Problem of the Month (April 2008)

Given a polyomino P and a positive integer n ≥ 2, what is the largest possible shape S so that n congruent non-overlapping copies of S can be packed inside P? (If P is a rectangle, or if n is a multiple of the area of P, then this problem is too easy, so we restrict our attention to the other cases.) Can you improve any of the results below? What about other polyforms? What about other shapes?

ANSWERS

The following people sent in best-known solutions this month: Károly Hajba, Andrew Bayly, Joe DeVincentis, George Sicherman, Gabriele Carelli, Livio Zucca, Jeremy Galvagni, Maurizio Morandi, and Gavin Theobald. I recently heard from Dick Hess, Yoshiyuki Kotani, and Robert Wainwright that they had considered this problem a decade ago. There was much improving upon other's ideas, so all these folks deserve credit.

Here are the best known non-trivial solutions:

Triomino In 4 Parts
1

Triomino In 5 Parts
.959+ (Enlarge)

Triomino In 7 Parts
.956+
(Károly Hajba) 		(Enlarge)

Triomino In 10 Parts
1
(Joe DeVincentis)

Triomino In 11 Parts
.984+
(Maurizio Morandi) 		(Enlarge)

Triomino In 13 Parts
.994+
(Károly Hajba) 		(Enlarge)

Triomino In 14 Parts
1
(Joe DeVincentis)

Triomino In 16 Parts
1

Triomino In 17 Parts
.991+
(Livio Zucca) 		(Enlarge)

Triomino In 19 Parts
95⁄96

Triomino In 20 Parts
1
(Joe DeVincentis)

Tetrominoes In 3 Parts
.962+
(Maurizio Morandi) 		(Enlarge)
15⁄16

Tetrominoes In 5 Parts
.972+
(Károly Hajba) 		(Enlarge)
.947+
(Maurizio Morandi) 		(Enlarge)

Tetrominoes In 6 Parts
.942+
(Maurizio Morandi) 		(Enlarge)

Tetrominoes In 7 Parts
.986+
(Maurizio Morandi) 		(Enlarge)
.985+
(Maurizio Morandi) 		(Enlarge)

Tetrominoes In 9 Parts
1
63⁄64
(Joe DeVincentis)

Tetrominoes In 10 Parts
.971+
(Maurizio Morandi) 		(Enlarge)

Tetrominoes In 11 Parts
.987+
(Maurizio Morandi) 		(Enlarge)
275⁄288
(Joe DeVincentis)

Tetrominoes In 13 Parts
.985+
(Maurizio Morandi) 		(Enlarge)
.981+
(Maurizio Morandi) 		(Enlarge)

Tetrominoes In 14 Parts
.979+
(Maurizio Morandi) 		(Enlarge)

Pentominoes In 2 Parts
.928+
(Dick Hess) 		(Enlarge)
9⁄10
(Dick Hess)
(Dick Hess)
.871+
(Dick Hess) 		(Enlarge)
.864+
(Dick Hess) 		(Enlarge)

Pentominoes In 3 Parts
1
(Dick Hess)
1–ε
(Andrew Bayly)
15⁄16
(Dick Hess)
(Dick Hess)
(Dick Hess)
.925+
(Maurizio Morandi) 		(Enlarge)
9⁄10
(Dick Hess)
(Dick Hess)
(Dick Hess)
(Dick Hess)
.853+
(Dick Hess) 		(Enlarge)

Pentominoes In 4 Parts
1
.944+
(Maurizio Morandi) 		(Enlarge)
9⁄10
(Dick Hess)
(Dick Hess)

Pentominoes In 6 Parts
1
(Dick Hess)
(Dick Hess)
(Dick Hess)
1–ε
(Livio Zucca)
.977+
(Maurizio Morandi) 		(Enlarge)
.976+
(Maurizio Morandi) 		(Enlarge)
24⁄25
(Dick Hess)
(Livio Zucca)
.933+
(Maurizio Morandi) 		(Enlarge)
(Maurizio Morandi) 		(Enlarge)
.908+
(Gavin Theobald) 		(Enlarge)

Pentominoes In 7 Parts
.981+
(Maurizio Morandi) 		(Enlarge)
.970+
(Livio Zucca) 		(Enlarge)
(Livio Zucca) 		(Enlarge)
.967+
(Maurizio Morandi) 		(Enlarge)
.966+
(Livio Zucca) 		(Enlarge)
.960+
(Livio Zucca) 		(Enlarge)
.956+
(Livio Zucca) 		(Enlarge)
.940+
(Livio Zucca) 		(Enlarge)
14⁄15
(Joe DeVincentis)

