Mathematics

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Note: this page is under construction.

This page lists some mathematical properties of multi-dimensional puzzles, mostly numbers of their positions.

There may be non-mathematicians reading this, so here is an introduction to these issues; however, some calculations may be of more advanced level:
If we have a pieces, we can permute them a! ways; this can be easily shown: suppose we remove all the pieces. If we are placing the first one, there are a ways to do so. For the second piece, there are, however, only a − 1, since one is already occupied by the first piece, and both these pieces have together a × (a − 1) permutations as there are a − 1 positions of the second piece per each of the a positions of the first piece. If we continue this way, it becomes clear that there are a × (a − 1) × (a − 2) × ... × 3 × 2 × 1 (we actually have no choice for the last a-th piece), which is conventionally denoted as a!.
Now, each of the a pieces can be oriented b if it stays in place. This means that there will be ba ways to only orient the pieces if we do not permute them, because there are b orientations of the first piece per b orientations of the second piece etc.
Multiplying these numbers should give us the total number of a puzzle’s positions (there are ba orientations per each of a! permutations), but it often happens that not all of them are reachable by using legal moves, and we have to divide this figure due to constraints (if the pieces’ permutations have for example even parity, the permutation constraint c = 2 because only half of the permutations are attainable (even). With regard to orientations, we can say that all but last pieces have ba − 1 orientations in total and it may happen that the last piece cannot reach all orientations, so it has only b/d, where d is the orientation constraint. Also, if there are some pieces that are not distinguishable from each other and we swap them, the change will not be visible, and we therefore regard them as the same position. If there are sets of e indistinguishable pieces, we have to divide by e!a/e as a consequence, because the e! possible permutations of a set are not distinguishable and there are (logically) a/e such sets).

The general structure of the data presented here is of this form:

  • n-coloured pieces type X: count (a); number of orientations (b); permutation constraint (c); orientation constraint (d); indistinguishability constraint (e)

The position count of a row is then a!/(c × e!a/e) × (ba)/d.
Number of positions of the whole puzzle is the product of position counts of all its rows.

The pieces are divided first by number of colours and then by types, which are determined by orbits – a piece in a given type can reach the positions of all other pieces in that type by legal moves.
The types are listed in such order that they go “from centre”.
They are named based on which feature of the shape are they in, so for example on tesseract, “1-coloured type 1.3” means that it is on face (1) of a cube and on that face it is in the corner (3). “Two-coloured type 2.2” signifies that it is on edge of a square and that it is alternative (2; just to distinguish between it and type 2.1, because they behave differently). Subscripts are added to number pieces which would get the same type.
When listing general properties of a class of puzzles, it is first noted how many times does the type appear.

Values in parentheses are a “common constraint”, and are counted as one. This happens when more types of pieces have a given parity together, so that one may for example perform only odd permutations of both or even permutations of both. This would result in c = 2, counted only once despite applying to two types.
When is a whole type or number of some pieces is in parentheses and italics, it means that (some of) those pieces are there, but are immobile. By “mobile”, I mean permutable and/or orientable, that is, mobile are pieces that can change their state.
Numbers of pieces in square brackets denote the impossibility of permuting this type of pieces.

Some puzzles have no fixed reference points, and it is necessary to include a “puzzle orientation constraint”, because we counted all its positions in all of the puzzle’s orientations. This constraint is equal to the number of orientations of the whole m-dimensional shape. This can also be viewed as fixing one piece in place.

Numbers in this page are named according to Conway’s and Guy’s naming scheme extended in Saibian’s fashion when necessary.

Calculated by Jakub Štepo unless stated otherwise. Please note that some of the results may be unverified, as they are based on theoretical predictions rather than actual solving.

MagicCube4D

{3,3,3}

  • Shape: regular 5-cell (pentachoron)
  • Cells (colours): 5 regular tetrahedra {3,3}
  • Faces: 10 equilateral triangles {3}
  • Edges: 10
  • Vertices: 5

Length 2

  • 4-coloured: type 1: [5]; 12; 1; 1; 1
  • (5-coloured: 1)
  • Total pieces: 5 (6)
  • Total stickers: 25

Number of positions:
125 =
= 248 832 ≈
≈ 2.49 × 105
= 248 thousand 832

Length 3

  • 3-coloured: type 1: 10; 6; 2; 2; 1
  • 4-coloured: 10
    • Type 1: [5]; 12; 1; 1; 1
    • Type 2: [5]; 12; 1; 1; 1
  • Total pieces: 20
  • Total stickers: 70

Number of positions:
10!/2 × 610/2 × (125)2 =
= 3 396 471 743 308 934 991 052 800 ≈
≈ 3.40 × 1024
≈ 3 septillion 396 sextillion (short scale) / 3 quadrillion 396 trilliard (long scale)

Length 4

  • 2-coloured: type 1: 10; 2; 2; 2; 1
  • 3-coloured: 30
    • Type 1: 10; 6; 2; 2; 1
    • Type 2: 20; 3; 2; 3; 1
  • 4-coloured: 10
    • Type 1: [5]; 12; 1; 1; 1
    • Type 2: [5]; 12; 1; 1; 1
  • Total pieces: 50
  • Total stickers: 150

Number of positions:
10!/2 × 210/2 × 10!/2 × 610/2 × 20!/2 × 320/3 × (125)2 =
= 4 460 971 667 252 991 547 434 208 214 041 871 442 189 607 102 945 689 600 000 000≈
≈ 4.46 × 1060
≈ 4 novemdecillion 461 octodecillion (short scale) / 4 decillion 461 nonilliard (long scale)

