.. -*- coding: utf-8 -*- ==================== Notes on Polyhexes ==================== :Author: David Goodger :Date: $Date: 2015-02-24 14:21:02 -0600 (Tue, 24 Feb 2015) $ :Revision: $Revision: 600 $ :Web site: http://puzzler.sourceforge.net/ :Copyright: © 1998-2015 by David J. Goodger :License: `GPL 2 <../COPYING.html>`__ .. image:: images/puzzler.png :align: center .. sidebar:: Also see: * `Pentahex Puzzles & Solutions `_ * `Polyhex Puzzles & Solutions `_ * `An Introduction to Polyhexes `_ * `Polyform Puzzler: Puzzles & Solutions `_ * `Polyform Puzzler FAQ `_ (`polyform details `__, `numbers of polyforms `__, `interpreting solution files `__) .. contents:: Polyform Counts =============== Units are regular hexagons. ===== ========== ====== ====== ====== ========= ====== ====== Order Name Free Units Sum One-sided Units Sum ===== ========== ====== ====== ====== ========= ====== ====== 1 monohex 1 1 1 1 1 1 2 dihex 1 2 3 1 2 3 3 trihex 3 9 12 3 9 12 4 tetrahex 7 28 40 10 40 52 5 pentahex 22 110 150 33 165 217 6* hexahex 82 492 642 147 882 1099 7* heptahex 333 620 8* octahex 1448 2821 ===== ========== ====== ====== ====== ========= ====== ====== "*" indicates that pieces with holes exist. E.g. one hexahex contains a central hole (1 hexagon). Shapes ====== Triangles:: T(n) = n(n + 1)/2 Hexagons:: H(n) = 3n(n - 1) + 1 Hexagrams:: Hg(n) = H(n) + 6T(n-1) = 6n² - 6n + 1 == === ==== ==== n T H Hg == === ==== ==== 1 1 1 1 2 3 7 13 3 6 19 37 4 10 37 73 5 15 61 121 6 21 91 181 7 28 127 253 8 36 169 337 9 45 217 433 10 55 271 541 11 66 331 661 12 78 397 793 13 91 469 937 14 105 547 1093 15 120 631 1261 16 136 721 1441 17 153 817 1633 18 171 919 1837 19 190 1027 2053 20 210 1141 2281 21 231 1261 2521 22 253 1387 2773 23 276 1519 3037 24 300 1657 3313 25 325 1801 3601 26 351 1951 3901 27 378 2107 4213 28 406 2269 4537 29 435 2437 4873 30 465 2611 5221 31 496 2791 5581 32 528 2977 5953 33 561 3169 6337 34 595 3367 6733 35 630 3571 7141 36 666 3781 7561 37 703 3997 7993 38 741 4219 8437 39 780 4447 8893 40 820 4681 9361 == === ==== ==== Parallelograms & Staggered Rectangles:: P(m,n) = m * n Rs(m,n) = P(m,n) === === === === === === === === === === === === === === === === P m=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === === === === === === === === === === === === === === === n=1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 ... 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3 ... ... 9 12 15 18 21 24 27 30 33 36 39 42 45 4 ... ... ... 16 20 24 28 32 36 40 44 48 52 56 60 5 ... ... ... ... 25 30 35 40 45 50 55 60 65 70 75 6 ... ... ... ... ... 36 42 48 54 60 66 72 78 84 90 7 ... ... ... ... ... ... 49 56 63 70 77 84 91 98 105 8 ... ... ... ... ... ... ... 64 72 80 88 96 104 112 120 9 ... ... ... ... ... ... ... ... 81 90 99 108 117 126 135 10 ... ... ... ... ... ... ... ... ... 100 110 120 130 140 150 11 ... ... ... ... ... ... ... ... ... ... 121 132 143 154 165 12 ... ... ... ... ... ... ... ... ... ... ... 144 156 168 180 13 ... ... ... ... ... ... ... ... ... ... ... ... 169 182 195 14 ... ... ... ... ... ... ... ... ... ... ... ... ... 196 210 15 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 225 === === === === === === === === === === === === === === === === === ==== === === === === === === === === === === === === === === P m=16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 === ==== === === === === === === === === === === === === === === n=1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 3 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 4 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 5 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 6 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 7 112 119 126 133 140 147 154 161 168 175 182 189 196 203 210 8 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 