Pentominoes In 8 Parts
1
(Erich Friedman)
.980+ (Enlarge)
.971+ (Enlarge)
.966+ (Enlarge) (Enlarge)
.965+ (Enlarge)
.960+ (Enlarge) (Enlarge)
24⁄25
(Joe DeVincentis)
.958+ (Enlarge)
19⁄20
(Joe DeVincentis)
(most by Maurizio Morandi)

Pentominoes In 9 Parts
1
(Livio Zucca)
.993+
(Livio Zucca) 		(Enlarge)
.984+
(Maurizio Morandi) 		(Enlarge)
.979+
(Maurizio Morandi) 		(Enlarge)
.971+
(Maurizio Morandi) 		(Enlarge)
.968+
(Livio Zucca) 		(Enlarge)
(Maurizio Morandi) 		(Enlarge)
(Maurizio Morandi) 		(Enlarge)
153⁄160
(Joe DeVincentis)
.949+
(Maurizio Morandi) 		(Enlarge)
(Maurizio Morandi) 		(Enlarge)

Pentominoes In 11 Parts
44⁄45
(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)

(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)

(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)
31⁄32
(Joe DeVincentis)

Pentominoes In 12 Parts
1
(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)

(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)
(Joe DeVincentis)

Hexominoes In 2 Parts
1–ε
(Andrew Bayly)
(Andrew Bayly)
(Gavin Theobald)
(Gavin Theobald)
11⁄12
5⁄6

Hexominoes In 3 Parts
1–ε
(Andrew Bayly)
(Andrew Bayly)
(Andrew Bayly)
(Andrew Bayly)
.878+
(Gavin Theobald) 		(Enlarge)
7⁄8
5⁄6
5⁄6–ε
(George Sicherman)
.821+
(Joe DeVincentis) 		(Enlarge)

Hexominoes In 4 Parts
1–ε
(Andrew Bayly)
11⁄12
(Livio Zucca)
.900+
(Maurizio Morandi) 		(Enlarge)
5⁄6

Hexominoes In 5 Parts
.9725+
(Maurizio Morandi) 		(Enlarge)
35⁄36
.968+
(Maurizio Morandi) 		(Enlarge)
15⁄16
(Gavin Theobald)

(Maurizio Morandi)
.931+
(Maurizio Morandi) 		(Enlarge)
.926+
(Maurizio Morandi) 		(Enlarge)
25⁄27
(Gavin Theobald)
(Gavin Theobald)
.916+
(Maurizio Morandi) 		(Enlarge)
175⁄192
(Gavin Theobald)
.910+
(Maurizio Morandi) 		(Enlarge)
65⁄72
(Gavin Theobald)
(Gavin Theobald)
85⁄96

Triangle in 5 Parts
.978+
(Károly Hajba) 		(Enlarge)

Triangle in 7 Parts
.980+
(Károly Hajba) 		(Enlarge)

Triamond in 5 Parts
29⁄30
(Maurizio Morandi) 		(Enlarge)

Triamond in 7 Parts
20⁄21
(Maurizio Morandi)

Triamond in 8 Parts
1
(Maurizio Morandi)

Pentiamonds in 2 Parts
14⁄15
(George Sicherman)
9⁄10
(George Sicherman)

Pentiamonds in 3 Parts
1
(Maurizio Morandi)
.939+
(Károly Hajba) 		(Enlarge)
14⁄15
(Maurizio Morandi)
9⁄10
(George Sicherman)

Pentiamonds in 4 Parts
1
(George Sicherman)
.951+
(Joe DeVincentis) 		(Enlarge)
19⁄20
(Joe DeVincentis)
.942+
(Maurizio Morandi) 		(Enlarge)

Pentiamonds in 6 Parts
1
(George Sicherman)
591⁄610
(Maurizio Morandi) 		(Enlarge)
29⁄30
(Maurizio Morandi) 		(Enlarge)
24⁄25
(Maurizio Morandi) 		(Enlarge)

Pentiamonds in 7 Parts
279⁄280
(Maurizio Morandi) 		(Enlarge)
.971+
(Maurizio Morandi) 		(Enlarge)
.957+
(Maurizio Morandi) 		(Enlarge)
14⁄15
(George Sicherman)