Length 5

  • 1-coloured: type 1: 5; 1; 2; 1; 1
  • 2-coloured: 40
    • Type 1: 10; 2; 2; 2; 1
    • Type 3: 30; 2; 1; 2; 3
  • 3-coloured: 50
    • Type 1: 10; 6; 2; 2; 1
    • Type 21: 20; 3; 2; 3; 1
    • Type 22: 20; 3; 2; 3; 1
  • 4-coloured: 10
    • Type 1: [5]; 12; 1; 1; 1
    • Type 2: [5]; 12; 1; 1; 1
  • Total pieces: 105
  • Total stickers: 275

Number of positions:
5!/2 × 10!/2 × 210/2 × 30!/(3!10) × 230/2 × 10!/2 × 610/2 × (20!/2 × (320)/3)2 × (125)2 =
= 891 244 004 975 919 897 976 748 360 350 536 026 444 717 921 800 196 028 281 830 709 220 726 284 058 861 218 760 784 054 113 171 564 134 400 000 000 000 000 000 000 ≈
≈ 8.91 × 10122
≈ 891 noventrigintillion 244 octotrigintillion (short scale) / 891 vigintillion 244 novendecilliard (long scale)

{4,3,3}

  • Shape: tesseract
  • Cells (colours): 8 cubes {4,3}
  • Faces: 24 squares {4}
  • Edges: 32
  • Vertices: 16

Length n, n ≥ 2:

  • 1-coloured: ((n − 2)3 − n mod 2) × 8 ((n − 2)3 × 8)
    • (Type 0: 8 n mod 2)
    • Type 1.1: 48; 1; 1; 1; 6; × (n − 3)/2 × n mod 2
    • Type 1.2.1: 192; 1; 1; 1; 24; × (n − 5)(n − 3)/2 × n mod 2
    • Type 1.2.2: 192; 1; 1; 1; 24; ×⌊(‘'n − 6)/2⌋⌊(n − 4)/2⌋⌊(n − 2)/2⌋/3
    • Type 1.3: 192; 1; 1; 1; 24; × ⌊(n − 4)/2⌋⌊(n − 2)/2⌋/2
    • Type 2.1: 96; 1; 1; 1; 12; × (n − 3)/2 × n mod 2
    • Type 2.2: 192; 1; 1; 1; 24; × ⌊(n − 4)/2⌋⌊(n − 2)/2⌋/2
    • Type 3: 64; 1; 1; 1; 8; × ⌊(n − 2)/2⌋
  • 2-coloured: (n − 2)2 × 24
    • Type 1: 24; 2; 2; 2; 1; × n mod 2
    • Type 2.1: 96; 2; 1; 2; 4; × (n − 3)/2 × n mod 2
    • Type 2.2: 192; 1; 1; 1; 4; × ⌊(n − 4)/2⌋⌊(n − 2)/2⌋/2
    • Type 3: 2; 1; 2; 4; × ⌊(n − 2)/2⌋
  • 3-coloured: (n − 2) × 32
    • Type 1: 32; 6; 2; 2; 1; × n mod 2
    • Type 2: 64; 3; 2; 3; 1; × ⌊(n − 2)/2⌋
  • 4-coloured: 16; 12; 2; 3; 1; × 1
  • Puzzle orientation constraint: 192; × (n + 1) mod 2
  • Total pieces: n4 − (n − 2)4 - n mod 2 (n4 − (n − 2)4)
  • Total stickers: 8n3

Number of positions:
((((48! × 96!2 × 296)/(6!8 × 12!8 × 4!24 × 2))(n − 3)/2 × (24! × 32! × 224 × 632)/(23))n mod 2 × (192!/(24!8))(n − 5)(n − 3)/2 × n mod 2 + ⌊(n − 4)/2⌋⌊(n − 2)/2⌋⌊n/2⌋/3 × ((64!2 × 364)/(8!8 × 2 × 3))⌊(n − 2)/2⌋ × (192!/(4!48))⌊(n − 4)/2⌋⌊(n − 2)/2⌋/2 × (16! × 1216)/(2 × 3))/(192(n + 1) mod 2)

Length 2

  • 4-coloured: 16; 12; 2; 3; 1
  • Puzzle orientation constraint: 192
  • Total pieces: 16
  • Total stickers: 64

Number of positions:
(16!/2 × 126/3)/192 =
= 3 357 894 533 384 932 272 635 904 000 ≈
≈ 3.36 × 1027
≈ 3 octillion 358 septillion (short scale) / 3 quadrilliard 358 quadrillion (long scale)

Length 3

For more details, see Mathematics/Length-3 Tesseract.

