# Problem of the Month (May 2017)

Given a polyomino, how can we pack n non-overlapping copies, not necessarily the same size, inside a square to cover the maximum area?

Solutions were received from Bryce Herdt, Maurizio Morandi, George Sicherman, and Berend van der Zwaag.

Triominoes
 3 / 8 = .375 2 / 3 = .666+ 1 3 / 4 = .750 15 / 16 = .937+ 63 / 64 = .984+ 255 / 256 = .996+ 1023 / 1024 = .999+ 1

Tetrominoes
 8 / 25 = .320 229 / 450 = .508+ (MM) 3 / 4 = .750 1 1 / 2 = .500 17 / 25 = .680 3 / 4 = .750 1 4 / 9 = .444+ 8 / 9 = .888+ 76 / 81 = .938+ 1 40 / 81 = .493+ 16 / 25 = .640 3 / 4 = .750 13 / 16 = .812+ 1093 / 1293 = .843+ 1133 / 1293 = .874+ 7223 / 8100 = .891+ 2347 / 2592 = .905+

Pentominoes
 5 / 18 = .277+ .444+ 217 / 360 = .602+ (MM) 4 / 5 = .800 1 5 / 9 = .555+ 25 / 36 = .694+ 15 / 16 = .937+ 35 / 36 = .972+ 80 / 81 = .987+ 255 / 256 = .996+ 575 / 576 = .998+ 1 65 / 196 = .331+ 5 / 8 = .625+ 205 / 256 = .800+ 65 / 72 = .902+ (MM) 15 / 16 = .937+ 35 / 36 = .972+ 5027 / 5120 = .981+ 1 5 / 9 = .555+ 65 / 81 = .802+ 665 / 729 = .912+ 6305 / 6561 = .960+ .982+ .992+ .996+ .998+ 2 / 5 = .400 .632+ (MM) 29 / 40 = .725 257 / 320 = .803+ .893+ (MM) 297 / 320 = .928+ (BH) 19 / 20 = .950 77 / 80 = .962+ 5 / 9 = .555+ 25 / 36 = .694+ 5 / 6 = .833+ 143 / 162 = .882+ 295 / 324 = .910+ 25 / 27 = .925+ 305 / 324 = .941+ 155 / 162 = .956+ 5 / 9 = .555+ 5 / 8 = .625 55 / 81 = .679+ 4 / 5 = .800 37 / 45 = .822+ 38 / 45 = .844+ 13 / 15 = .866+ 8 / 9 = .888+ 25 / 64 = .390+ 5 / 8 = .625 185 / 256 = .722+ 4 / 5 = .800 55 / 64 = .859+ 9 / 10 = .900 293 / 320 = .915+ 297 / 320 = .928+ 5 / 9 = .555+ 25 / 36 = .694+ 245 / 324 = .756+ 1045/1296 = .806+ (MM) 620 / 729 = .850+ (MM) 2525/2916 = .865+ (MM) 2605/2916 = .893+ (MM) 1325/1458 = .908+ (MM) 5 / 9 = .555+ 205 / 324 = .632+ 25 / 36 = .694+ 4 / 5 = .800 37 / 45 = .822+ 38 / 45 = .844+ 13 / 15 = .866+ 8 / 9 = .888+ 5 / 9 = .555+ 2 / 3 = .666+ 7 / 9 = .777+ 64 / 81 = .790+ 596 / 729 = .817+ 605 / 729 = .829+ 625 / 729 = .857+ 634 / 729 = .869+ 5 / 8 = .625 255 / 392 = .650+ 25 / 36 = .694+ 55 / 72 = .763+ 5 / 6 = .833+ 1475 / 1764 = .836+ 545 / 648 = .841+ 275 / 324 = .848+