Pentiamonds in 8 Parts
1
(George Sicherman)
.969+
(Maurizio Morandi) 		(Enlarge)
.966+
(Maurizio Morandi) 		(Enlarge)
.932+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 2 Parts
1–ε
(George Sicherman)

Hexiamonds in 3 Parts
1–ε
(George Sicherman)
(Andrew Bayly)
5⁄6
(Károly Hajba)

Hexiamonds in 4 Parts
.928+
(Maurizio Morandi) 		(Enlarge)
(Maurizio Morandi) 		(Enlarge)
.906+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 5 Parts
.950+
(Maurizio Morandi) 		(Enlarge)
14⁄15
(Maurizio Morandi)
.930+
(Maurizio Morandi) 		(Enlarge)
.927+
(Maurizio Morandi) 		(Enlarge)
25⁄27
(George Sicherman)
(Joe DeVincentis)
.926+
(Maurizio Morandi) 		(Enlarge)
.925+
(Jeremy Galvagni) 		(Enlarge)
.923+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 7 Parts
.955+ (Enlarge)
.950+ (Enlarge) (Enlarge)
.949+ (Enlarge)
.947+ (Enlarge)
.940+ (Enlarge)
.938+ (Enlarge)
14⁄15
(all by Maurizio Morandi)

Heptiamonds in 2 Parts
20⁄21
13⁄14
6⁄7
6⁄7–ε
(all by George Sicherman)

Heptiamonds in 3 Parts
1
(George Sicherman)
(George Sicherman)
20⁄21
.951+ (Enlarge)
.948+ (Enlarge) (Enlarge)
13⁄14
51⁄56
(Andrew Bayly) 		(Enlarge)
.910+ (Enlarge)
.910+ (Enlarge)
108⁄119 (Enlarge)
.900+ (Enlarge)
99⁄112 (Enlarge)
.883+ (Enlarge)
43⁄49 (Enlarge)
6⁄7
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)
(most by Maurizio Morandi)

Heptiamonds in 4 Parts
1
(George Sicherman)
.994+ (Enlarge)
.990+ (Enlarge)
27⁄28
20⁄21
(George Sicherman)
(George Sicherman)
.935+ (Enlarge)
.929+ (Enlarge)
13⁄14
.921+ (Enlarge)
.918+ (Enlarge)
.913+ (Enlarge)
.912+ (Enlarge)
.909+ (Enlarge)
19⁄21
.899+ (Enlarge)
44⁄49 (Enlarge) (Enlarge)
.894+ (Enlarge) (Enlarge)
6⁄7
(George Sicherman)
(George Sicherman)
(most by Maurizio Morandi)

Heptiamonds in 5 Parts
1
(Maurizio Morandi)
20⁄21
.937+
(Maurizio Morandi) 		(Enlarge)
.926+
(Maurizio Morandi) 		(Enlarge)
185⁄203
(Maurizio Morandi) 		(Enlarge)
25⁄28
55⁄63
(most by George Sicherman)

Heptiamonds in 6 Parts
1
(George Sicherman)
41⁄42
(Joe DeVincentis)
(Joe DeVincentis)
.967+
(Joe DeVincentis) 		(Enlarge)
.957+ (Enlarge) (Enlarge)
.956+ (Enlarge)
.953+ (Enlarge)
.952+ (Enlarge)
.950+ (Enlarge)
.945+ (Enlarge)
.945+ (Enlarge)
.943+ (Enlarge)
33⁄35
.942+ (Enlarge)
.942+ (Enlarge)
.938+ (Enlarge)
.937+ (Enlarge)
.934+ (Enlarge)
.932+ (Enlarge)
.926+ (Enlarge)
789⁄854 (Enlarge)
(most by Maurizio Morandi)

Heptiamonds in 8 Parts
1
(George Sicherman)
55⁄56 (Enlarge) (Enlarge)
48⁄49
335⁄343 (Enlarge)
334⁄343 (Enlarge)
.957+ (Enlarge)
328⁄343 (Enlarge) (Enlarge) (Enlarge) (Enlarge)
(Enlarge) (Enlarge) (Enlarge) (Enlarge)
.955+ (Enlarge)
20⁄21
72⁄77 (Enlarge) (Enlarge)
13⁄14
1954⁄2107 (Enlarge)
(most by Maurizio Morandi)

After Joe DeVincentis showed that (1-ε) of a particular hexomino could be covered by 3 pieces, Andrew Bayly proved that at least (1-ε) of any step polyomino could be covered by any number of pieces in this manner. His "proof by picture":

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/29/09.