  • (1-coloured: type 0: 8)
  • 2-coloured: type 1: 24; 2; (2); 2; 1
  • 3-coloured: type 1: 32; 6; (2); 2; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Total pieces: 72 (80)
  • Total stickers: 216

Number of positions:
(24! × 32!)/2 × 224/2 × 632/2 × 16!/2 × 1216/3 =
= 1 756 772 880 709 135 843 168 526 079 081 025 059 614 484 630 149 557 651 477 156 021 733 236 798 970 168 550 600 274 887 650 082 354 207 129 600 000 000 000 000 ≈
≈ 1.76 × 10120
≈ 1 noventrigintillion 757 octotrigintillion (short scale) / 1 vigintillion 757 novendecilliard (long scale)

Symmetry

Here are numbers of positions symmetric under some conjugacy class, using Greg Egan’s notation:

  • e: 1 756 772 880 709 135 843 168 526 079 081 025 059 614 484 630 149 557 651 477 156 021 733 236 798 970 168 550 600 274 887 650 082 354 207 129 600 000 000 000 000
  • (1,−)4: 11 497 557 803 313 571 701 881 319 062 903 855 825 682 866 660 890 902 528 000 000
  • (1,−)2: 6 271 395 165 443 766 382 844 355 852 493 012 268 554 290 905 940 492 288 000 000
  • (2,+): 426 893 024 140 465 883 454 209 890 713 600
  • (1,−)2(2,+): 71 148 837 356 744 313 909 034 981 785 600
  • (2,+)2: 106 723 256 035 116 470 863 552 472 678 400
  • (2,−)2: 149 318 932 510 565 866 258 198 948 868 881 244 489 387 878 712 868 864 000 000
  • (1,−)(2,−): 127 750 642 259 039 685 576 459 100 698 931 731 396 476 296 232 121 139 200 000
  • (3,+): 1 237 680 706 117 919 967 859 807 513 199 071 199 232 000
  • (1,−)(3,−): 43 129 799 915 034 095 124 480
  • (4,+): 230 844 665 274 826 752
  • (1,−): 1 856 873 273 785 608 466 117 989 769 149 838 721 779 822 477 836 435 975 045 120 000 000
  • (1,−)3: 137 970 693 639 762 860 422 575 828 754 846 269 908 194 399 930 690 830 336 000 000
  • (2,−): 11 911 481 795 714 655 997 805 044 354 212 748 848 156 298 016 980 992 000 000
  • (1,−)2(2,−): 34 492 673 409 940 715 105 643 957 188 711 567 477 048 599 982 672 707 584 000 000
  • (1,−)(2,+): 426 893 024 140 465 883 454 209 890 713 600
  • (2,−)(2,+): 213 446 512 070 232 941 727 104 945 356 800
  • (3,−): 32 347 349 936 275 571 343 360
  • (1,−)(3,+): 1 572 081 206 902 992 767 287 296
  • (4,−): 1 280 679 072 421 397 650 362 629 672 140 800

Dividing their sum by 384 (the total number of symmetries of the tesseract) gives us
4 574 929 376 846 707 924 918 036 664 273 502 759 412 720 391 014 473 055 557 864 106 301 875 758 650 990 456 653 060 234 022 928 953 153 029 428 983 365 632 ≈
≈ 4.57 × 10117
≈ 4 octotrigintillion 575 septentrigintillion (short scale) / 4 novendecilliard 575 novendecillion (long scale)
essentially different positions of this puzzle up to symmetry.

Antisymmetry

The number of purely antisymmetric (without additional symmetry operations; self-inverse, order 2) positions of this puzzle is found to be equal to
1 514 851 187 547 945 564 174 052 809 349 480 746 221 364 817 706 402 235 357 461 479 424 ≈
≈ 1.51 × 1066
≈ 1 unvigintillion 515 vigintillion (short scale) / 6 undecillion 515 decilliard (long scale).

Length 4

  • 1-coloured: type 3: 64; 1; 1; 1; 8
  • 2-coloured: type 3: 96; 2; 1; 2; 4
  • 3-coloured: type 2: 64; 3; 2; 3; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Puzzle orientation constraint: 192
  • Total pieces: 240
  • Total stickers: 512

Number of positions:
(64!/(8!8) × 96!/(4!24) × 296/2 × 64!/2 × 364/3 × 16!/2 × 1216/3)/192 =
= 130 465 639 524 605 309 368 634 620 044 528 122 859 025 488 438 611 959 323 482 221 544 701 493 566 589 669 139 598 204 956 926 940 147 059 366 252 849 247 482 898 636 104 705 417 194 760 866 897 307 590 845 202 461 293 100 468 293 214 262 958 591 194 739 437 727 430 945 469 384 490 361 714 647 847 550 801 897 750 293 894 453 665 815 572 829 257 758 907 425 128 919 808 862 616 259 604 997 210 112 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 1.30 × 10344
≈ 130 tredecicentillion 466 duodecicentillion (short scale) / 130 septenquinquagintillion 466 sesquinquagintilliard (long scale)

Length 5

  • 1-coloured: 208 (216)
    • (Type 0: 8)
    • Type 1.1: 48; 1; 1; 1; 6
    • Type 2.1: 96; 1; 1; 1; 12
    • Type 3: 64; 1; 1; 1; 8
  • 2-coloured: 216
    • Type 1: 24; 2; (2); 2; 1
    • Type 2.1: 96; 2; 1; 2; 4
    • Type 3: 96; 2; 1; 2; 4
  • 3-coloured: 96
    • Type 1: 32; 6; (2); 2; 1
    • Type 2: 64; 3; 2; 3; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Total pieces: 536 (544)
  • Total stickers: 1 000

