Difference between revisions of "Mathematics"

Revision as of 09:53, 11 February 2019

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 a short 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 b^a 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 b^a 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 these constraints (if the pieces’ permutations have given parity, the permutation constraint c = 2 because only half of the permutations are attainable. With regard to orientations, we can say that all but last pieces have b (a − 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 e colours and f pieces per each one, we can freely permute the f pieces of the same colour, and, as a consequence, we have to divide by f!^e.

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

• n-colour pieces type m: count (a); number of orientations (b); permutation constraint (c); orientation constraint (d); indistinguishability constraint (f!^e)

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

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 results in c = 2, counted only once despite applying to more types. When is a whole type type in parentheses, it means that 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 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.

Calculated by Jakub Štepo. I do not guarantee the correctness of my results, but there should not be any mistakes.

MagicCube4D

{3,3,3}

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

Length 2

• 4-colour: [5]; 12; 1; 1
• (5-colour: 1)
• Total mobile pieces: 5
• Total stickers: 25

Number of positions:
12^5 =
= 248 832

Length 3

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

Number of positions:
10!/2 × (6^10)/2 × (12^5)^2 =
= 3 396 471 743 308 934 991 052 800 ≈
≈ 3.40×10^24
≈ 3 septillion 396 sextillion (short scale) / 3 quadrillion 396 trilliard (long scale)

Length 4

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

Number of positions:
10!/2 × (2^10)/2 × 10!/2 × (6^10)/2 × 20!/2 × (3^20)/3 × (12^5)^2 =
= 4 460 971 667 252 991 547 434 208 214 041 871 442 189 607 102 945 689 600 000 000 ≈
≈ 4.46×10^60
≈ 4 novemdecillion 461 octodecillion (short scale) / 4 decillion 461 nonilliard (long scale)

Length 5

• 1-colour: 5; 1; 2; 1
• 2-colour: 40
  • Type 1: 10; 2; 2; 2
  • Type 2; 30; 2; 2; 2; 3!^10
• 3-colour: 50
  • Type 1: 10; 6; 2; 2
  • Type 2: 20; 3; 2; 3
  • Type 3: 20; 3; 2; 3
• 4-colour: 10
  • Type 1: [5]; 12; 1; 1
  • Type 2: [5]; 12; 1; 1
• Total mobile pieces: 105
• Total stickers: 275

Number of positions:
5!/2 × 10!/2 × (2^10)/2 × 30!/(2 × 3!^10) × (2^30)/2 × 10!/2 × (6^10)/2 × (20!/2 × (3^20)/3)^2 × (12^5)^2 =
= 445 622 002 487 959 948 988 374 180 175 268 013 222 358 960 900 098 014 140 915 354 610 363 142 029 430 609 380 392 027 056 585 782 067 200 000 000 000 000 000 000 ≈
≈ 4.46×10^122
≈ 445 noventrigintillion 622 octotrigintillion (short scale) / 445 vigintillion 622 novendecilliard (long scale)

{4,3,3}

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

Length 2

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

Number of positions:
(16!/2 × (12^6)/3)/192 =
= 3 357 894 533 384 932 272 635 904 000 ≈
≈ 3.36×10^27
≈ 3 octillion 358 septillion (short scale) / 3 quadrilliard 358 quadrillion (long scale)

Length 3

• (1-colour: 8)
• 2-colour: 24; 2; (2); 2
• 3-colour: 32; 6; (2); 2
• 4-colour: 16; 12; 2; 3
• Total mobile pieces: 72
• Total stickers: 216

Number of positions:
(24! × 32!)/2 × (2^24)/2 × (6^32)/2 × 16!/2 × (12^16)/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 ≈
≈ 1.76×10^120
≈ 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,−): 63 875 321 129 519 842 788 229 550 349 465 865 698 238 148 116 060 569 600 000
• (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
4 574 929 376 846 707 924 918 036 664 273 502 759 412 720 391 014 473 055 557 863 939 959 893 650 526 399 862 305 272 865 622 237 030 657 852 043 408 965 632 ≈
≈ 4.57×10^117
≈ 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 ≈
≈ 1.51×10^66
≈ 1 unvigintillion 515 vigintillion (short scale) / 6 undecillion 515 decilliard (long scale).

