# Complement Pair Pattern Groups

The groups are classified by the complement pair patterns of the squares.
The lines joining the complement pairs are not shown here.
Pair numbers are indicated by the same letter, (a, b, ...).

## Group Form

The group form of a square is determined as follows:

• the square is listed as a 1-dimensional array of 25 cells, indexed 0 to 24
• a sequence is made showing the square index of each complement number,
(number 13 gets its own index)
• such a sequence is made for each aspect of the square
• the lowest order sequence is chosen as the group form of the square

Example square and and its 1-dimensional array:

```      2  3 14 22 24
8 23 19 11  4
25 20  1  9 10
13 12 16 18  6
17  7 15  5 21

2 3 14 22 24 8 23 19 11 4 25 20 1 9 10 13 12 16 18 6 17 7 15 5 21
```

The 8 aspects of the square:

```      2  3 14 22 24   17 13 25  8  2   21  5 15  7 17   24  4 10  6 21
8 23 19 11  4    7 12 20 23  3    6 18 16 12 13   22 11  9 18  5
25 20  1  9 10   15 16  1 19 14   10  9  1 20 25   14 19  1 16 15
13 12 16 18  6    5 18  9 11 22    4 11 19 23  8    3 23 20 12  7
17  7 15  5 21   21  6 10  4 24   24 22 14  3  2    2  8 25 13 17

24 22 14  3  2   21  6 10  4 24   17  7 15  5 21    2  8 25 13 17
4 11 19 23  8    5 18  9 11 22   13 12 16 18  6    3 23 20 12  7
10  9  1 20 25   15 16  1 19 14   25 20  1  9 10   14 19  1 16 15
6 18 16 12 13    7 12 20 23  3    8 23 19 11  4   22 11  9 18  5
21  5 15  7 17   17 13 25  8  2    2  3 14 22 24   24  4 10  6 21
```

The corresponding sequences of the pair indexes:

``` 4  6 16  9  0 18  1 21 22  3 12 19 10 20 17 15  2 14  5 11 13  7  8 24 23
17  1 12 16 24 13 14 21  9  8 18 22  2  5  6 20  3  0 10 23 15  7 11 19  4
1  0 16 17 11 13 19 10 22  9  7  4 14  5 12 21  2  3 23  6 24 15  8 18 20
20  5 13 17  9  1 14 24 21  4 18 19 22  2  6 16 15  3 10 11  0  8 12 23  7
4  5 18  8  0  1 22 23  3 16 17 24 14 15 12 13  9 10  2 19 21 20  6  7 11
5 17 11  9 24  0 23 20 10  3  8  2 22 15 16 13 14  1 19 18  7 21 12  6  4
13 17 18  4  3  5 22 14 15 11 12  9 10  0  7  8 21  1  2 23 24 16  6 19 20
20 18 12  3 17  6  5 23 10 11  8  9  2 22 16 21 14  4  1 24  0 15 13  7 19
```

The group form of the square is:

``` 1 0 16 17 11 13 19 10 22 9 7 4 14 5 12 21 2 3 23 6 24 15 8 18 20
```

Pairs having a number at index 1, 3, 5, 9, 15, 19, 21 or 23 are shown in color.

## Group Order

The groups are numbered by size in descending order, i.e., from big to small.
Groups of the same size are numbered in descending order of their group forms.

## Groups

There are 101,774,553 groups. The biggest group has 228,960 squares. The 87,430,664 smallest groups each have 2, (complementary), squares.

There are 617 group sizes.