Number of positions:
48!/(6!8) × 96!/(12!8) × 64!/(8!8) × (24! × 32!)/2 × 224/2 × 632/2 × (96!/(4!24) × 296/2)2 × 64!/2 × 364/3 × 16!/2 × 1216/3 =
= 123 657 056 923 899 002 698 227 805 778 387 808 933 769 666 084 597 331 170 345 244 675 638 825 481 620 700 008 237 306 084 142 730 598 637 705 860 008 300 844 182 287 747 674 018 136 874 315 751 080 178 664 887 107 264 876 848 935 590 538 625 767 958 284 656 419 396 560 246 923 935 065 962 447 405 384 165 866 873 326 263 467 921 778 683 862 961 389 770 831 926 039 889 601 733 193 275 112 578 283 448 018 613 526 925 847 925 558 456 540 351 327 099 176 534 335 451 141 045 209 002 537 535 755 031 468 961 150 691 008 214 712 492 137 716 092 251 416 854 303 972 448 469 954 444 917 129 644 451 683 375 275 906 483 623 456 408 625 743 663 232 956 462 751 569 098 735 992 247 230 927 473 597 130 714 467 427 915 529 825 001 467 413 803 400 014 037 257 220 682 520 596 555 932 663 885 324 005 539 599 667 276 944 926 310 400 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 1.24 × 10701
≈ 123 duotrigintaducentillion 657 untrigintaducentillion (short scale) / 123 sedecicentilliard 657 sedecicentillion (long scale)

Length 6

  • 1-coloured: 512
    • Type 31: 64; 1; 1; 1; 8
    • Type 1.3: 192; 1; 1; 1; 24
    • Type 2.2: 192; 1; 1; 1; 24
    • Type 32: 64; 1; 1; 1; 8
  • 2-coloured: 384
    • Type 31: 96; 2; 1; 2; 4
    • Type 2.2: 192; 1; 1; 1; 4
    • Type 32: 96; 2; 1; 2; 4
  • 3-coloured: 128
    • Type 21: 64; 3; 2; 3; 1
    • Type 22: 64; 3; 2; 3; 1
  • 4-coloured 16; 12; 2; 3; 1
  • Puzzle orientation constraint: 192
  • Total pieces: 1 040
  • Total stickers: 1 728

Number of positions:
((64!/(8!8))2 × (192!/(24!8))2 × (96!/(4!24) × 296/2)2 × 192!/(4!48) × (64!/2 × 364/3)2 × 16!/2 × 1216/3)/192 =

[Expand]

≈ 2.64 × 101 283
≈ 264 sesvigintiquadringentillion 343 quinquavigintiquadringentillion (short scale) / 264 tredeciducentilliard 343 tredeciducentillion (long scale)

Length 7

  • 1-coloured: 992 (1 000)
    • (Type 0: 8)
    • Type 1.11: 48; 1; 1; 1; 6
    • Type 2.11: 96; 1; 1; 1; 12
    • Type 31: 64; 1; 1; 1; 8
    • Type 1.12: 48; 1; 1; 1; 6
    • Type 1.2.1: 192; 1; 1; 1; 24
    • Type 1.3: 192; 1; 1; 1; 24
    • Type 2.12: 96; 1; 1; 1; 12
    • Type 2.2: 192; 1; 1; 1; 24
    • Type 32: 64; 1; 1; 1; 8
  • 2-coloured: 600
    • Type 1: 24; 2; (2); 2; 1
    • Type 2.11: 96; 2; 1; 2; 4
    • Type 31: 96; 2; 1; 2; 4
    • Type 2.12: 96; 2; 1; 2; 4
    • Type 2.2: 192; 1; 1; 1; 4
    • Type 32: 96; 2; 1; 2; 4
  • 3-coloured: 160
    • Type 1: 32; 6; (2); 2; 1
    • Type 21: 64; 3; 2; 3; 1
    • Type 22: 64; 3; 2; 3; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Total pieces: 1 768 (1 776)
  • Total stickers: 2 744

Number of positions:
(48!/(6!8))2 × (96!/(12!8))2 × (64!/(8!8))2 × (192!/(24!8))3 × (24! × 32!)/2 × 224/2 × 632/2 × (96!/(4!24) × 296/2)4 × 192!/(4!48) × (64!/2 × 364/3)2 × 16!/2 × 1216/3 =

[Expand]

≈ 7.34 × 102 070
≈ 7 novemoctogintasescentillion 337 octooctogintasescentillion (short scale) / 7 quinquaquadragintatrecentillion 337 quattuorquadragintatrecentilliard (long scale)

Length 8

  • 1-coloured: 1 728
    • Type 31: 64; 1; 1; 1; 8
    • Type 1.31: 192; 1; 1; 1; 24
    • Type 2.21: 192; 1; 1; 1; 24
    • Type 32: 64; 1; 1; 1; 8
    • Type 1.32: 192; 1; 1; 1; 24
    • Type 1.2.21: 192; 1; 1; 1; 24
    • Type 1.2.22: 192; 1; 1; 1; 24
    • Type 1.33: 192; 1; 1; 1; 24
    • Type 2.22: 192; 1; 1; 1; 24
    • Type 2.23: 192; 1; 1; 1; 24
    • Type 33: 64; 1; 1; 1; 8
  • 2-coloured: 864
    • Type 31: 96; 2; 1; 2; 4
    • Type 2.21: 192; 1; 1; 1; 4
    • Type 32: 96; 2; 1; 2; 4
    • Type 2.22: 192; 1; 1; 1; 4
    • Type 2.23: 192; 1; 1; 1; 4
    • Type 33: 96; 2; 1; 2; 4
  • 3-coloured: 192
    • Type 21: 64; 3; 2; 3; 1
    • Type 22: 64; 3; 2; 3; 1
    • Type 23: 64; 3; 2; 3; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Puzzle orientation constraint: 192
  • Total pieces: 2 800
  • Total stickers: 4 096