For more details, see [[Mathematics/{4,3,3}]].

Length 4

• 1-colour: 64; 1; 1; 1; 8!^8
• 2-colour: 96; 2; 1; 2; 4!^24
• 3-colour: 64; 3; 2; 3
• 4-colour: 16; 12; 2; 3
• Puzzle orientation constraint: 192
• Total mobile pieces: 240
• Total stickers: 512

Number of positions:
(64!/(8!^8) × 96!/(4!^24) × (2^96)/2 × 64!/2 × (3^64)/3 × 16!/2 × (12^16)/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 ≈
≈ 1.30×10^344
≈ 130 tredecicentillion 466 duodecicentillion (short scale) / 130 septenquinquagintillion 466 sesquinquagintilliard (long scale)

Length 5

• 1-colour: 208 (216)
  • (Type 0: 8)
  • Type 1A: 48; 1; 1; 1; 6!^8
  • Type 1Ba: 96; 1; 1; 1; 12!^8
  • Type 1Bb: 64; 1; 1; 1; 8!^8
• 2-colour: 216
  • Type 1: 24; 2; (2); 2
  • Type 2A: 96; 2; 1; 2; 4!^24
  • Type 2B: 96; 2; 1; 2; 4!^24
• 3-colour: 96
  • Type 1: 32; 6; (2); 2
  • Type 2: 64; 3; 2; 3
• 4-colour: 16; 12; 2; 3
• Total mobile pieces: 536
• Total stickers: 1000

Number of positions:
48!/(6!^8) × 96!/(12!^8) × 64!/(8!^8) × (24! × 32!)/2 × (2^24)/2 × (6^32)/2 × (96!/(4!^24) × (2^96)/2)^2 × 64!/2 × (3^64)/3 × 16!/2 × (12^16)/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 ≈
≈ 1.24×10^701
≈ 123 duotrigintaducentillion 657 untrigintaducentillion (short scale) / 123 sedecicentilliard 657 sedecicentillion (long scale)

Length 6

• 1-colour: 512
  • Type 1: 64; 1; 1; 1; 8!^8
  • Type 2A: 192; 1; 1; 1; 24!^8
  • Type 2Ba: 192; 1; 1; 1; 24!^8
  • Type 2Bb: 64; 1; 1; 1; 8!^8
• 2-colour: 384
  • Type 1: 96; 2; 1; 2; 4!^24
  • Type 2A: 192; 2; 1; 2; 8!^24
  • Type 2B: 96; 2; 1; 2; 4!^24
• 3-colour: 128
  • Type 1: 64; 3; 2; 3
  • Type 2: 64; 3; 2; 3
• 4-colour 16; 12; 2; 3
• Puzzle orientation constraint: 192
• Total mobile pieces: 1040
• Total stickers: 1728

