%
Cubie: UL 42020 41600 41711 41734 41687 41423 41381 41758 41948 41768 41575 41619 42031 41594 41505 41576 41748 41424 41639 41448 41512 41966 41487 41846
0.38%
Cubie: DF 41791 41359 41877 41871 41452 41755 41747 41613 41921 41657 41825 41555 41387 41680 41727 41825 41448 41485 41301 41916 41440 41812 41704 41852
0.39%
Cubie: DR 41443 41934 41748 41646 41509 41602 41859 42172 41336 41876 41888 41652 41650 41976 41670 41609 41618 41247 41841 41607 41370 41597 41307 41843
0.43%
Cubie: DB 41419 41826 41651 41791 41769 41776 41622 41432 41903 41785 41673 41694 41688 41558 41737 41567 41552 41748 41485 41705 41411 41640 42091 41477
0.30%
Cubie: DL 41933 42010 41497 41243 41653 41528 41560 41928 41538 41607 41513 41556 41783 41591 41955 41624 41967 41432 41724 41825 41975 41443 41602 41513
0.42%
Cubie: FR 41573 41594 41956 41691 41795 41807 41183 41532 41805 41613 41733 41545 41803 41767 41953 41419 41874 41278 41560 41752 41377 41802 41790 41798
0.40%
Cubie: FL 41634 41592 41206 41723 41796 41586 41656 41736 41434 41666 41913 41897 41765 41671 41512 41689 41374 41900 41309 41543 41818 41704 42058 41818
0.36%
Cubie: BR 41469 41702 41626 41969 41576 41603 41706 41969 41984 41649 41224 41785 41500 41825 41817 41636 41367 41664 41350 41765 41406 41736 41657 42015
0.39%
Cubie: BL 42010 41537 41623 41471 41374 41762 41583 41692 41475 41493 41792 41896 41266 41710 41735 41816 41780 41924 41950 41512 41643 41378 41729 41849
0.40%
Cubie: UFR 41691 41706 42079 41708 41366 41484 41522 41656 41684 41750 41544 41791 41628 41747 41992 41349 41321 41513 41738 41817 41870 41670 41840 41534
0.35%
Cubie: URB 41442 41510 41638 41553 41994 41294 41477 41505 42014 41463 41359 41614 42151 41863 41789 41700 41707 41652 41693 41413 42083 41643 41676 41767
0.42%
Cubie: UBL 41530 41729 41758 41605 41569 41643 41549 41453 41633 41720 41787 41481 41616 41625 41800 41896 41366 41639 41726 41891 41877 41556 41837 41714
0.28%
Cubie: ULF 41413 41669 41632 41968 41514 41669 41854 42068 41619 41775 41618 41804 41289 41416 41357 41905 41746 41594 41426 41788 41715 42234 41479 41448
0.44%
Cubie: DRF 41924 41562 41084 41633 41781 41853 41710 41542 42080 41442 41781 41589 41584 41752 41968 41439 41922 41933 41478 41457 42028 41381 41489 41588
0.48%
Cubie: DFL 41601 41633 41517 41731 42055 41998 41831 41925 41189 41610 41682 41599 42103 41227 41521 41801 41831 41602 41790 41583 41552 41582 41753 41284
0.43%
Cubie: DLB 41605 41609 41611 41589 41483 41509 41843 41483 41583 41930 41757 41633 41626 41703 41806 41523 41506 41763 42008 41542 41794 41557 41781 41756
0.29%
Cubie: DBR 41827 41970 41743 41283 41752 41183 41498 41711 41543 41757 41568 41438 41434 41726 41735 41711 41591 41808 41678 41933 41895 41779 41760 41677
0.36%
AVE: 0.38%

Last edited:

#### Lucas Garron

##### Member
If you're interested in a more thorough analysis, I suggest looking at Stefan Pochmann's 3x33x scramble analyzer:
http://www.stefan-pochmann.info/spocc/other_stuff/tools/

There are some good ways to generalize this to handle e.g. take into account move cancellation or analyze other puzzles, but unfortunately I haven't seen many people doing that.