Number of positions:
((64!/(8!8))3 × (192!/(24!8))8 × (96!/(4!24) × 296/2)3 × (192!/(4!48))3 × (64!/2 × 364/3)3 × 16!/2 × 1216/3)/192 =

[Expand]

≈ 7.30 × 103 177
≈ 7 millioctoquinquagintillion 299 milliseptenquinquagintillion (short scale) / 7 novemvigintiquingentilliard 299 novemvigintiquingentillion (long scale)

Length 9

  • 1-coloured: 2 736 (2 744)
    • (Type 0: 8)
    • Type 1.11: 48; 1; 1; 1; 6
    • Type 2.11: 96; 1; 1; 1; 12
    • Type 31: 64; 1; 1; 1; 8
    • Type 1.12: 48; 1; 1; 1; 6
    • Type 1.2.11: 192; 1; 1; 1; 24
    • Type 1.31: 192; 1; 1; 1; 24
    • Type 2.12: 96; 1; 1; 1; 12
    • Type 2.21: 192; 1; 1; 1; 24
    • Type 32: 64; 1; 1; 1; 6
    • Type 1.13: 48; 1; 1; 1; 6
    • Type 1.2.12: 192; 1; 1; 1; 24
    • Type 1.32: 192; 1; 1; 1; 24
    • Type 1.2.13: 192; 1; 1; 1; 24
    • Type 1.2.21: 192; 1; 1; 1; 24
    • Type 1.2.22: 192; 1; 1; 1; 24
    • Type 1.33: 192; 1; 1; 1; 24
    • Type 2.13: 96; 1; 1; 1; 12
    • Type 2.22: 192; 1; 1; 1; 24
    • Type 2.23: 192; 1; 1; 1; 24
    • Type 33: 64; 1; 1; 1; 8
  • 2-coloured: 1 176
    • Type 1: 24; 2; (2); 2; 1
    • Type 2.11: 96; 2; 1; 2; 4
    • Type 31: 96; 2; 1; 2; 4
    • Type 2.12: 96; 2; 1; 2; 4
    • Type 2.21: 192; 2; 1; 2; 4
    • Type 32: 96; 2; 1; 2; 4
    • Type 2.13: 96; 2; 1; 2; 4
    • Type 2.22: 192; 2; 1; 2; 4
    • Type 2.23: 192; 2; 1; 2; 4
    • Type 33: 96; 2; 1; 2; 4
  • 3-coloured: 224
    • Type 1: 32; 6; (2); 2
    • Type 21: 64; 3; 2; 3; 1
    • Type 22: 64; 3; 2; 3; 1
    • Type 23: 64; 3; 2; 3; 1
  • 4-coloured: 16; 12; 2; 3; 1
  • Total pieces: 4 152 (4 160)
  • Total stickers: 5 832

Number of positions:
(48!/(6!8))3 × (96!/(12!8))3 × (64!/(8!8))3 × (192!/(24!8))11 × (24! × 32!)/2 × 224/2 × 632/2 × (96!/(4!24) × 296/2)6 × (192!/(4!48))3 × (64!/2 × 364/3)3 × 16!/2 × 1216/3 =

[Expand]

≈ 2.88 × 104 562
≈ 287 millinovendeciquingentillion 721 millioctodeciquingentillion (short scale) / 287 sexagintaseptingentillion 721 novenquinquagintaseptingentilliard (long scale)

{3}×{3}

  • Shape: uniform triangular duoprism
  • Cells (colours): 6
  • Faces: 15 (9 squares, 6 triangles)
  • Edges: 18
  • Vertices: 9

Length 2

  • (2-coloured: 6)
  • 4-coloured: 9; 1; 2; 1; 1
  • Total pieces: 9 (15)
  • Total stickers: 48

Number of positions:
9!/2 =
= 181 440 ≈
≈ 1.81 × 105
= 181 thousand 440

Length 3

  • 1-coloured: 18
    • Type 11: 9; 1; 1; 1; 3
    • Type 12: 9; 1; 1; 1; 3
  • 2-coloured: 27
    • Type 11: 9; 2; 1; 2; 3
    • Type 12: 9; 2; 1; 2; 3
    • Type 2: 9; 1; (2); 1; 1
  • 3-coloured: 18
    • Type 11: 9; 2; (2); 2; 1
    • Type 12: 9; 2; (2); 2; 1
  • 4-coloured: 9; 2; (2); 2; 1
*Puzzle orientation constraint: 18
  • Total pieces: 72
  • Total stickers: 162

Number of positions: ((9!/(3!3))2 × (9!/(3!3) × 29/2)2 × (9! × 9!2 × 9!)/2 × (29/2)2 × 29/2)/18 =
= 4 218 777 141 356 540 340 690 364 512 335 403 417 600 000 000 ≈
≈ 4.22 × 1045
≈ 4 quattuordecillion 219 tredecillion (short scale) / 4 septilliard 219 septillion (long scale)

Magic120Cell

Calculated by David Smith.

{5,3,3}

  • Shape: regular 120-cell (hecatonicosachoron)
  • Cells (colours): 120 regular dodecahedra {5,3}
  • Faces: 720 regular pentagons {5}
  • Edges: 1 200
  • Vertices: 600

Length 3

  • (1-coloured: 120)
  • 2-coloured: 720; 2; 2; 2; 1
  • 3-coloured: 1 200; 6; 2; 2; 1
  • 4-coloured: 600; 12; 2; 3; 1
  • Total pieces: 2 520 (2 640)
  • Total stickers: 7 560

Number of positions:
720!/2 × 2720/2 × 1200!/2 × 61200/2 × 600!/2 × 12600/3 =

[Expand]

≈ 2.34 × 108 126
≈ 234 duomilliseptenseptingentillion 350 duomilliseseptingentillion (short scale) / 234 milliquattuorquinquagintatrecentillion 350 millitresquinquagintatrecentilliard (long scale)

MagicCube5D

Calculated by David Smith.