Number of positions:
((64!/(8!^8))^2 × (192!/(24!^8))^2 × (96!/(4!^24) × (2^96)/2)^2 × 192!/(24!^8) × 2^192 × (64!/2 × (3^64)/3)^2 × 16!/2 × (12^16)/3)/192 =
= 4 330 563 586 781 524 771 753 221 225 538 402 895 653 388 384 512 732 580 964 855 890 366 682 053 812 694 249 885 251 815 291 282 459 189 648 971 632 660 257 088 554 076 996 985 058 715 088 036 992 192 728 975 917 814 718 029 299 052 083 846 038 648 754 825 049 995 663 272 249 254 128 117 192 731 901 634 400 308 947 476 030 539 549 978 320 057 004 945 663 595 047 113 628 963 904 290 898 903 827 146 814 392 616 906 490 655 289 199 893 119 261 891 206 161 900 906 257 483 955 915 710 224 366 923 373 245 271 718 733 079 279 765 899 738 315 643 452 777 113 421 178 368 067 350 615 865 043 174 293 537 175 058 193 468 860 436 495 299 974 819 750 245 204 191 457 021 371 616 500 111 770 611 406 679 134 450 672 458 586 190 379 569 036 167 736 875 335 003 539 441 335 137 258 422 220 372 546 747 114 002 551 126 680 815 988 245 824 985 433 407 088 692 697 333 561 262 003 577 523 082 417 655 617 950 186 228 379 563 306 510 562 816 109 381 188 782 556 022 182 951 264 812 583 181 338 476 758 843 656 815 450 582 577 953 344 774 452 140 231 512 418 155 651 907 136 814 773 135 453 283 225 784 924 643 619 592 218 809 435 178 694 962 677 052 687 103 134 823 206 815 491 915 961 670 677 118 240 910 078 761 237 466 908 849 289 680 931 298 048 694 186 676 188 140 069 600 568 474 994 332 865 234 729 589 265 917 305 767 727 982 276 258 846 202 979 725 332 236 033 627 934 977 999 457 218 799 923 734 902 706 512 208 406 549 078 005 906 541 138 275 332 514 367 170 021 753 614 862 178 186 282 726 864 469 846 272 572 675 589 510 222 452 910 044 531 916 800 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 4.33×10^1296
≈ 4 untrigintaquadringentillion 331 trigintaquadringentillion (short scale) / 4 sedeciducentillion 331 quinquadeciducentilliard (long scale)

Length 7

• 1-colour: 992 (1000)
  • (Type 0: 8)
  • Type 1A: 48; 1; 1; 1; 6!^8
  • Type 1Ba: 96; 1; 1; 1; 12!^8
  • Type 1Bb: 64; 1; 1; 1; 8!^8
  • Type 2A: 48; 1; 1; 1; 6!^8
  • Type 2Ba: 192; 1; 1; 1; 24!^8
  • Type 2Bb: 192; 1; 1; 1; 24!^8
  • Type 2Ca: 96; 1; 1; 1; 12!^8
  • Type 2Cb: 192; 1; 1; 1; 24!^8
  • Type 2Cc: 64; 1; 1; 1; 8!^8
• 2-colour: 600
  • Type 1: 24; 2; (2); 2
  • Type 2A: 96; 2; 1; 2; 4!^24
  • Type 2B: 96; 2; 1; 2; 4!^24
  • Type 3A: 96; 2; 1; 2; 4!^24
  • Type 3B: 192; 2; 1; 2; 8!^24
  • Type 3C: 96; 2; 1; 2; 4!^24
• 3-colour: 160
  • Type 1: 32; 6; (2); 2
  • Type 2: 64; 3; 2; 3
  • Type 3: 64; 3; 2; 3
• 4-colour: 16; 12; 2; 3
• Total mobile pieces: 1768
• Total stickers: 2744