9 144 153 162 171 180 189 198 207 216 225 234 243 252 261 270 10 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 11 176 187 198 209 220 231 242 253 264 275 286 297 308 319 330 12 192 204 216 228 240 252 264 276 288 300 312 324 336 348 360 13 208 221 234 247 260 273 286 299 312 325 338 351 364 377 390 14 224 238 252 266 280 294 308 322 336 350 364 378 392 406 420 15 240 255 270 285 300 315 330 345 360 375 390 405 420 435 450 === ==== === === === === === === === === === === === === === === ==== ==== === === === === === === === === === === === === === === P m=16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ==== ==== === === === === === === === === === === === === === === n=16 256 272 288 304 320 336 352 368 384 400 416 432 448 464 480 17 ... 289 306 323 340 357 374 391 408 425 442 459 476 493 510 18 ... ... 324 342 360 378 396 414 432 450 468 486 504 522 540 19 ... ... ... 361 380 399 418 437 456 475 494 513 532 551 570 20 ... ... ... ... 400 420 440 460 480 500 520 540 560 580 600 21 ... ... ... ... ... 441 462 483 504 525 546 567 588 609 630 22 ... ... ... ... ... ... 484 506 528 550 572 594 616 638 660 23 ... ... ... ... ... ... ... 529 552 575 598 621 644 667 690 24 ... ... ... ... ... ... ... ... 576 600 624 648 672 696 720 25 ... ... ... ... ... ... ... ... ... 625 650 675 700 725 750 26 ... ... ... ... ... ... ... ... ... ... 676 702 728 754 780 27 ... ... ... ... ... ... ... ... ... ... ... 729 756 783 810 28 ... ... ... ... ... ... ... ... ... ... ... ... 784 812 840 29 ... ... ... ... ... ... ... ... ... ... ... ... ... 841 870 30 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 900 ==== ==== === === === === === === === === === === === === === === Inset Rectangles:: Ri(m,n) = P(m,n) - int(m/2) === === === === === === === === === === === === === === === === Ri m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 === === === === === === === === === === === === === === === === n=2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24 3 ... 8 10 13 15 18 20 23 25 28 30 33 35 38 40 4 ... ... 14 18 21 25 28 32 35 39 42 46 49 53 56 5 ... ... ... 23 27 32 36 41 45 50 54 59 63 68 72 6 ... ... ... ... 33 39 44 50 55 61 66 72 77 83 88 7 ... ... ... ... ... 46 52 59 65 72 78 85 91 98 104 8 ... ... ... ... ... ... 60 68 75 83 90 98 105 113 120 9 ... ... ... ... ... ... ... 77 85 94 102 111 119 128 136 10 ... ... ... ... ... ... ... ... 95 105 114 124 133 143 152 11 ... ... ... ... ... ... ... ... ... 116 126 137 147 158 168 12 ... ... ... ... ... ... ... ... ... ... 138 150 161 173 184 13 ... ... ... ... ... ... ... ... ... ... ... 163 175 188 200 14 ... ... ... ... ... ... ... ... ... ... ... ... 189 203 216 15 ... ... ... ... ... ... ... ... ... ... ... ... ... 218 232 16 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 248 === === === === === === === === === === === === === === === === === ==== === === === === === === === === === === === === === === Ri m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 === ==== === === === === === === === === === === === === === === n=2 26 27 29 30 32 33 35 36 38 39 41 42 44 45 47 3 43 45 48 50 53 55 58 60 63 65 68 70 73 75 78 4 60 63 67 70 74 77 81 84 88 91 95 98 102 105 109 5 77 81 86 90 95 99 104 108 113 117 122 126 131 135 140 6 94 99 105 110 116 121 127 132 138 143 149 154 160 165 171 7 111 117 124 130 137 143 150 156 163 169 176 182 189 195 202 8 128 135 143 150 158 165 173 180 188 195 203 210 218 225 233 9 145 153 162 170 179 187 196 204 213 221 230 238 247 255 264 10 162 171 181 190 200 209 219 228 238 247 257 266 276 285 295 11 179 189 200 210 221 231 242 252 263 273 284 294 305 315 326 12 196 207 219 230 242 253 265 276 288 299 311 322 334 345 357 13 213 225 238 250 263 275 288 300 313 325 338 350 363 375 388 14 230 243 257 270 284 297 311 324 338 351 365 378 392 405 419 15 247 261 276 290 305 319 334 348 363 377 392 406 421 435 450 16 264 279 295 310 326 341 357 372 388 403 419 434 450 465 481 === ==== === === === === === === === === === === === === === === ==== ==== === === === === === === === === === === === === === === Ri m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ==== ==== === === === === === === === === === === === === === === n=17 281 297 314 330 347 363 380 396 413 429 446 462 479 495 512 18 ... 