{4,3,3,3}

  • Shape: Penteract
  • 4-faces (colours): 10 tesseracts {4,3,3}
  • Cells: 40 cubes {4,3}
  • Faces: 80 squares {4}
  • Edges: 80
  • Vertices: 32

Length 2

  • 5-coloured: 32; 60; 2; 1; 1
  • Puzzle orientation constraint: 1 920
  • Total pieces: 32
  • Total stickers: 160

Number of positions:
(32!/2 × 6032)/1920 =
= 54 535 655 175 308 197 058 635 263 389 110 963 213 764 726 777 446 400 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 5.45 × 1088
≈ 54 octovigintillion 536 septemvigintillion (short scale) / 54 quattuordecilliard 536 quattuordecillion

Length 3

  • (1-coloured: type 1: 10)
  • 2-coloured: type 1: 40; 2; (2); 2; 1
  • 3-coloured: type 1: 80; 6; (2); 2; 1
  • 4-coloured: type 1: 80; 24; 2; 2; 1
  • 5-coloured: 32; 60; 2; 1; 1
  • Total pieces: 232 (242)
  • Total stickers: 800

Number of positions:
(40! × 80!)/2 × 240/2 × 680/2 × 80!/2 × 2480/2 × 32!/2 × 6032 =
= 701 667 712 402 950 678 588 563 925 537 442 843 125 814 486 474 172 376 339 080 083 735 282 432 570 880 422 175 614 251 163 058 229 250 653 847 841 202 640 036 019 428 140 364 685 715 598 365 298 331 873 395 846 086 528 536 260 972 280 760 386 269 552 019 118 684 785 923 871 866 118 371 825 759 785 012 234 146 827 079 564 220 427 338 910 666 898 674 313 780 003 300 502 236 858 905 700 554 243 767 722 706 512 968 255 467 907 689 651 857 607 094 055 701 717 148 055 663 687 118 563 692 897 948 419 085 505 315 326 824 962 012 039 175 406 034 820 217 915 303 954 177 226 545 938 524 363 992 267 629 090 384 186 791 766 814 569 267 200 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 7.02 × 10560
≈ 701 quinquaoctogintacentillion 668 quattuoroctogintacentillion (short scale) / 701 trenonagintillion 668 duononagintilliard (long scale)

Length 4

  • 1-coloured: type 5: 160; 1; 1; 1; 16
  • 2-coloured: type 4: 320; 2; 1; 2; 8
  • 3-coloured: type 3: 320; 6; 1; 2; 4
  • 4-coloured: type 2: 160; 12; 2; 3; 1
  • 5-coloured: 32; 60; 2; 1; 1
  • Puzzle orientation constraint: 1 920
  • Total pieces: 992
  • Total stickers: 2 560

Number of positions:
(160!/(16!10) × 320!/(8!40) × 2320/2 × 320!/(4!80) × 6320/2 × 160!/2 × 12160/3 × 32!/2 × 6032)/ 1920 =

[Expand]

≈ 3.29 × 102 075
≈ 329 nonagintasescentillion 259 novemoctogintasescentillion (short scale) / 329 quinquaquadragintatrecentilliard 259 quinquaquadragintatrecentillion (long scale)

Length 5

  • 1-coloured: 800 (810)
    • (Type 1: 10)
    • Type 2.1: 80; 1; 1; 1; 8
    • Type 3.1: 240; 1; 1; 1; 24
    • Type 4.1: 320; 1; 1; 1; 32
    • Type 5: 160; 1; 1; 1; 16
  • 2-coloured: 1 080
    • Type 1: 40; 2; (2); 2; 1
    • Type 2.1: 240; 2; 1; 2; 6
    • Type 3.1: 480; 2; 1; 2; 12
    • Type 4: 320; 2; 1; 2; 8
  • 3-coloured: 720
    • Type 1: 80; 6; (2); 2; 1
    • Type 2.1: 320; 6; 1; 2; 4
    • Type 3: 320; 6; 1; 2; 4
  • 4-coloured: 240
    • Type 1: 80; 24; 2; 2; 1
    • Type 2: 160; 12; 2; 3; 1
  • 5-coloured: 32; 60; 2; 1; 1
  • Total pieces: 2 872 (2 882)
  • Total stickers: 6 520

Number of positions:
80!/(8!10) × 240!/(24!10) × 320!/(32!10) × 160!/(16!10) × (40! × 80!)/2 × 240/2 × 680/2 × 240!/(6!40) × 2240/2 × 480!/(12!40) × 2480/2 × 320!/(8!40) × 2320/2 × (320!/(4!80) × 6320/2)2 × 80!/2 × 2480/2 × 160!/2 × 12160/3 × 32!/2 × 6032 =

[Expand]

≈ 2.32 × 105 267
≈ 231 milliquattuorquinquagintaseptingentillion 742 millitresquinquagintaseptingentillion (short scale) / 231 septenseptuagintaoctingentilliard 742 septenseptuagintaoctingentillion (long scale)