Number of positions:
(48!/(6!^8))^2 × (96!/(12!^8))^2 × (64!/(8!^8))^2 × (192!/(24!^8))^3 × (24! × 32!)/2 × (2^24)/2 × (6^32)/2 × (96!/(4!^24) × (2^96)/2)^4 × 192!/(8!^24) × (2^192)/2 × (64!/2 × (3^64)/3)^2 × 16!/2 × (12^16)/3 =
= 120 204 420 262 829 797 162 433 788 919 585 455 757 204 805 471 800 349 179 170 259 140 241 084 037 126 862 235 334 757 004 155 458 806 899 430 992 342 308 808 995 176 512 130 635 779 825 935 530 398 806 387 940 433 384 505 189 091 725 286 236 191 252 318 683 414 214 787 470 502 609 610 517 282 849 855 626 585 956 033 459 665 523 049 390 647 991 775 586 700 657 741 349 643 597 253 242 616 624 817 277 542 197 445 002 087 552 822 043 265 955 990 515 717 639 936 270 166 304 619 581 025 290 323 597 571 358 759 174 290 461 782 064 624 612 055 601 144 214 294 228 786 967 379 542 444 011 455 701 652 062 905 728 494 743 529 666 464 877 593 555 610 305 818 629 355 758 752 510 639 383 756 741 327 547 512 086 292 498 955 661 005 792 054 480 736 079 010 227 809 116 951 027 221 904 223 055 707 648 314 052 055 477 421 906 905 289 505 370 361 120 147 122 173 579 103 416 507 661 748 018 695 784 564 799 352 956 393 190 079 203 036 419 189 948 248 477 878 049 622 771 456 379 023 414 067 102 043 661 614 635 607 834 761 191 465 168 363 350 168 674 013 400 450 115 957 116 152 351 707 757 447 051 290 408 418 185 671 839 886 399 403 181 603 633 061 426 419 690 487 267 117 919 459 677 797 784 911 222 398 435 900 117 426 919 421 024 706 608 858 996 402 208 516 057 084 322 625 811 549 693 944 605 404 438 543 784 212 751 100 229 464 403 751 680 650 480 940 153 700 025 567 728 438 079 548 898 782 769 129 950 148 465 515 885 995 611 511 631 608 074 802 316 156 887 221 829 931 050 520 296 403 051 243 790 768 240 970 019 554 523 795 346 801 637 621 576 218 289 232 316 961 943 149 681 473 200 551 321 603 075 487 657 751 232 622 559 464 294 599 320 283 967 704 302 870 691 764 899 327 706 273 156 736 470 358 347 648 904 903 142 233 794 512 726 116 109 151 832 357 567 239 146 375 565 492 114 510 643 603 512 414 850 178 450 805 385 551 320 188 193 586 448 762 466 992 548 202 016 870 057 740 388 571 969 940 150 609 666 211 877 013 629 528 042 973 697 119 635 627 710 214 720 949 176 473 675 120 739 414 599 671 229 234 422 489 806 227 168 015 622 648 468 200 892 625 010 634 255 390 021 079 648 947 768 836 693 491 025 320 641 635 329 586 697 630 429 400 325 405 252 766 348 724 199 577 647 165 752 375 360 727 626 753 528 177 284 870 630 326 269 441 828 106 175 020 545 861 506 382 637 439 043 253 401 709 360 991 008 939 122 977 258 825 091 947 860 006 122 086 682 980 739 561 226 686 391 673 347 813 637 808 731 491 824 948 407 728 384 094 391 223 980 156 720 291 239 702 453 813 248 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 1.20×10^2084
≈ 120 trenonagintasescentillion 204 duononagintasescentillion (short scale) / 120 septenquadragintatrecentillion 204 sesquadragintatrecentilliard (long scale)

Length 8

• 1-colour: 1728
  • Type 1: 64; 1; 1; 1; 8!^8
  • Type 2A: 192; 1; 1; 1; 24!^8
  • Type 2Ba: 192; 1; 1; 1; 24!^8
  • Type 2Bb: 64; 1; 1; 1; 8!^8
  • Type 3A: 192; 1; 1; 1; 24!^8
  • Type 3Ba: 384; 1; 1; 1; 48!^8
  • Type 3Bb: 192; 1; 1; 1; 24!^8
  • Type 3Ca: 192; 1; 1; 1; 24!^8
  • Type 3Cb: 192; 1; 1; 1; 24!^8
  • Type 3Cc: 64; 1; 1; 1; 8!^8
• 2-colour: 864
  • Type 1: 96; 2; 1; 2; 4!^24
  • Type 2A: 192; 2; 1; 2; 8!^24
  • Type 2B: 96; 2; 1; 2; 4!^24
  • Type 3A: 192; 2; 1; 2; 8!^24
  • Type 3B: 192; 2; 1; 2; 8!^24
  • Type 3C: 96; 2; 1; 2; 4!^24
• 3-colour: 192
  • Type 1: 64; 3; 2; 3
  • Type 2: 64; 3; 2; 3
  • Type 3: 64; 3; 2; 3
• 4-colour: 16; 12; 2; 3
• Puzzle orientation constraint: 192
• Total mobile pieces: 2800
• Total stickers: 4096

