# Number Of Combinations

How many combinations does the Rubik’s cube have? It’s easy to find out how many the 3x3x3 has, but when I looked, there were precious few pages that showed the number of combinations for all the sizes from 2x2x2 to 7x7x7. So, I made this page, listing the numbers of combinations for various sized cubes. There’s also a javascript calculator in case you want to figure it out for larger sizes.

• The 2x2x2 Rubik’s cube (called the Pocket Cube) has 3674160 combinations. This is a manageable number. If you fiddle with the 2x2x2 cube randomly, eight hours a day continuously, you’ll solve it by pure chance roughly two or three times per year. Assuming your cube – or your wrist – doesn’t break in the meantime. Mind you, four months to solve the 2×2 cube is somewhat slower than the world record.
• The original 3x3x3 Rubik’s cube has 43 252 003 274 489 856 000 combinations, or 43 quintillion. Again, as pointed out on this website’s main page, this is a manageably imaginable number. It’s a little less than the square of the earth’s population, for example.
• The 4x4x4 Rubik’s cube (called the Master Cube, or Rubik’s Revenge – not sure who he was avenging, I must say) has 7 401 196 841 564 901 869 874 093 974 498 574 336 000 000 000 combinations (that is, 7.4 quattuordecillion, if you really wanted to know). To understand how big this number is, imagine you had this many teaspoons of sugar (say you’re planning a really big tea party). The sugar would fill the solar system out to about 3.5 times the orbit of Pluto. It would also weigh about 70 times as much as our galaxy, and instantly collapse into a black hole with an explosion that would wipe out the Milky way, the Magellanic clouds, and probably wake up some sleepy Andromedans as well. Think about that next time you twist the 4×4.
• As if that’s not enough, the 5x5x5 Rubik’s cube (called the Professor’s Cube) has 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 combinations (aka 283 trevigintillion). This is getting uncomfortably close to the number of atoms in the known universe.
• Recently, a Greek engineer Panagiotis Verdes figured out how to make 6×6 and 7×7 cubes. The V-Cube 6 (a 6x6x6 Rubik’s cube) has 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000 000 000 000 000 000 000 combinations. This is a ridiculously humungous number, of course, but…
• The 7x7x7 Rubik’s cube (the V-Cube 7) has 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 combinations. As I point out in this movie, that’s more combinations than eight independent 3x3x3 cubes. And yet, some people can still solve the 7x7x7 in just a few minutes. Amazing!

Nobody’s built and marketed any larger cube than that, although Panagiotis Verdes promises that some larger ones are in the pipeline. In the meantime, there are software programs that will let you play with any sized cube. These larger sizes are no harder to solve than the 6×6 and 7×7, just more tedious.

• An 8x8x8 cube would have 35 173 780 923 109 452 777 509 592 367 006 557 398 539 936 328 978 098 352 427 605 879 843 998 663 990 903 628 634 874 024 098 344 287 402 504 043 608 416 113 016 679 717 941 937 308 041 012 307 368 528 117 622 006 727 311 360 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 combinations.
• A 9x9x9 cube would have 14 170 392 390 542 612 915 246 393 916 889 970 752 732 946 384 514 830 589 276 833 655 387 444 667 609 821 068 034 079 045 039 617 216 635 075 219 765 012 566 330 942 990 302 517 903 971 787 699 783 519 265 329 288 048 603 083 134 861 573 075 573 092 224 082 416 866 010 882 486 829 056 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 combinations.
• A 10x10x10 cube would have 82 983 598 512 782 362 708 769 381 780 036 344 745 129 162 094 677 382 883 567 691 311 764 021 348 095 163 778 336 143 207 042 993 152 056 079 271 030 423 741 110 902 768 732 457 008 486 832 096 777 758 106 509 177 169 197 894 747 758 859 723 340 177 608 764 906 985 646 389 382 047 319 811 227 549 112 086 753 524 742 719 830 990 076 805 422 479 380 054 016 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 combinations. Youch!

If you want to find the number of combinations for a larger-sized cube, you can do so using the form below. I should warn you that if you type in a ridiculous size for the cube (say more than about 30), it will bog down your computer, not mine.

Note – due to bugs in Internet Explorer, you may need to use Google Chrome or Mozilla Firefox to get this form to fix – until I can figure out a way to work around the bugs. Anyway, Chrome and Firefox are much better than IE, so it’d be a mistake for you not to get them.

 Enter N :

Enjoy!

## 33 thoughts on “Number Of Combinations”

1. Brian Flynn says:

I think the math is off on this. The puzzle is not a 3x3x3 puzzle. It’s 8 corner pieces that can change positions and be oriented 3 ways combined with an independent 10 side piece puzzle that can be oriented in 2 ways. I don’t understand why, but part of the geometry of the puzzle dictates that no piece can rotate independently – any piece that does must rotate another piece. I think this effectively cuts the number of possibilities in half.

To simplify this down to the 2×2 puzzle, you have 8 pieces to make up the puzzle. Each puzzle piece can have 3 orientations. So the answer is the number of different places to put the puzzle pieces times the number of different combinations of orientations that the pieces can have (divided by 2 because of the rotation dependency).

