.. -*- coding: utf-8 -*- ====================== Notes on Polyiamonds ====================== :Author: David Goodger :Date: $Date: 2016-12-05 18:35:43 -0600 (Mon, 05 Dec 2016) $ :Revision: $Revision: 644 $ :Web site: http://puzzler.sourceforge.net/ :Copyright: © 1998-2016 by David J. Goodger :License: `GPL 2 <../COPYING.html>`__ .. image:: images/puzzler.png :align: center .. sidebar:: Also see: * `Hexiamond Puzzles & Solutions `_ * `Heptiamond Puzzles & Solutions `_ * `Polyiamond Puzzles & Solutions `_ * `An Introduction to Polyiamonds `_ * `Polyform Puzzler: Puzzles & Solutions `_ * `Polyform Puzzler FAQ `_ (`polyform details `__, `numbers of polyforms `__, `interpreting solution files `__) .. contents:: Polyform Counts =============== Units are equilateral triangles. ===== ========== ====== ====== ====== ========= ====== ====== Order Name Free Units Sum One-sided Units Sum ===== ========== ====== ====== ====== ========= ====== ====== 1 moniamond 1 1 1 1 1 1 2 diamond 1 2 3 1 2 3 3 triamond 1 3 6 1 3 6 4 tetriamond 3 12 18 4 16 22 5 pentiamond 4 20 38 6 30 52 6 hexiamond 12 72 110 19 114 166 7 heptiamond 24 168 278 43 301 467 8 octiamond 66 528 806 120 960 1427 ===== ========== ====== ====== ====== ========= ====== ====== Parity Data =========== In the table below, the "Min-D" and "Max-D" columns record, respectively, the minimum and maximum disparities that can be supported by these polyiamonds: the minimum and maximum discrepancies in parity of the puzzle's unit triangles, i.e. the minimum and maximum differences between the number of z=0 and z=1 triangles. ===== ========== ====== ====== ===== ===== ========= ====== ===== ===== Order Name Free Units Min-D Max-D One-sided Units Min-D Max-D ===== ========== ====== ====== ===== ===== ========= ====== ===== ===== 1 moniamond 1 1 1 1 1 1 1 1 2 diamond 1 2 0 0 1 2 0 0 3 triamond 1 3 1 1 1 3 1 1 4 tetriamond 3 12 2 2 4 16 2 2 5 pentiamond 4 20 0 4 6 30 0 6 6 hexiamond 12 72 0 4 19 114 0 8 7 heptiamond 24 168 0 26 43 301 1 45 8 octiamond 66 528 ? ? 120 960 ? ? ===== ========== ====== ====== ===== ===== ========= ====== ===== ===== Shapes ====== Triangles:: T(n) = n² Hexagons:: H(n) = 6T(n) = 6n² Hexagrams:: Hg(n) = H(n) + 6T(n) = 2H(n) = 12n² == ========== ========= ========== n n-Triangle n-Hexagon n-Hexagram == ========== ========= ========== 1 1 6 12 2 4 24 48 3 9 54 108 4 16 96 192 5 25 150 300 6 36 216 432 7 49 294 588 8 64 384 768 9 81 486 972 10 100 600 1200 11 121 726 1452 12 144 864 1728 13 169 1014 2028 14 196 1176 2352 15 225 1350 2700 16 256 1536 3072 17 289 1734 3468 18 324 1944 3888 19 361 2166 4332 20 400 2400 4800 21 441 2646 5292 22 484 2904 5808 23 529 3174 6348 24 576 3456 6912 == ========== ========= ========== Parallelograms:: P(m,n) = 2mn === === === === === === === === === === === === === === === === P m=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === === === === === === === === === === === === === === === n=1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 2 ... 8 12 16 20 24 28 32 36 40 44 48 52 56 60 3 ... ... 18 24 30 36 42 48 54 60 66 72 78 84 90 4 ... ... ... 32 40 48 56 64 72 80 88 96 104 112 120 5 ... ... ... ... 