Number of positions:
((64!/(8!^8))^3 × (192!/(24!^8))^5 × 384!/(48!^8) × (96!/(4!^24) × (2^96)/2)^3 × (192!/(8!^24) × (2^192)/2)^3 × (64!/2)^3 × ((3^64)/3)^3 × 16!/2 × (12^16)/3)/192 =
= 2 721 581 823 080 873 052 859 657 844 142 014 134 015 658 403 678 494 295 364 865 414 829 788 995 830 454 165 128 239 497 743 501 201 101 594 431 668 779 730 953 156 729 631 579 937 971 719 056 751 144 401 352 961 570 367 304 955 773 354 070 352 438 779 264 240 874 518 343 498 698 109 108 071 550 490 239 249 369 299 779 425 733 239 725 882 216 107 369 817 932 322 734 919 453 500 268 860 563 607 119 758 967 852 177 439 306 546 586 704 562 888 044 269 567 455 128 918 682 222 002 990 492 082 331 472 916 603 942 065 542 310 359 514 016 236 954 391 659 683 606 457 748 064 297 708 757 514 219 352 850 969 091 827 919 644 440 920 100 105 068 708 379 140 500 474 324 190 706 764 162 803 024 206 607 547 576 137 245 621 532 878 521 415 636 519 361 472 025 540 560 621 965 578 403 361 089 606 752 419 683 485 105 247 191 293 769 994 013 775 891 523 566 093 685 242 790 304 855 558 043 285 651 052 444 928 975 835 213 649 345 749 772 692 523 491 296 371 617 220 672 464 524 735 868 482 287 471 366 996 437 163 771 142 347 067 997 144 590 276 890 459 798 601 555 369 492 373 757 240 383 940 146 560 615 580 308 437 660 368 129 779 651 604 683 772 170 228 191 643 855 422 475 634 102 541 463 937 701 345 517 916 041 666 146 427 274 478 848 175 138 731 787 259 015 144 862 351 409 681 116 878 374 857 920 519 299 361 150 730 306 175 799 626 394 739 470 181 222 619 561 526 552 062 480 425 479 046 565 472 457 218 276 858 261 229 363 508 486 532 218 072 521 655 635 799 226 729 127 256 273 871 147 861 479 580 568 919 256 509 899 397 580 761 316 499 643 509 071 715 498 565 100 821 044 085 755 263 076 790 306 322 739 772 795 272 530 600 430 061 324 907 786 186 720 689 379 252 904 602 234 870 779 261 219 313 675 470 095 759 903 775 149 605 927 858 536 465 393 281 829 122 279 014 997 040 033 995 794 965 221 939 364 014 873 890 559 311 256 609 963 621 251 390 692 205 375 851 820 846 920 367 319 017 156 776 441 154 845 968 832 373 847 534 855 109 913 104 569 716 217 279 807 860 602 795 780 554 520 459 609 699 938 724 223 660 713 949 038 974 976 745 149 188 061 902 861 568 698 115 784 364 684 946 154 710 382 794 352 073 679 769 326 849 308 872 400 227 894 557 336 755 497 557 394 411 839 017 324 802 740 655 275 318 335 035 292 904 821 514 083 627 084 358 203 665 924 789 215 733 105 400 999 682 824 463 527 159 968 274 643 049 727 497 634 527 660 511 796 626 474 114 863 866 103 492 676 623 482 861 459 788 001 638 794 964 748 043 780 518 381 236 055 493 149 699 159 448 035 950 269 712 347 881 705 892 922 519 418 840 453 633 134 579 855 706 443 565 857 328 371 447 075 522 864 543 635 187 455 011 611 760 065 956 973 645 704 440 092 639 429 973 621 238 034 854 115 014 636 291 924 564 143 167 508 485 141 817 204 330 539 409 885 398 144 379 068 761 137 360 537 031 241 990 259 288 449 874 949 069 064 103 319 717 457 440 819 478 738 975 025 648 107 081 924 347 619 557 153 129 980 495 268 525 720 509 031 511 680 852 433 550 828 341 074 384 869 248 370 046 989 275 320 615 723 095 914 501 628 975 295 527 450 877 777 440 712 016 521 092 974 806 473 928 849 523 186 344 703 225 141 596 447 790 800 176 278 637 020 278 394 278 694 626 509 750 465 781 544 612 404 393 605 567 806 276 095 473 698 660 070 105 647 847 424 739 951 272 855 114 900 415 140 158 269 690 881 260 962 836 750 832 971 075 479 901 249 985 839 740 137 119 149 041 557 802 309 956 919 999 637 643 463 145 545 665 040 664 840 208 557 009 630 517 513 874 079 938 964 009 424 310 837 644 840 596 915 481 244 071 134 176 090 303 414 256 274 518 042 384 682 848 681 814 331 807 171 862 476 912 826 536 666 748 876 636 579 342 316 890 251 860 785 707 404 727 303 341 925 008 580 879 859 726 078 054 020 520 677 926 943 734 937 847 611 785 216 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 2.72×10^3057
≈ 2 millioctodecillion 722 milliseptendecillion (short scale) / 2 novenquingentilliard 722 novenquingentillion (long scale)