315 333 350 368 385 403 420 438 455 473 490 508 525 543 19 ... ... 352 370 389 407 426 444 463 481 500 518 537 555 574 20 ... ... ... 390 410 429 449 468 488 507 527 546 566 585 605 21 ... ... ... ... 431 451 472 492 513 533 554 574 595 615 636 22 ... ... ... ... ... 473 495 516 538 559 581 602 624 645 667 23 ... ... ... ... ... ... 518 540 563 585 608 630 653 675 698 24 ... ... ... ... ... ... ... 564 588 611 635 658 682 705 729 25 ... ... ... ... ... ... ... ... 613 637 662 686 711 735 760 26 ... ... ... ... ... ... ... ... ... 663 689 714 740 765 791 27 ... ... ... ... ... ... ... ... ... ... 716 742 769 795 822 28 ... ... ... ... ... ... ... ... ... ... ... 770 798 825 853 29 ... ... ... ... ... ... ... ... ... ... ... ... 827 855 884 30 ... ... ... ... ... ... ... ... ... ... ... ... ... 885 915 31 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 946 ==== ==== === === === === === === === === === === === === === === Rounded Rectangles:: Rr(m,n) = Ri(m,n) - (m % 2) === === === === === === === === === === === === === === === === Rr m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 === === === === === === === === === === === === === === === === n=2 3 4 6 7 9 10 12 13 15 16 18 19 21 22 24 3 ... 7 10 12 15 17 20 22 25 27 30 32 35 37 40 4 ... ... 14 17 21 24 28 31 35 38 42 45 49 52 56 5 ... ... ... 22 27 31 36 40 45 49 54 58 63 67 72 6 ... ... ... ... 33 38 44 49 55 60 66 71 77 82 88 7 ... ... ... ... ... 45 52 58 65 71 78 84 91 97 104 8 ... ... ... ... ... ... 60 67 75 82 90 97 105 112 120 9 ... ... ... ... ... ... ... 76 85 93 102 110 119 127 136 10 ... ... ... ... ... ... ... ... 95 104 114 123 133 142 152 11 ... ... ... ... ... ... ... ... ... 115 126 136 147 157 168 12 ... ... ... ... ... ... ... ... ... ... 138 149 161 172 184 13 ... ... ... ... ... ... ... ... ... ... ... 162 175 187 200 14 ... ... ... ... ... ... ... ... ... ... ... ... 189 202 216 15 ... ... ... ... ... ... ... ... ... ... ... ... ... 217 232 16 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 248 === === === === === === === === === === === === === === === === === ==== === === === === === === === === === === === === === === Rr m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 === ==== === === === === === === === === === === === === === === n=2 25 27 28 30 31 33 34 36 37 39 40 42 43 45 46 3 42 45 47 50 52 55 57 60 62 65 67 70 72 75 77 4 59 63 66 70 73 77 80 84 87 91 94 98 101 105 108 5 76 81 85 90 94 99 103 108 112 117 121 126 130 135 139 6 93 99 104 110 115 121 126 132 137 143 148 154 159 165 170 7 110 117 123 130 136 143 149 156 162 169 175 182 188 195 201 8 127 135 142 150 157 165 172 180 187 195 202 210 217 225 232 9 144 153 161 170 178 187 195 204 212 221 229 238 246 255 263 10 161 171 180 190 199 209 218 228 237 247 256 266 275 285 294 11 178 189 199 210 220 231 241 252 262 273 283 294 304 315 325 12 195 207 218 230 241 253 264 276 287 299 310 322 333 345 356 13 212 225 237 250 262 275 287 300 312 325 337 350 362 375 387 14 229 243 256 270 283 297 310 324 337 351 364 378 391 405 418 15 246 261 275 290 304 319 333 348 362 377 391 406 420 435 449 16 263 279 294 310 325 341 356 372 387 403 418 434 449 465 480 === ==== === === === === === === === === === === === === === === ==== ==== === === === === === === === === === === === === === === Rr m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ==== ==== === === === === === === === === === === === === === === n=17 280 297 313 330 346 363 379 396 412 429 445 462 478 495 511 18 ... 