I’m still struggling with the right math, but I think this it. There are 8! different ways to arrange the 8 corner pieces. Each piece can have 3 orientations, which is 3^8 different ways to combine them. So, I get a much larger number at 40320 * 6561 / 2 = 132,269,760 different positions that the 2×2 puzzle can have.

Expanding this to the 3×3 is actually the 2×2 puzzle times a 10 side piece / 2 orientation puzzle calculated the same and still with the same rotation dependency. So that is the 2×2 answer * (3,628,800 * 1024 / 2) which is 245,750,018,605,056,000 (245.75 quadrillion), significantly less than a 3x3x3 grid answer. But still really big.

Not 100% sure on this, but my best guess – I know the 3x3x3 is out because that isn’t how the cube is made up.

1. TheRubikZone says:

Well, for the 2x2x2, the eight corner pieces can’t be all independently rotated – you can choose an arbitrary rotation for seven of them, and then the 8th is determined. So that’s 8! x 3^7.

And for the 3x3x3, why do you say there are ten edge pieces? Surely you mean twelve?

2. I am sure about your calculation for 2*2*2 cube but we are wrong 😂
And i had alrady calculate about this i had get the number 4377107200 i think 2*2*2 cube cane have these combination but I don’t know. I will try it again. I will get the correct answer then i will calculate about 3*3*3 cube. I am saying good luck to me. 😂😂🤣

3. r says:

Apparently “The 2.5×2.5×2.5 cube has 7 338 606 900 448 337 036 789 808 927 714 722 374 232 054 265 556 867 083 216 987 563 605 831 302 280 124 954 511 318 020 360 031 738 952 123 850 689 949 301 067 016 810 098 821 782 952 622 119 827 191 343 811 714 153 308 174 763 344 519 829 765 740 744 966 395 281 285 250 907 162 807 467 469 411 206 193 386 044 382 177 920 656 626 556 592 406 989 586 319 048 297 532 171 232 195 980 524 608 549 470 654 382 377 337 629 683 696 121 250 485 203 852 784 671 253 465 552 723 661 921 197 323 812 277 759 207 873 014 075 438 873 998 083 588 200 340 665 392 813 884 974 033 548 770 683 966 539 756 225 190 776 056 150 183 488 334 560 717 199 712 278 700 837 482 528 510 188 957 224 739 072 757 519 973 451 653 988 332 028 901 106 023 947 653 492 013 122 471 978 273 081 760 332 246 552 295 633 523 901 575 290 421 730 800 211 251 366 731 783 357 835 649 641 682 382 571 868 916 849 108 307 173 498 957 116 523 350 145 921 674 054 778 826 517 754 760 789 955 404 188 878 815 975 983 885 256 352 902 632 633 873 020 662 232 301 766 339 543 018 349 634 680 668 362 283 438 183 702 813 228 324 979 071 338 435 402 301 910 372 014 568 598 527 037 916 456 835 524 456 894 773 921 188 526 831 393 779 186 341 836 593 724 125 208 353 829 100 559 876 542 082 775 503 510 870 799 705 631 415 086 583 990 681 065 679 299 057 922 416 132 958 730 491 243 560 060 356 773 368 590 520 847 306 709 113 021 275 033 399 321 328 976 746 385 764 292 971 315 981 681 607 381 127 459 115 526 117 981 354 672 920 424 342 659 383 996 123 548 655 976 674 499 673 057 309 342 414 853 636 864 217 778 175 504 956 701 789 630 256 547 841 517 726 192 968 093 645 888 423 683 924 945 743 234 939 386 117 107 128 406 986 222 138 769 394 079 967 354 094 120 661 761 182 197 876 722 276 455 719 218 131 307 283 520 932 301 820 550 597 642 037 727 788 159 125 117 410 163 828 735 097 141 678 436 524 639 308 956 176 346 087 213 881 095 179 847 923 513 780 445 012 559 343 543 898 821 960 560 224 764 734 838 087 057 026 877 946 627 114 988 892 818 954 861 787 433 838 722 015 223 207 055 499 970 664 774 362 372 935 907 468 804 546 985 510 049 528 366 673 121 962 024 926 187 502 872 847 716 756 747 755 025 480 612 185 841 109 985 394 393 975 911 931 963 302 893 074 202 797 834 753 563 717 887 841 509 506 339 744 669 847 853 737 346 773 669 138 655 220 124 613 894 314 393 600 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 combinations.”

2. Paul Drewitz says:

How about Apple Safari! Ask me and I’ll tell you it takes not even a full-second when entering 100!

3. kurt says:

Brian Flynn’s math is way off.

4. Sam Allen says:

A commonly asked question is “How many ways can a 3 x 3 x 3 Rubik’s Cube be arranged” and the answer is usually given as 43,252,003,274,489,856,000. I should like to put the question in a different way, “Given a Rubik’s cube in its solved position, in how many ways can you scramble it?”