Length 9

• 1-colour: 2736 (2744)
  • (Type 0: 8)
  • Type 1A: 48; 1; 1; 1; 6!^8
  • Type 1Ba: 96; 1; 1; 1; 12!^8
  • Type 1Bb: 64; 1; 1; 1; 8!^8
  • Type 2A: 48; 1; 1; 1; 6!^8
  • Type 2Ba: 192; 1; 1; 1; 24!^8
  • Type 2Bb: 192; 1; 1; 1; 24!^8
  • Type 2Ca: 96; 1; 1; 1; 12!^8
  • Type 2Cb: 192; 1; 1; 1; 24!^8
  • Type 2Cc: 64; 1; 1; 1; 6!^8
  • Type 3A: 48; 1; 1; 1; 6!^8
  • Type 3Ba: 192; 1; 1; 1; 24!^8
  • Type 3Bb: 192; 1; 1; 1; 24!^8
  • Type 3Ca: 192; 1; 1; 1; 24!^8
  • Type 3Cb: 384; 1; 1; 1; 48!^8
  • Type 3Cc: 192; 1; 1; 1; 24!^8
  • Type 3Da: 96; 1; 1; 1; 12!^8
  • Type 3Db: 192; 1; 1; 1; 24!^8
  • Type 3Dc: 192; 1; 1; 1; 24!^8
  • Type 3Dd: 64; 1; 1; 1; 8!^8
• 2-colour: 1176
  • Type 1: 24; 2; (2); 2
  • Type 2A: 96; 2; 1; 2; 4!^24
  • Type 2B: 96; 2; 1; 2; 4!^24
  • Type 3A: 96; 2; 1; 2; 4!^24
  • Type 3B: 192; 2; 1; 2; 8!^24
  • Type 3C: 96; 2; 1; 2; 4!^24
  • Type 4A: 96; 2; 1; 2; 4!^24
  • Type 4B: 192; 2; 1; 2; 8!^24
  • Type 4C: 192; 2; 1; 2; 8!^24
  • Type 4D: 96; 2; 1; 2; 4!^24
• 3-colour: 224
  • Type 1: 32; 6; (2); 2
  • Type 2: 64; 3; 2; 3
  • Type 3: 64; 3; 2; 3
  • Type 4: 64; 3; 2; 3
• 4-colour: 16; 12; 2; 3
• Total mobile pieces: 4152
• Total stickers: 5832