315 332 350 367 385 402 420 437 455 472 490 507 525 542 19 ... ... 351 370 388 407 425 444 462 481 499 518 536 555 573 20 ... ... ... 390 409 429 448 468 487 507 526 546 565 585 604 21 ... ... ... ... 430 451 471 492 512 533 553 574 594 615 635 22 ... ... ... ... ... 473 494 516 537 559 580 602 623 645 666 23 ... ... ... ... ... ... 517 540 562 585 607 630 652 675 697 24 ... ... ... ... ... ... ... 564 587 611 634 658 681 705 728 25 ... ... ... ... ... ... ... ... 612 637 661 686 710 735 759 26 ... ... ... ... ... ... ... ... ... 663 688 714 739 765 790 27 ... ... ... ... ... ... ... ... ... ... 715 742 768 795 821 28 ... ... ... ... ... ... ... ... ... ... ... 770 797 825 852 29 ... ... ... ... ... ... ... ... ... ... ... ... 826 855 883 30 ... ... ... ... ... ... ... ... ... ... ... ... ... 885 914 31 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 945 ==== ==== === === === === === === === === === === === === === === Trapezoids:: Tr(m,n) = T(m) - T(m - n) n/2(2m - n + 1) === === === === === === === === === === === === === === === === Tr m=3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 === === === === === === === === === === === === === === === === n=2 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 3 ... 9 12 15 18 21 24 27 30 33 36 39 42 45 48 4 ... ... 14 18 22 26 30 34 38 42 46 50 54 58 62 5 ... ... ... 20 25 30 35 40 45 50 55 60 65 70 75 6 ... ... ... ... 27 33 39 45 51 57 63 69 75 81 87 7 ... ... ... ... ... 35 42 49 56 63 70 77 84 91 98 8 ... ... ... ... ... ... 44 52 60 68 76 84 92 100 108 9 ... ... ... ... ... ... ... 54 63 72 81 90 99 108 117 10 ... ... ... ... ... ... ... ... 65 75 85 95 105 115 125 11 ... ... ... ... ... ... ... ... ... 77 88 99 110 121 132 12 ... ... ... ... ... ... ... ... ... ... 90 102 114 126 138 13 ... ... ... ... ... ... ... ... ... ... ... 104 117 130 143 14 ... ... ... ... ... ... ... ... ... ... ... ... 119 133 147 15 ... ... ... ... ... ... ... ... ... ... ... ... ... 135 150 16 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 152 === === === === === === === === === === === === === === === === === ==== === === === === === === === === === === === === === === Tr m=18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 === ==== === === === === === === === === === === === === === === n=2 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 3 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 4 66 70 74 78 82 86 90 94 98 102 106 110 114 118 122 5 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 6 93 99 105 111 117 123 129 135 141 147 153 159 165 171 177 7 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203 8 116 124 132 140 148 156 164 172 180 188 196 204 212 220 228 9 126 135 144 153 162 171 180 189 198 207 216 225 234 243 252 10 135 145 155 165 175 185 195 205 215 225 235 245 255 265 275 11 143 154 165 176 187 198 209 220 231 242 253 264 275 286 297 12 150 162 174 186 198 210 222 234 246 258 270 282 294 306 318 13 156 169 182 195 208 221 234 247 260 273 286 299 312 325 338 14 161 175 189 203 217 231 245 259 273 287 301 315 329 343 357 15 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 16 168 184 200 216 232 248 264 280 296 312 328 344 360 376 392 === ==== === === === === === === === === === === === === === === ==== ==== === === === === === === === === === === === === === === Tr m=18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ==== ==== === === === === === === === === === === === === === === n=17 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 18 ... 189 207 225 243 261 279 297 315 333 351 369 387 405 423 19 ... ... 209 228 247 266 285 304 323 342 361 380 399 418 437 20 ... ... ... 230 250 270 290 310 330 350 370 390 410 430 450 21 ... ... ... ... 