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, 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+

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.