Length 6

  • 1-coloured: 2 560
    • Type 51: 160; 1; 1; 1; 16
    • Type 2.4: 640; 1; 1; 1; 64
    • Type 3.3: 960; 1; 1; 1; 96
    • Type 4.2: 640; 1; 1; 1; 64
    • Type 52: 160; 1; 1; 1; 16
  • 2-coloured: 2 560
    • Type 41: 320; 2; 1; 2; 8
    • Type 2.3: 960; 2; 1; 2; 24
    • Type 3.2: 960; 2; 1; 2; 24
    • Type 42: 320; 2; 1; 2; 8
  • 3-coloured: 1 280
    • Type 31: 320; 6; 1; 2; 4
    • Type 2.2: 640; 3; 1; 3; 4
    • Type 32: 320; 6; 1; 2; 4
  • 4-coloured: 320
    • Type 21: 160; 12; 2; 3; 1
    • Type 22: 160; 12; 2; 3; 1
  • 5-coloured: 32; 60; 2; 1; 1
  • Puzzle orientation constraint: 1 920
  • Total pieces: 6 752
  • Total stickers: 12 960

Number of positions:
((160!/(16!10))2 × (640!/(64!10))2 × 960!/(96!10) × (320!/(8!40) × 2320/2)2 × (960!/(24!40) × 2960/2)2 × (320!/(4!80) × 6320/2)2 × 640!/(4!160) × 3640/3 × (160!/2 × 12160/3)2 × 32!/2 × 6032)/ 1920 =

[Expand]

≈ 3.49 × 1011 441
≈ 348 tremilliduodecioctingentillion 978 tremilliundecioctingentillion (short scale) / 348 millisenongentilliard 978 millisenongentillion (long scale)

Length 7

  • 1-coloured: 6 240 (6 250)
    • (Type 1: 10)
    • Type 2.11: 80; 1; 1; 1; 8
    • Type 3.11: 240; 1; 1; 1; 24
    • Type 4.11. 320; 1; 1; 1; 32
    • Type 51: 160; 1; 1; 1; 16
    • Type 2.12: 80; 1; 1; 1; 8
    • Type 2.2.1: 480; 1; 1; 1; 48
    • Type 2.3.1: 960; 1; 1; 1; 96
    • Type 2.4: 640; 1; 1; 1; 64
    • Type 3.12: 240; 1; 1; 1; 24
    • Type 3.2.1: 960; 1; 1; 1; 96
    • Type 3.3: 960; 1; 1; 1; 96
    • Type 4.12: 320; 1; 1; 1; 32
    • Type 4.2. 640; 1; 1; 1; 64
    • Type 52: 160; 1; 1; 1; 16
  • 2-coloured: 5 000
    • Type 1: 40; 2; (2); 2; 1
    • Type 2.11: 240; 2; 1; 2; 6
    • Type 3.11: 480; 2; 1; 2; 12
    • Type 41: 320; 2; 1; 2; 8
    • Type 2.12: 240; 2; 1; 2; 6
    • Type 2.2.1: 960; 2; 1; 2; 24
    • Type 2.3: 960; 2; 1; 2; 24
    • Type 3.12: 480; 2; 1; 2; 12
    • Type 3.2: 960; 2; 1; 2; 24
    • Type 42: 320; 2; 1; 2; 4
  • 3-coloured: 2 000
    • Type 1: 80; 6; (2); 2; 1
    • Type 2.11: 320; 6; 1; 2; 4
    • Type 31: 320; 6; 1; 2; 4
    • Type 2.12: 320; 6; 1; 2; 4
    • Type 2.2: 640; 3; 1; 3; 4
    • Type 32: 320; 6; 1; 2; 4
  • 4-coloured: 400
    • Type 1: 80; 24; 2; 2; 1
    • Type 21: 160; 12; 2; 3; 1
    • Type 22. 160; 12; 2; 3; 1
  • 5-coloured: 32; 60; 2; 1; 1
  • Total pieces: 13 672 (13 682)
  • Total stickers: 24 010

Number of positions:
(80!/(8!10))2 × (240!/(24!10))2 × (320!/(32!10))2 × (160!/(16!10))2 × 480!/(48!10) × (960!/(96!10))3 × (640!/(64!10))2 × (40! × 80!)/2 × 240/2 × 680/2 × (240!/(6!40) × 2240/2)2 × (480!/(12!40) × 2480/2)2 × (320!/(8!40) × 2320/2)2 × (960!/(24!40) × 2960/2)3 × (320!/(4!80) × 6320/2)4 × 640!/(4!160) × 3640/3 × 80!/2 × 2480/2 × (160!/2 × 12160/3)2 × 32!/2 × 6032 =

[Expand]

≈ 2.29 × 1021 503
≈ 228 septemillisesexagintacentillion 762 septemilliquinquasexagintacentillion (short scale) / 228 tremillitresoctogintaquingentilliard 762 tremillitresoctogintaquingentillion (long scale)

Magic Cube 7D

{4,3,3,3,3}

  • Shape: hexeract
  • 5-faces (colours): 12 penteracts {4,3,3,3}
  • 4-faces: 60 tesseracts {4,3,3}
  • Cells: 160 cubes {4,3}
  • Faces: 240 squares {4}
  • Edges: 192
  • Vertices: 64

Length 3

  • (1-coloured: type 1: 12)
  • 2-coloured: type 1: 60; 2; (2); 2; 1
  • 3-coloured: type 1: 160; 6; (2); 2; 1
  • 4-coloured: type 1: 240; 24; 2; 2; 1
  • 5-coloured: type 1: 192; 120; 2; 2; 1
  • 6-coloured: 64; 360; 2; 1; 1
  • Total pieces: 716 (728)
  • Total stickers: 2 916