252 273 294 315 336 357 378 399 420 441 462 22 ... ... ... ... ... 275 297 319 341 363 385 407 429 451 473 23 ... ... ... ... ... ... 299 322 345 368 391 414 437 460 483 24 ... ... ... ... ... ... ... 324 348 372 396 420 444 468 492 25 ... ... ... ... ... ... ... ... 350 375 400 425 450 475 500 26 ... ... ... ... ... ... ... ... ... 377 403 429 455 481 507 27 ... ... ... ... ... ... ... ... ... ... 405 432 459 486 513 28 ... ... ... ... ... ... ... ... ... ... ... 434 462 490 518 29 ... ... ... ... ... ... ... ... ... ... ... ... 464 493 522 30 ... ... ... ... ... ... ... ... ... ... ... ... ... 495 525 31 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 527 ==== ==== === === === === === === === === === === === === === === Elongated Hexagons:: He(m,n) = H(n) + (m - n)(2n - 1) === === === === === === === === === === === === === === === === He m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 === === === === === === === === === === === === === === === === n=2 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 3 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 4 23 30 37 44 51 58 65 72 79 86 93 100 107 114 121 5 34 43 52 61 70 79 88 97 106 115 124 133 142 151 160 6 47 58 69 80 91 102 113 124 135 146 157 168 179 190 201 7 62 75 88 101 114 127 140 153 166 179 192 205 218 231 244 8 79 94 109 124 139 154 169 184 199 214 229 244 259 274 289 9 98 115 132 149 166 183 200 217 234 251 268 285 302 319 336 10 119 138 157 176 195 214 233 252 271 290 309 328 347 366 385 11 142 163 184 205 226 247 268 289 310 331 352 373 394 415 436 12 167 190 213 236 259 282 305 328 351 374 397 420 443 466 489 13 194 219 244 269 294 319 344 369 394 419 444 469 494 519 544 14 223 250 277 304 331 358 385 412 439 466 493 520 547 574 601 15 254 283 312 341 370 399 428 457 486 515 544 573 602 631 660 16 287 318 349 380 411 442 473 504 535 566 597 628 659 690 721 === === === === === === === === === === === === === === === === Semi-regular Hexagons:: Hs(m,n) = T(m + 2n - 2) - 3T(n - 1) === === === === === === === === === === === === === === === === Hs m=3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 === === === === === === === === === === === === === === === === n=2 12 18 25 33 42 52 63 75 88 102 117 133 150 168 187 3 ... 27 36 46 57 69 82 96 111 127 144 162 181 201 222 4 ... ... 48 60 73 87 102 118 135 153 172 192 213 235 258 5 ... ... ... 75 90 106 123 141 160 180 201 223 246 270 295 6 ... ... ... ... 108 126 145 165 186 208 231 255 280 306 333 7 ... ... ... ... ... 147 168 190 213 237 262 288 315 343 372 8 ... ... ... ... ... ... 192 216 241 267 294 322 351 381 412 9 ... ... ... ... ... ... ... 243 270 298 327 357 388 420 453 10 ... ... ... ... ... ... ... ... 300 330 361 393 426 460 495 11 ... ... ... ... ... ... ... ... ... 363 396 430 465 501 538 12 ... ... ... ... ... ... ... ... ... ... 432 468 505 543 582 13 ... ... ... ... ... ... ... ... ... ... ... 507 546 586 627 14 ... ... ... ... ... ... ... ... ... ... ... ... 588 630 673 15 ... ... ... ... ... ... ... ... ... ... ... ... ... 675 720 16 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 768 === === === === === === === === === === === === === === === === Chevrons:: C(m,n) = P(m, (2n - 1)) === === == == === === === === === === === === === === === === C m=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === == == === === === === === === === === === === === === n=2 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 3 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 4 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 5 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 6 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 7 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 8 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 9 