Number of Groups of each Group Size
Group
Size
Number
of Groups
Group
Size
Number
of Groups
Group
Size
Number
of Groups
Group
Size
Number
of Groups
Group
Size
Number
of Groups
Group
Size
Number
of Groups
228960 1 1856 5 1028 8 660 8 416 94 204 230
174240 2 1848 4 1024 16 658 4 414 12 202 44
145632 2 1844 4 1020 4 656 21 412 36 200 358
90144 1 1840 4 1016 4 652 8 410 28 198 12
48544 3 1824 4 1012 4 650 4 408 36 196 176
45072 2 1808 8 1008 12 648 20 404 38 194 32
43776 1 1800 4 1004 8 644 8 402 4 192 498
26688 2 1796 4 1000 4 640 18 400 42 190 28
23232 2 1778 8 998 4 636 10 396 32 188 220
22536 4 1776 2 992 4 632 28 394 4 186 40
20336 2 1772 4 988 4 630 4 392 68 184 302
19728 4 1768 4 984 12 628 16 390 12 182 76
18784 2 1760 2 980 16 626 4 388 40 180 162
16896 2 1736 2 976 8 624 20 386 12 178 48
14464 2 1728 10 972 4 622 4 384 60 176 352
14104 4 1696 18 968 4 620 16 382 12 174 64
13808 4 1680 4 960 14 618 4 380 14 172 208
12544 2 1672 8 956 4 616 66 378 16 170 32
12416 4 1664 4 952 14 614 4 376 62 168 352
12192 2 1634 4 948 4 612 4 374 8 166 80
11568 2 1628 4 944 12 610 8 372 56 164 176
10848 2 1624 4 936 8 608 38 370 20 162 64
10272 2 1616 4 932 8 604 24 368 72 160 514
9776 4 1608 6 928 14 600 34 366 4 158 76
9004 2 1592 6 920 8 596 20 364 64 156 284
8352 2 1584 2 916 4 594 8 362 20 154 92
7200 8 1576 2 912 20 592 22 360 82 152 500
7008 2 1552 4 910 4 590 4 358 20 150 72
6656 2 1548 2 904 4 588 16 356 84 148 282
6622 4 1536 2 900 4 584 26 354 12 146 80
6456 4 1532 4 898 8 580 26 352 128 144 715
6432 2 1520 12 896 29 578 8 350 8 142 128
6400 2 1512 4 892 8 576 34 348 60 140 328
6272 1 1508 4 888 14 572 16 346 24 138 120
6096 1 1504 4 884 12 570 4 344 124 136 870
5936 6 1484 8 880 23 568 28 342 12 134 84
5536 2 1480 12 878 4 566 12 340 54 132 516
5264 4 1474 4 876 8 564 14 336 126 130 180
5088 2 1468 4 872 4 562 8 334 12 128 1125
4992 4 1456 4 868 8 560 34 332 26 126 140
4944 8 1452 16 864 22 558 4 330 16 124 534
4904 2 1448 4 856 4 556 8 328 76 122 100
4400 2 1446 4 852 12 554 4 326 36 120 918
4376 2 1444 4 850 4 552 20 324 76 118 128
4320 2 1442 16 848 34 550 8 322 28 116 702
4256 7 1438 4 844 4 548 36 320 157 114 200
4224 4 1432 4 840 26 546 8 318 20 112 1130
4104 2 1426 8 836 8 544 46 316 64 110 192
4096 2 1424 4 834 4 542 4 314 12 108 902
4048 2 1416 4 832 16 540 40 312 132 106 320
3976 4 1408 20 828 4 538 4 310 20 104 1058
3960 8 1400 8 826 8 536 22 308 50 102 280
3696 4 1380 4 824 8 532 10 306 8 100 1458
3632 4 1376 8 820 8 530 4 304 134 98 372
3480 4 1368 4 818 8 528 81 302 12 96 2135
3328 2 1352 6 808 20 526 8 300 60 94 436
3312 2 1344 8 804 8 524 18 298 12 92 1246
3248 8 1336 8 800 20 522 4 296 74 90 440
3216 4 1328 12 796 8 520 36 294 12 88 2080
3120 2 1324 4 792 26 518 20 292 80 86 532
3068 4 1312 2 788 12 516 46 290 8 84 1796
3064 2 1296 10 784 16 512 48 288 162 82 580
3024 4 1294 8 780 8 508 18 286 16 80 3156
2928 4 1288 18 778 4 506 8 284 82 78 788
2876 4 1276 8 776 28 504 30 282 32 76 2276
2844 4 1272 8 772 8 502 4 280 64 74 792
2736 4 1264 8 770 12 500 30 278 16 72 4171
2728 8 1260 4 768 22 498 8 276 104 70 1072
2688 4 1256 4 766 4 496 22 274 16 68 2938
2672 6 1248 2 764 4 492 24 272 152 66 1228
2640 4 1240 12 762 4 490 12 270 8 64 5806
2608 4 1236 4 760 34 488 65 268 86 62 1500
2596 6 1232 6 756 8 486 4 266 12 60 4360
2592 2 1224 4 752 22 484 18 264 104 58 1852
2576 4 1220 4 750 8 482 12 262 28 56 7452
2496 2 1216 14 748 8 480 76 260 104 54 2432
2432 12 1212 4 744 22 476 24 258 8 52 6492
2428 4 1208 8 740 8 474 4 256 193 50 3332
2376 8 1204 4 736 8 472 54 254 8 48 10913
2336 4 1200 4 732 4 470 8 252 110 46 3624
2272 8 1198 4 728 8 468 18 250 20 44 9976
2208 8 1192 4 724 16 466 4 248 104 42 5208
2200 4 1176 4 722 4 464 64 246 4 40 17918
2176 8 1156 4 720 24 462 8 244 110 38 7096
2172 4 1148 4 716 16 460 20 242 16 36 20852
2160 4 1144 4 712 10 456 66 240 214 34 9464
2136 4 1136 12 708 16 454 8 238 32 32 37725
2112 4 1132 2 706 4 452 26 236 132 30 12612
2092 6 1128 12 704 12 450 16 234 16 28 30346
2080 8 1124 6 702 4 448 122 232 236 26 20136
2064 2 1104 6 700 4 446 4 230 32 24 66514
2016 12 1102 4 696 20 444 46 228 62 22 31356
1984 16 1100 10 692 12 442 4 226 36 20 103524
1968 4 1096 4 688 24 440 40 224 263 18 58760
1960 9 1088 16 686 4 438 4 222 24 16 230507
1952 10 1084 6 684 12 436 28 220 142 14 114724
1944 1 1068 8 680 32 432 58 218 16 12 290230
1936 10 1064 6 678 8 430 4 216 258 10 378612
1920 2 1056 15 676 24 428 22 214 24 8 2170810
1904 4 1054 8 672 52 426 4 212 76 6 991212
1888 2 1048 8 668 12 424 56 210 32 4 9634186
1872 4 1040 26 664 12 420 48 208 327 2 87430664
1864 4 1032 2 662 4 418 8 206 36 - -

##### REFERENCES

Heinz, H.D. and J.R. Hendricks "Magic Square Lexicon: Illustrated"