The cube has six faces and so there are 6! ways of choosing the order in which to twist the faces. Each face has four positions, so I get 6! x 4^6 = 2,949,120 ways of scrambling the cube. I feel that this should be an upper limit to the number of legal combinations of the squares, since the only way to get a legal combination is by twisting the faces.

1. TheRubikZone says:

You’ve assumed you can only twist each face once – but that’s not the case. After you do your first twists, you can do more on faces already twisted to access even more combinations.

1. Rubix Master says:

I’m confused, how’d you do this?

2. Nicholas Anyonymouslastnamelol says:

Yes I completely agree

3. Scott says:

I agree. If the cube has pictures or figures you will see that the center pieces are rotated from their original position. Just the combination of the center pieces is 4096. Ignoring this factor significantly reduces the number of patterns for a cube with solid colors

5. Boris says:

Hi,
I am working on a school subject about this topic and I came up with my own formula.
Can you send me your formule for N? I am curious if we came up with the same formula…

Boris

6. tomas says:

i try 1000,my computer die

7. Jack Weber says:

The 100×100 Rubixcube would would have a LUDICROUS number of possible combinations! Try it for yourself.

8. Oshimimers says:

Wait… has anybody else tried a decimal number like 2.5? It comes out to a huge number, which i don’t think makes any sense

9. Isn’t there one thing that, so far, all the websites I’ve seen claiming 43 quintillion combinations have missed? I.e. that the whole cube itself has various orientations, and therefore though one combination may look totally different to another on the face of it, if you turn the whole cube to left or right or up/down it could be the same as another combination in that position. Surely this must limit the 43 quint coms somewhat?
Mark Townsend

10. Robert Ambalong says:

Are you talking about the properly assembled Rubiks cube that can be solved? If you do, your computation of the number of combinations is wrong, because it includes combinations where it is not possible to exist. As an example, if you dismantle the cube and put the pieces back randomly, there is a greater chance that it cannot be solved. In this case, any combinations derived from this improperly assembled cube is not counted as a valid combination.

11. Ally says:

How many methods are their to solve the rubicks cube and how can I learn them?

12. Ally says:

Also what’s the formula to figure this out?

13. Richard L. says:

Hi, I have a question. Does the math accounts for orientation of the cube? What I mean is, if I put a solved cube on a table with white on top and red facing me. I rotate the cube 90 degree and now have blue facing me. In the math, is that considered a different combination? We can look at the same solved cube 24 different ways. Is that 24 combinations? If I do one move. Is that only one new combination or 24 new combinations?

Also on a 3x3x3, do the different rotation positions of the center piece counts as different combinations?

14. Layne says:

If you merely spin the sides of a cube, you will create different combinations. You can also disassemble and reassemble a cube one piece at a time to create many more combinations. A cube that has been modified by reassembly cannot be solved. The limitation of only “scrambling” a cube, reduces​ the true number of possible combinations. See the Numberphile video published Sep. 7,2012. Cheers!

15. RandomCuber says:

Post needs an update. We have the MF8 and MF9, Yuxin Huanglong 10×10 And 11×11, Moyu 13×13 and 15×15, and the largest mass-produced is the Yuxin Huanglong 17×17. Also, we have the largest cube which is 33×33 built by Greg’s Puzzle, and before that was the 22×22 built by someone else I forgot the name of.

Yeah this post needs a *slight* update.

16. Geckomayhem says:

Are you taking into account the fact that there are nine of each colour? So essentially, if one green piece is in one spot, if another green piece were to take its place it couldn’t be counted as a different combination: it’s still green.

17. Rubix Master says:

Here is how I would go about solving the permutations. For a two by two. There are 8 corner pieces which can be rearranged in 8! ways since a piece goes in one spot and another piece has that number of spots – 1 since it is now filled.

So 8! would have to be multiplied by 3^8 since all 8 pieces can be orientated in 3 ways. Then, divide that number by 3 since that is the number of parities the corner piece can be in, 2 of which are impossible.

This gives us an equation of 8! * 3^8 ÷ 3 leaving us with 88179840.

I do not understand how this website has done it but this looks good to me 🙂

18. Cube Science says:

This is how to permutate.

Think of a 2 by 2 cube. It has 8 corners which can be rearranged in 8! combinations. We then multiply by 3^7 since the 7 pieces can individually rotate. We then divide this by 24 because there are 24 possible parities.
(8! * 3^7) / 24

A 3 by 3 cube uses the same 8! * 3^7 because the corners will always hold this property of rearrangement and individually rotating. We multiply this by 12! and 2^10 because it has a fixed center and 12 edge pieces which can be rearranged in any order and 10 pieces can individually flip.
(8! * 3^7 * 12! * 2^10)

I will explain more on how it works later. Most of it is about dividing the edge and center count to groups of 24 and dividing because of the groups having 4 of each 6 color.

19. Joe Harrison says:

How did you come up with the number for solving a 2x2x2 with a random walk?

20. Sylwia says:

I entrered 50… I think thats infinity

21. The generator wasn’t able to load for me, how many digits (hopefully the full numbers if there’s a website or something) are in a 100x100x100 rubik cube?