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 10 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 11 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 12 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 13 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 14 27 54 81 108 135 162 189 216 243 270 297 324 351 378 405 15 29 58 87 116 145 174 203 232 261 290 319 348 377 406 435 16 31 62 93 124 155 186 217 248 279 310 341 372 403 434 465 === === == == === === === === === === === === === === === === Butterflies:: B(m,n) = C(m,n) - (n - 1)² === === === === === === === === === === === === === === === === B m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 === === === === === === === === === === === === === === === === n=2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 3 ... 11 16 21 26 31 36 41 46 51 56 61 66 71 76 4 ... ... 19 26 33 40 47 54 61 68 75 82 89 96 103 5 ... ... ... 29 38 47 56 65 74 83 92 101 110 119 128 6 ... ... ... ... 41 52 63 74 85 96 107 118 129 140 151 7 ... ... ... ... ... 55 68 81 94 107 120 133 146 159 172 8 ... ... ... ... ... ... 71 86 101 116 131 146 161 176 191 9 ... ... ... ... ... ... ... 89 106 123 140 157 174 191 208 10 ... ... ... ... ... ... ... ... 109 128 147 166 185 204 223 11 ... ... ... ... ... ... ... ... ... 131 152 173 194 215 236 12 ... ... ... ... ... ... ... ... ... ... 155 178 201 224 247 13 ... ... ... ... ... ... ... ... ... ... ... 181 206 231 256 14 ... ... ... ... ... ... ... ... ... ... ... ... 209 236 263 15 ... ... ... ... ... ... ... ... ... ... ... ... ... 239 268 16 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 271 === === === === === === === === === === === === === === === === === ==== === === === === === === === === === === === === === === B m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 === ==== === === === === === === === === === === === === === === n=2 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 3 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 4 110 117 124 131 138 145 152 159 166 173 180 187 194 201 208 5 137 146 155 164 173 182 191 200 209 218 227 236 245 254 263 6 162 173 184 195 206 217 228 239 250 261 272 283 294 305 316 7 185 198 211 224 237 250 263 276 289 302 315 328 341 354 367 8 206 221 236 251 266 281 296 311 326 341 356 371 386 401 416 9 225 242 259 276 293 310 327 344 361 378 395 412 429 446 463 10 242 261 280 299 318 337 356 375 394 413 432 451 470 489 508 11 257 278 299 320 341 362 383 404 425 446 467 488 509 530 551 12 270 293 316 339 362 385 408 431 454 477 500 523 546 569 592 13 281 306 331 356 381 406 431 456 481 506 531 556 581 606 631 14 290 317 344 371 398 425 452 479 506 533 560 587 614 641 668 15 297 326 355 384 413 442 471 500 529 558 587 616 645 674 703 16 302 333 364 395 426 457 488 519 550 581 612 643 674 705 736 === ==== === === === === === === === === === === === === === === ==== ==== === === === === === === === === === === === === === === B m=17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ==== ==== === === === === === === === === === === === === === === n=17 305 338 371 404 437 470 503 536 569 602 635 668 701 734 767 18 ... 341 376 411 446 481 516 551 586 621 656 691 726 761 796 19 ... ... 379 416 453 490 527 564 601 638 675 712 749 786 823 20 ... ... ... 419 458 497 536 575 614 653 692 731 770 809 848 21 ... ... ... ... 461 502 543 584 625 666 707 748 789 830 871 22 ... ... ... ... ... 505 548 591 634 677 720 763 806 849 892 23 ... ... ... ... ... ... 551 596 641 686 731 776 821 866 911 24 ... ... ... ... ... ... ... 599 646 693 740 787 834 881 928 25 ... ... ... ... ... ... ... ... 649 698 747 796 845 894 943 26 ... ... ... ... ... ... ... ... ... 701 752 803 854 905 956 27 ... ... ... ... ... ... ... ... ... ... 755 808 861 914 967 28 ... ... ... ... ... ... ... ... ... ... ... 