50 60 70 80 90 100 110 120 130 140 150 6 ... ... ... ... ... 72 84 96 108 120 132 144 156 168 180 7 ... ... ... ... ... ... 98 112 126 140 154 168 182 196 210 8 ... ... ... ... ... ... ... 128 144 160 176 192 208 224 240 9 ... ... ... ... ... ... ... ... 162 180 198 216 234 252 270 10 ... ... ... ... ... ... ... ... ... 200 220 240 260 280 300 11 ... ... ... ... ... ... ... ... ... ... 242 264 286 308 330 12 ... ... ... ... ... ... ... ... ... ... ... 288 312 336 360 13 ... ... ... ... ... ... ... ... ... ... ... ... 338 364 390 14 ... ... ... ... ... ... ... ... ... ... ... ... ... 392 420 15 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 450 === === === === === === === === === === === === === === === === Trapezoids:: Tr(m,n) = P(m,n) - T(n) = 2mn - n² === === === === === === === === === === === === === === === === Tr m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 === === === === === === === === === === === === === === === === n=1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 2 ... 8 12 16 20 24 28 32 36 40 44 48 52 56 60 3 ... ... 15 21 27 33 39 45 51 57 63 69 75 81 87 4 ... ... ... 24 32 40 48 56 64 72 80 88 96 104 112 5 ... ... ... ... 35 45 55 65 75 85 95 105 115 125 135 6 ... ... ... ... ... 48 60 72 84 96 108 120 132 144 156 7 ... ... ... ... ... ... 63 77 91 105 119 133 147 161 175 8 ... ... ... ... ... ... ... 80 96 112 128 144 160 176 192 9 ... ... ... ... ... ... ... ... 99 117 135 153 171 189 207 10 ... ... ... ... ... ... ... ... ... 120 140 160 180 200 220 11 ... ... ... ... ... ... ... ... ... ... 143 165 187 209 231 12 ... ... ... ... ... ... ... ... ... ... ... 168 192 216 240 13 ... ... ... ... ... ... ... ... ... ... ... ... 195 221 247 14 ... ... ... ... ... ... ... ... ... ... ... ... ... 224 252 15 ... ... ... ... ... ... ... ... ... ... ... ... ... ... 255 === === === === === === === === === === === === === === === === Elongated hexagons:: He(m,n) = 2P(m,n) + 2T(n) = 4mn + 2n² = 2n(2m + n) === === === === === === === === === === ==== ==== ==== ==== ==== He m=1 2 3 4 5 6 7 8 9 10 11 12 13 14 === === === === === === === === === === ==== ==== ==== ==== ==== n=1 6 10 14 18 22 26 30 34 38 42 46 50 54 58 2 16 24 32 40 48 56 64 72 80 88 96 104 112 120 3 30 42 54 66 78 90 102 114 126 138 150 162 174 186 4 48 64 80 96 112 128 144 160 176 192 208 224 240 256 5 70 90 110 130 150 170 190 210 230 250 270 290 310 330 6 96 120 144 168 192 216 240 264 288 312 336 360 384 408 7 126 154 182 210 238 266 294 322 350 378 406 434 462 490 8 160 192 224 256 288 320 352 384 416 448 480 512 544 576 9 198 234 270 306 342 378 414 450 486 522 558 594 630 666 10 240 280 320 360 400 440 480 520 560 600 640 680 720 760 11 286 330 374 418 462 506 550 594 638 682 726 770 814 858 12 336 384 432 480 528 576 624 672 720 768 816 864 912 960 13 390 442 494 546 598 650 702 754 806 858 910 962 1014 1066 14 448 504 560 616 672 728 784 840 896 952 1008 1064 1120 1176 15 510 570 630 690 750 810 870 930 990 1050 1110 1170 1230 1290 === === === === === === === === === === ==== ==== ==== ==== ==== Semiregular hexagons:: Hs(m,n) = T(m+2n) - 3T(n) = m² + 4mn + 4n² - 3n² = m² + 4mn + n² === === === === === === === === === === === === === ==== ==== Hs m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === === === === === === === === === === === === ==== ==== n=1 13 22 33 46 61 78 97 118 141 166 193 222 253 286 2 ... 