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?


ANSWERS

Solutions were received from Bryce Herdt, Maurizio Morandi, and George Sicherman.

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

193 / 216 = .893+

197 / 216 = .912+ (MM)

67 / 72 = .930+ (MM)

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+


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/15/17.