Number of positions:
(60! × 160!)/2 × 260/2 × 6160/2 × 240!/2 × 24240/2 × 192!/2 × 120192/2 × 64!/2 × 36064 =

[Expand]

≈ 1.18 × 102 315
≈ 117 septuagintaseptingentillion 831 novensexagintaseptingentillion (short scale) / 117 quinquaoctogintatrecentilliard 831 quinquaoctogintatrecentillion (long scale)

Length 4

  • 1-coloured: type 6: 384; 1; 1; 1; 32
  • 2-coloured: type 5: 960; 2; 1; 2; 16
  • 3-coloured: type 4: 1 280; 6; 1; 2; 8
  • 4-coloured: type 3: 960; 24; 1; 2; 4
  • 5-coloured: type 2: 384; 60; 2; 1; 1
  • 6-coloured: 64; 360; 2; 1; 1
  • Puzzle orientation constraint: 23 040
  • Total pieces: 4 032
  • Total stickers: 12 288

Number of positions:
(384!/(32!12) × 960!/(16!60) × 2960/2 × 1280!/(8!160) × 61280/2 × 960!/(4!240) × 24!960/2 × 384!/2 × 60384 × 64!/2 × 36064)/23040 =

[Expand]

≈ 1.11 × 1032 737
≈ 11 decimilliundecinongentillion 148 decimillidecinongentillion (short scale) / 11 quinquamillisesquinquagintaquadringentillion 148 quinquamilliquinquaquinquagintaquadringentilliard (long scale)

Length 5

  • 1-coloured: 2 904 (2 916)
    • (Type 1: 12)
    • Type 2: 120; 1; 1; 1; 10
    • Type 3: 480; 1; 1; 1; 40
    • Type 4: 960; 1; 1; 1; 80
    • Type 5: 960; 1; 1; 1; 80
    • Type 6: 384; 1; 1; 1; 32
  • 2-coloured: 4 860
    • Type 1: 60; 2; (2); 2; 1
    • Type 2: 480; 2; 1; 2; 8
    • Type 3: 1 440; 2; 1; 2; 24
    • Type 4; 1 920; 2; 1; 2; 32
    • Type 5: 960; 2; 1; 2; 16
  • 3-coloured: 4 320
    • Type 1: 160; 6; (2); 2; 1
    • Type 2: 960; 6; 1; 2; 6
    • Type 3: 1 920; 6; 1; 2; 12
    • Type 4: 1 280; 6; 1; 2; 8
  • 4-coloured: 2 160
    • Type 1: 240; 24; 2; 2; 1
    • Type 2: 960; 24; 1; 2; 4
    • Type 3: 960; 24; 1; 2; 4
  • 5-coloured: 576
    • Type 1: 192; 120; 2; 2; 1
    • Type 2: 384; 60; 2; 1; 1
  • 6-coloured: 64; 360; 2; 1; 1
  • Total pieces: 14 884 (14 896)
  • Total stickers: 37 500

Number of positions:
120!/(10!12) × 480!/(40!12) × (960!/(80!12))2 × 384!/(32!12) × (60! × 160!)/2 × 260/2 × 6160/2 × 480!/(8!60) × 2480/2 × 1440!/(24!60) × 21440/2 × 1920!/(32!60) × 21920/2 × 960!/(16!60) × 2960/2 × 960!/(6!160) × 6960/2 × 1920!/(12!160) × 61920/2 × 1280!/(8!160) × 61280/2 × 240!/2 × 24240/2 × (960!/(4!240) × 24960/2)2 × 192!/2 × 120192/2 × 384!/2 × 60384 × 64!/2 × 36064 =
≈ 6.69 × 10^35 515
≈ 66 undecimilliseptentrigintaoctingentillion 861 undecimillisestrigintaoctingentillion (short scale) / 66 quinquamillinovendecinongentillion 861 quinquamillioctodecinongentilliard (long scale)

{4,3,3,3,3,3}

  • Shape: hepteract
  • 6-faces (colours): 14 hexeracts {4,3,3,3,3}
  • 5-faces: 84 penteracts {4,3,3,3}
  • 4-faces: 280 tesseracts {4,3,3}
  • Cells: 560 cubes {4,3}
  • Faces: 672 squares {4}
  • Edges: 448
  • Vertices: 128

Length 3

  • (1-coloured: 14)
  • 2-coloured: 84; 2; (2); 2; 1
  • 3-coloured: 280; 6; (2); 2; 1
  • 4-coloured: 560; 24; 2; 2; 1
  • 5-coloured: 672; 120; 2; 2; 1
  • 6-coloured: 448; 720; 2; 2; 1
  • 7-coloured: 128; 2 520; 2; 1; 1
  • Total pieces: 2 172 (2 186)
  • Total stickers: 10 206

Number of positions:
(84! × 280!)/2 × 284/2 × 6280/2 × 560!/2 × 24560/2 × 672!/2 × 120672/2 × 448!/2 × 720448/2 × 128!/2 × 2520128 =

[Expand]

≈ 3.37 × 108 935
≈ 33 duomilliseptenseptuagintanongentillion 743 duomilliseseptuagintanongentillion (short scale) / 33 millinovemoctogintaquadringentillion 743 millioctooctogintaquadringentilliard (long scale)

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