Number of positions:
(48!/(6!^8))^3 × (96!/(12!^8))^3 × (64!/(8!^8))^3 × (192!/(24!^8))^9 × 384!/(48!^8) × (24! × 32!)/2 × (2^24)/2 × (6^32)/2 × (96!/(4!^24) × (2^96)/2)^6 × (192!/(8!^24) × (2^192)/2)^3 × (64!/2 × (3^64)/3)^3 × 16!/2 × (12^16)/3 =
= 578 107 776 180 430 388 102 837 597 507 554 738 026 218 295 608 889 456 750 918 842 950 288 390 048 717 405 663 907 101 699 838 386 699 596 153 953 108 196 281 321 063 690 868 672 377 796 000 032 226 057 971 684 348 744 227 545 396 296 423 449 111 583 259 404 479 996 155 786 834 140 762 234 882 028 558 232 532 744 152 515 647 922 425 971 155 483 154 558 532 182 955 325 618 048 601 984 806 649 571 823 948 428 568 057 547 750 447 127 147 340 826 525 549 050 107 519 088 281 458 280 359 145 972 938 767 485 553 113 456 888 883 434 936 055 857 095 534 792 538 769 607 988 856 585 188 190 598 397 868 018 772 353 755 752 000 477 286 080 993 420 548 243 557 522 434 106 757 475 413 506 099 748 713 273 305 700 273 742 618 781 790 024 378 496 669 235 086 458 318 361 394 186 205 015 093 055 081 640 560 911 299 123 183 880 109 358 519 572 116 369 310 100 343 422 361 015 733 609 156 720 157 574 239 190 917 694 564 780 834 852 369 318 409 343 764 857 284 113 803 237 322 377 642 890 980 883 160 291 376 298 419 656 035 805 327 753 316 529 527 261 623 545 072 578 909 776 792 537 623 005 534 100 018 818 714 759 564 305 793 635 408 897 883 816 419 997 579 218 522 144 566 838 255 501 372 981 484 551 183 971 611 330 545 041 193 462 506 917 385 184 721 695 683 492 906 363 493 883 547 451 021 099 052 080 408 235 777 038 829 913 181 555 859 547 967 676 965 148 112 788 063 933 412 674 429 642 584 218 129 408 102 213 498 760 153 415 594 122 684 759 912 802 543 816 957 010 293 897 433 107 987 061 364 831 062 169 656 430 565 316 379 190 570 468 423 791 910 249 551 630 787 337 005 574 584 383 066 189 914 180 975 049 860 173 081 004 288 621 230 589 282 273 023 070 974 106 469 919 760 271 532 969 845 987 686 462 812 728 313 999 400 052 405 132 102 168 577 195 125 058 177 615 687 275 856 783 166 621 887 347 797 657 401 972 282 507 125 448 486 530 157 828 361 034 388 454 726 485 926 606 987 318 243 861 523 743 790 520 965 805 653 038 463 322 266 930 123 456 381 364 697 258 944 541 834 336 489 361 465 169 484 845 260 492 050 252 012 248 688 288 393 493 581 825 974 510 264 883 094 479 929 571 691 371 121 437 567 462 603 006 625 812 319 786 568 763 971 111 579 101 870 369 944 625 425 416 752 364 932 917 708 548 394 894 921 518 859 241 449 136 545 283 395 101 136 034 100 595 025 609 301 140 830 995 011 757 247 050 119 802 905 884 676 323 450 216 469 321 381 595 348 036 708 745 895 776 612 221 437 606 741 734 041 091 814 873 021 339 711 678 009 140 018 871 160 303 177 019 179 784 503 907 656 889 307 065 099 703 832 746 548 108 424 581 645 764 735 581 652 925 345 839 979 515 703 977 592 201 660 710 586 398 969 585 476 593 500 903 008 309 042 220 097 508 683 101 661 466 733 906 967 003 113 923 284 096 356 950 770 064 015 068 197 708 461 685 468 587 698 975 600 097 792 929 