811 866 921 976 29 ... ... ... ... ... ... ... ... ... ... ... ... 869 926 983 30 ... ... ... ... ... ... ... ... ... ... ... ... ... 929 988 31 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 991 ==== ==== === === === === === === === === === === === === === === Potential Puzzles ================= *Puzzles not otherwise noted below have not been implemented or solved.* *Initial numbers are the counts of unit hexagons in the puzzles.* 12: polyhexes of order 1 to 3 * P(6,2) & Rs * P(4,3) & Rs * Ri(8,2) * Rr(5,3) * Tr(5,3) * Hs(3,2) * C(4,2) 28: tetrahexes * Potential: * P(7,4) & Rs * Ri(11,3) * Ri(8,4) * He(9,2) * C(4,4) * No solutions: * T(7) (= Tetrahex7x7Triangle) * P(14,2) & Rs (not possible) * H(4) - P(3,3) (= TetrahexesHexagon_x1) * 2P(4,4) overlapping, - 3 units (= TetrahexesTwoDiamonds_x1) * 4H(2) around 1 empty unit (= TetrahexesRosettes_x1) * 4H(2) in symmetrical trefoil (not possible) * 4H(2) in a row (not possible) * TetrahexesTrefoil_x1 ... _x4 37: trihexes + tetrahexes * He(12,2) * Rr(15,3) * Ri(15,3) - 1 unit * Rr(11,4) - 1 unit * Rr(7,6) - 1 unit * Tr(11,4) - 1 unit * 2H(3) overlapping (= He(7,3) - 2 units) 40: polyhexes of order 1 to 4 * Ri(16,3) * Rr(9,5) * Tr(10,5) * C(8,3) * B(7,4) * Hs(6,3) - 6 units (= T(3)) * T(10) - 2T(3) - T(2) "tree" (design by Dan Klarskov, email 2012-02-20) * He(3,5) - 3 units (2 corners & center, design by Dan Klarskov, email 2012-02-20) * Ri(9,5) - 1 unit (design by Dan Klarskov, email 2012-02-20) * Ri(5,5) + 2Tr(4,3) - 1 unit "eye" (design by Dan Klarskov, email 2012-02-20) * "Snowflake ring" (email from Dan Klarskov, 2012-02-27) * Hg(3) + 3P(2,1) - 3 holes, similar to tri-lobed crown (design by Dan Klarskov, email 2012-04-24) * H(4) + 4 units (2 corners & centers of 2 faces) - 1 unit central hole (design by Dan Klarskov, email 2012-04-24) 40: one-sided tetrahexes * P(10,4) & Rs * P(8,5) & Rs * and everything under polyhexes of order 1 to 4 49: one-sided trihexes + tetrahexes * P(7,7) & Rs * T(10) - 6 units * Ri(14,4) * Rr(11,5) * Rr(9,6) * Tr(10,7) * He(16,2) * He(9,3) * C(7,4) 52: one-sided polyhexes of order 1 to 4 * P(13,4) & Rs * Ri(8,7) * Rr(15,4) * Tr(10,8) * He(4,5) * Hs(8,2) * C(4,7) * B(7,6) 110: pentahexes * Rs(11,10) * Rs(22,5) * Ri(20,6) * Rr(13,9) * Rr(17,7) * Tr(15,11) * Tr(29,4) * Tr(24,5) * C(10,6) * B(14,5) * B(17,4) 150: polyhexes of order 1 to 5 * Potential: * Rs(50,3) * Rs(30,5) * Rs(25,6) * Rs(15,10) * Ri(13,12) * C(10,8) * C(6,13) * No solutions: * P(50,3) (= Polyhex12345_3x50) 165: one-sided pentahexes * P(33,5) & Rs * P(55,3) & Rs * P(15,11) & Rs * Hs(10,6) * C(15,6) * C(11,8) * H(8) - 4 units * Hg(6) - 16 units 217: one-sided polyhexes of order 1 to 5 * Rr(15,15) * C(7,16) * B(22,6) * T(22) - T(8) * T(21) - T(5) + 1 unit * T(22) - H(4) + 1 unit 486: hexahexes less the O06 piece (which has a hole) * P(27,18) * Tr(31,27) * He(10,15) 492: hexahexes, complete (all puzzles which include the O06 piece must have at least one single-hexagon hole) (Hexahexes on the Poly Pages: http://www.recmath.com/PolyPages/PolyPages/index.htm?6hexes.html) * Potential: * T(31) - 4 units (1 central & 3 nearby = HexahexesTriangle1; 0 solutions? or just too huge a solution space to search?) * T(31) - 4 units (3 corner tips & 1 central = HexahexesTriangle2; solutions exist, see Poly Pages) * He(12,14) - 1 unit (from Poly Pages) * H(15) - H(7) - 12 units * H(16) - H(9) - 12 units * H(17) - H(11) + 6 units * Hg(10) - H(4) - 12 units * No solutions: impossible since O06 needs a hole: * H(23) - H(19) * Ri(24,21) * Tr(32,24) * 2Hs(15,5) * Hg(11) - H(8) 642: polyhexes of order 1 to 6 882: one-sided hexahexes * 7(H(7) - 1 central unit); see `Hexhexes (by Peter F. Esser)`_ 1099: one-sided polyhexes of order 1 to 6 Links ===== * Kadon's `Hexnut `__ & `Hexnut II `__ * Andrew Clarke's `Poly Pages `__ * `Hexhexes (by Peter F. Esser) `_