Hexominoes
 2 / 3 = .666+ 5 / 6 = .833+ 7 / 8 = .875 230 / 243 = .946+ (MM) 232 / 243 = .954+ (MM) 26 / 27 = .962+ (MM) 2 / 3 = .666+ 3 / 4 = .750 22 / 27 = .814+ 24 / 25 = .960 74 / 75 = .986+ 99 / 100 = .990 3 / 8 = .375 3 / 4 = .750 27 / 32 = .843+ 15 / 16 = .937+ 123 / 128 = .960+ 63 / 64 = .984+ 12 / 25 = .480 3 / 4 = .750 27 / 32 = .843+ 24 / 25 = .960 612 / 625 = .979+ 99 / 100 = .990 2 / 3 = .666+ 20 / 27 = .740+ 22 / 27 = .814+ 8 / 9 = .888+ 218 / 243 = .897+ (MM) 220 / 243 = .905+ (MM) 39 / 98 = .397+ 540 / 961 = .561+ (MM) 320 / 507 = .631+ (MM) 20 / 27 = .740+ 409 / 540 = .757+ (MM) 209 / 270 = .774+ (MM) 3 / 8 = .375 30 / 49 = .612+ (MM) 627 / 980 = .639+ (MM) 327 / 490 = .667+ (MM) 17 / 24 = .708+ (MM) 36 / 49 = .734+ (MM) 39 / 98 = .397+ 3 / 4 = .750 27 / 32 = .843+ 24 / 25 = .960 2391 / 2450 = .975+ 99 / 100 = .990 2 / 3 = .666+ 3 / 4 = .750 22 / 27 = .814+ 8 / 9 = .888+ 224 / 243 = .921+ 905 / 972 = .931+ (MM) 12 / 49 = .244+ .398+ .532+ 818 / 1225 = .667+ (GS) 5 / 6 = .833+ 1 2 / 3 = .666+ 5 / 6 = .833+ 7 / 8 = .875 220 / 243 = .905+ (BZ) 203 / 216 = .939+ (BZ) 16813/17496 = .960+ (BZ) 2 / 3 = .666+ 20 / 27 = .740+ 22 / 27 = .814+ 8 / 9 = .888+ 218 / 243 = .897+ (MM) 220 / 243 = .905+ (MM) 150 / 529 = .283+ 1284/2645 = .485+ (MM) 18 / 25 = .720 61 / 75 = .813+ (MM) 1687 / 1875 = .899+ 1 51 / 128 = .398+ 3 / 4 = .750 1587 / 2048 = .774+ 819 / 1024 = .799+ 1689 / 2048 = .824+ 435 / 512 = .849+ 51 / 128 = .398+ 3 / 4 = .750 1101 / 1352 = .814+ 297 / 338 = .878+ 3641 / 4056 = .897+ 11 / 12 = .916+ 204 / 625 = .326+ 84 / 169 = .497+ (MM) 132 / 225 = .586+ (MM) 2 / 3 = .666+ (MM) 1318 / 1875 = .702+ (MM) 36 / 49 = .734+ (MM) 15 / 32 = .468+ 3 / 4 = .750 27 / 32 = .843+ 447 / 512 = .873+ (MM) 231 / 256 = .902+ 471 / 512 = .919+ (MM) 2 / 3 = .666+ 5 / 6 = .813+ 1 39 / 98 = .397+ 3 / 4 = .750 333 / 392 = .849+ (MM) 24 / 25 = .960 2391 / 2450 = .975+ 1 2 / 3 = .666+ 41 / 53 = .759+ (MM) 5 / 6 = .833+ (MM) 8 / 9 = .888+ 224 / 243 = .921+ 4097/4374 = .936+ (MM) 15 / 32 = .468+ 540 / 841 = .642+ (GS) .672+ (MM) 20 / 27 = .740+ 2465/3267 = .754+ (MM) 2510/3267 = .768+ (MM) 204 / 625 = .326+ .504+ (MM) 18 / 25 = .720 .780+ (MM) .818+ (MM) 21 / 25 = .840 (MM) 3 / 8 = .375 30 / 49 = .612+ (GS) 627 / 980 = .639+ (MM) 20 / 27 = .740+ 409 / 540 = .757+ (MM) 209 / 270 = .774+ (MM) 3 / 8 = .375 3 / 4 = .750 19 / 24 = .791+ 24 / 25 = .960 39 / 40 = .975 99 / 100 = .990 39 / 98 = .397+ 3 / 4 = .750 27 / 32 = .843+ 15 / 16 = .937+ 243 / 256 = .949+ .955+ (MM) 3 / 8 = .375 3 / 4 = .750 27 / 32 = .843+ 24 / 25 = .960 39 / 40 = .975 99 / 100 = .990 5 / 12 = .416+ 3 / 4 = .750 149 / 192 = .776+ 77 / 96 = .802+ 53 / 64 = .828+ 41 / 48 = .854+ 156 / 361 = .432+ .547+ (MM) 36 / 49 = .734+ (GS) 132 / 175 = .754+ 948 / 1225 = .773+ 532 / 675 = .788+ (MM) 5 / 12 = .416+ 2 / 3 = .666+ (GS) 149 / 216 = .689+ (MM) 3 / 4 = .750 (MM) ? ? 12 / 25 = .480 = .551+ (MM) .623+ (MM) 2 / 3 = .666+ (GS) ? ? 12 / 25 = .480 540 / 961 = .561+ (MM) .610+ (MM) 120 / 169 = .710+ ? ? 1 / 3 = .333+ .533+ (MM) 18 / 25 = .720 39 / 50 = .780 5 / 6 = .833+ (MM) .880+ (MM) 1 / 3 = .333+ .533+ (MM) .618+ (MM) .692+ (MM) .743+ (MM) 516 / 625 = .825+ (MM) 3 / 8 = .375 .579+ (MM) .699+ (MM) .715+ (MM) .731+ (MM) ? 156 / 361 = .432+ 3 / 4 = .750 51 / 64 = .796+ 27 / 32 = .843+ 169 / 192 = .880+ 11 / 12 = .916+

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