37 52 69 88 109 132 157 184 213 244 277 312 349 3 ... ... 73 94 117 142 169 198 229 262 297 334 373 414 4 ... ... ... 121 148 177 208 241 276 313 352 393 436 481 5 ... ... ... ... 181 214 249 286 325 366 409 454 501 550 6 ... ... ... ... ... 253 292 333 376 421 468 517 568 621 7 ... ... ... ... ... ... 337 382 429 478 529 582 637 694 8 ... ... ... ... ... ... ... 433 484 537 592 649 708 769 9 ... ... ... ... ... ... ... ... 541 598 657 718 781 846 10 ... ... ... ... ... ... ... ... ... 661 724 789 856 925 11 ... ... ... ... ... ... ... ... ... ... 793 862 933 1006 12 ... ... ... ... ... ... ... ... ... ... ... 937 1012 1089 13 ... ... ... ... ... ... ... ... ... ... ... ... 1093 1174 14 ... ... ... ... ... ... ... ... ... ... ... ... ... 1261 === === === === === === === === === === === === === ==== ==== Chevrons:: C(m,n) = P(m,2n) = 4mn === === === === === === === === === === === === === === === === C m=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === === === === === === === === === === === === === === === n=1 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 2 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 3 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 4 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 5 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 6 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 7 28 56 84 112 140 168 196 224 252 280 308 336 364 392 420 8 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 9 36 72 108 144 180 216 252 288 324 360 396 432 468 504 540 10 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 11 44 88 132 176 220 264 308 352 396 440 484 528 572 616 660 12 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720 13 52 104 156 208 260 312 364 416 468 520 572 624 676 728 780 14 56 112 168 224 280 336 392 448 504 560 616 672 728 784 840 15 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 === === === === === === === === === === === === === === === === Butterflies:: B(m,n) = 2Tr(m,n) = 4mn - 2n² === === === === === === === === === === === === === === === B m=2 3 4 5 6 7 8 9 10 11 12 13 14 15 === === === === === === === === === === === === === === === n=1 6 10 14 18 22 26 30 34 38 42 46 50 54 58 2 ... 16 24 32 40 48 56 64 72 80 88 96 104 112 3 ... ... 30 42 54 66 78 90 102 114 126 138 150 162 4 ... ... ... 48 64 80 96 112 128 144 160 176 192 208 5 ... ... ... ... 70 90 110 130 150 170 190 210 230 250 6 ... ... ... ... ... 96 120 144 168 192 216 240 264 288 7 ... ... ... ... ... ... 126 154 182 210 238 266 294 322 8 ... ... ... ... ... ... ... 160 192 224 256 288 320 352 9 ... ... ... ... ... ... ... ... 198 234 270 306 342 378 10 ... ... ... ... ... ... ... ... ... 240 280 320 360 400 11 ... ... ... ... ... ... ... ... ... ... 286 330 374 418 12 ... ... ... ... ... ... ... ... ... ... ... 336 384 432 13 ... ... ... ... ... ... ... ... ... ... ... ... 390 442 14 ... ... ... ... ... ... ... ... ... ... ... ... ... 