783 379 863 741 099 169 997 858 955 109 165 920 100 877 920 719 036 449 730 517 316 056 386 417 819 802 544 275 434 064 116 013 062 112 604 673 296 940 053 863 523 861 131 551 981 440 420 274 903 552 765 459 517 488 037 487 966 113 952 924 210 761 443 526 887 609 135 869 934 195 247 498 933 658 706 962 405 008 359 372 099 144 889 360 951 628 561 540 083 119 644 162 763 218 663 520 572 601 778 687 916 334 125 160 814 707 710 623 675 587 924 097 665 124 060 604 459 777 751 217 423 331 353 653 456 219 397 302 275 188 358 432 388 133 798 893 328 811 322 208 861 906 802 677 284 776 829 689 657 466 578 215 137 761 700 422 958 510 576 561 988 008 058 309 997 608 435 746 275 786 730 144 752 588 184 544 940 496 898 680 990 226 352 446 204 944 647 647 571 895 854 862 936 604 793 235 975 254 283 361 507 398 590 921 862 098 816 266 869 784 568 521 880 131 270 564 581 185 555 926 818 644 827 935 944 639 977 780 952 135 464 937 940 555 510 872 208 803 056 898 289 624 700 480 313 813 492 327 562 661 385 558 886 372 963 506 928 704 832 210 773 668 893 641 772 111 885 062 745 313 524 445 976 971 134 297 043 758 451 266 673 953 302 916 011 558 529 767 946 549 164 994 297 649 977 958 928 456 721 400 062 439 009 235 239 172 447 184 736 943 364 493 696 792 346 756 719 610 907 726 044 652 653 010 102 253 130 952 766 365 361 399 023 646 205 823 544 038 232 849 234 666 468 016 840 839 213 417 302 337 514 586 436 028 069 741 645 295 425 914 609 666 445 195 670 680 805 573 298 213 686 235 677 302 128 357 337 670 813 187 386 524 459 298 617 830 628 047 324 013 434 030 645 715 364 542 230 845 131 571 813 072 888 934 457 581 851 548 557 631 790 685 026 494 918 897 180 984 366 287 654 179 026 414 529 548 284 774 952 016 571 853 806 820 599 396 481 472 153 026 484 166 578 362 688 075 179 867 770 284 849 747 961 352 130 700 380 645 759 538 780 109 763 376 753 871 736 723 218 590 252 406 162 499 643 984 031 388 772 771 184 857 394 627 784 684 134 521 004 447 326 059 304 619 296 238 338 044 618 979 707 854 574 719 683 804 299 076 074 056 878 215 627 518 932 962 504 648 038 573 465 594 485 531 443 357 480 260 229 266 506 296 517 508 268 205 538 262 687 058 686 660 661 393 537 424 795 035 427 237 275 895 494 326 633 382 221 547 655 078 354 273 176 301 410 541 431 303 921 079 553 180 358 687 768 878 644 916 902 139 836 742 487 579 307 453 098 372 115 874 651 856 274 945 773 791 535 711 723 203 788 822 040 798 945 340 260 383 492 383 885 250 108 586 114 814 714 987 457 902 754 909 381 708 885 922 719 711 843 658 091 487 236 795 287 075 782 874 922 728 950 826 199 225 189 655 779 617 525 526 093 729 543 564 958 479 499 835 236 818 446 204 356 587 627 055 046 459 146 184 957 235 578 181 350 384 924 142 897 732 242 050 143 183 281 660 750 060 351 899 441 157 971 724 206 080 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ≈
≈ 5.78×10^4607
≈ 578 milliquattuortrigintaquingentillion 108 millitrestrigintaquingentillion (short scale) / 578 septensexagintaseptingentilliard 108 septensexagintaseptingentillion (long scale)