448 === === === === === === === === === === === === === === === Potential Puzzles ================= *Puzzles not otherwise noted below have not been implemented or solved.* *Initial numbers are the counts of unit equilateral triangles in the puzzles.* 6: polyiamonds of order 1 to 3 * H(1) * P(3,1) * B(2,1) 18: polyiamonds of order 1 to 4 * P(3,3) * He(4,1) * B(5,1) * hexgrid T(2) = 3H(1) 22: one-sided polyiamonds of order 1 to 4 * P(11,1) * He(5,1) * Hs(3,1) * B(6,1) 38: polyiamonds of order 1 to 5 52: one-sided polyiamonds of order 1 to 5 * P(13,2) * Tr(14,2) * C(13,1) 72: Hexiamonds * Potential: * hexgrid Ri(8,2) * hexgrid Rr(5,3) * hexgrid C(4,2) * No solutions: * P(12,3) (= Hexiamonds3x12) * Tr(9,6) (= Hexiamonds6x9Trapezium) * B(10,2) (= Hexiamonds4x10LongButterfly) * H(4) - central H(2) (= HexiamondsRing) * H(4) - pinwheel-6 hole (= HexiamondsHexagon3) * H(4) - trefoil hole (4H(1); = HexiamondsHexagon4) * H(4) - pinwheel-3 hole (= HexiamondsHexagon5) * H(4) - trefoil hole (3C(2,1); = HexiamondsHexagon6) * H(2) * 3 (= Hexiamonds3Hexagons) * T(9) - T(3) (= HexiamondsTriangleRing_x, HexiamondsV_9x9) * T(6) * 2 (= HexiamondsTwoTriangles) * H(2) + 3P(4,2) (= HexiamondsSpinner_x1) * H(2) + 6P(2,2) (= HexiamondsSpinner_x2) * T(6) + 3P(3,2) (= HexiamondsSpinner_x3) * hexgrid H(3) - H(2) (= HexiamondsHexgridHexagonRing_x) * hexgrid Hg(2) - H(1) (= HexiamondsHexgridHexagramRing_x) * hexgrid T(5) - T(2) (= HexiamondsHexgridTriangleRing_x1 & _x2) * hexgrid P(6,2) (= HexiamondsHexgrid6x2_x) 92: pentiamonds & hexiamonds - 4 congruent groups (1 pentiamond & 3 hexiamonds each) -- combined in a symmetrical shape? * H(4) - 4 units (e.g. 4 doubled hexiamonds less 1 unit each) * T(10) - 8 units (e.g. 4(T(5) - 2)) * 4(H(2) - 1) * P(7,7) - 6 units * P(8,6) - 4 units (e.g. 4 doubled hexiamonds less 1 unit each; 4(P(4,3) - 1)) 108: 18 of 19 one-sided hexiamonds * Potential: 110: polyiamonds of order 1 to 6 114: One-sided hexiamonds * Potential: * Tr(14,5) - 1 * B(8,6) - H(1) * C(various) - H(1) * He(14,2) - H(1) * He(2,6) - H(1) * Tr(11,10) - H(1) * Tr(13,6) - H(1) * P(15,4)/P(12,5)/P(10,6) - H(1) * instead of H(1), B(2,1) * hexgrid Rr(13,2) * hexgrid Rs(4,3) * hexgrid Rs(6,2) * No solutions: * P(19,3) (= OneSidedHexiamonds19x3) 166: one-sided polyiamonds of order 1 to 6 168: Heptiamonds * Potential: * hexgrid Rs(14,2) * hexgrid Ri(11,3) * hexgrid Ri(8,4) * hexgrid C(4,4) * more designs from Johannes H. Hindriks (see email 2012-05-30) * No solutions: * hexgrid H(4) - 9 units, spread out (= HeptiamondsHexgridHexagon_x1) * hexgrid 4H(2) trefoil (= HeptiamondsHexgridTrefoil_x1) * hexgrid 4H(2) in a row (= HeptiamondsHexgridRosettes_x1) * H(5) surrounded by 30 individual unit triangles, with hole(s) of 12 unit triangles (HeptiamondsJaggedHexagon_x1, _x2, _x3) 278: polyiamonds of order 1 to 7 * H(7) - T(4) * P(14,10) - 2 units * {C(14,5), C(10,7), C(7,10), C(5,14)} - 2 units 301: One-sided heptiamonds * He(4,9) - 5 units * B(13,9) - 5 units 467: one-sided polyiamonds of order 1 to 7 * Hs(12,6) - 1 unit * C(13,9) - 1 unit * C(9,13) - 1 unit Links ===== * `MathPuzzle.com `_ * `The Poly Pages `__ * `Kadon's Iamond Ring `__