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 later 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 2-20 Parts
Parts234567891011
Shape
Coverage111.959+
Maurizio
Morandi
(Enlarge) 		1.983+
Maurizio
Morandi
(Enlarge) 		111
Joe
DeVincentis 		.984+
Maurizio
Morandi
(Enlarge)

Parts121314151617181920
Shape
Coverage 1.994+
Károly
Hajba
(Enlarge) 		1
Joe
DeVincentis		11 .991+
Livio
Zucca
(Enlarge)		1 95⁄961
Joe
DeVincentis

Tetrominoes In 2-14 Parts
Parts2345678
Shape
Coverage 1.962+
Maurizio
Morandi
(Enlarge) 		1.972+
Károly
Hajba
(Enlarge) 		1.985+
Maurizio
Morandi
(Enlarge) 		1
Shape
Coverage 115⁄161 .947+
Maurizio
Morandi
(Enlarge) 		.942+
Maurizio
Morandi
(Enlarge) 		.986+
Maurizio
Morandi
(Enlarge)		1

Parts91011121314
Shape
Coverage 11.987+
Maurizio
Morandi
(Enlarge) 		1.985+
Maurizio
Morandi
(Enlarge)		1
Shape
Coverage .987+
Maurizio
Morandi
(Enlarge) 		.971+
Maurizio
Morandi
(Enlarge) 		.978+
Maurizio
Morandi
(Enlarge)		1 .981+
Maurizio
Morandi
(Enlarge) 		.979+
Maurizio
Morandi
(Enlarge)

Pentominoes In 2-12 Parts
Parts23467891112
Shape
Coverage 9⁄10
Dick Hess 		15⁄16
Dick Hess 		1 .976+
Maurizio Morandi
Enlarge 		.981+
Maurizio Morandi
Enlarge 		.966+
Maurizio Morandi
Enlarge 		.994+
Maurizio Morandi
Enlarge 		44⁄45
Joe DeVincentis 		1
Joe DeVincentis
Shape
Coverage .928+
Dick Hess
Enlarge 		1
Dick Hess 		1 .977+
Maurizio Morandi
Enlarge 		.966+
Livio Zucca
Enlarge 		.960+
Maurizio Morandi
Enlarge 		1 .985+
Maurizio Morandi
Enlarge 		1
Shape
Coverage .871+
Dick Hess
Enlarge 		9⁄10
Dick Hess
1 .933+
Maurizio Morandi
Enlarge 		.956+
Livio Zucca
Enlarge 		19⁄20
Joe DeVincentis 		.968+
Livio Zucca
Enlarge 		44⁄45
Joe DeVincentis 		1
Joe DeVincentis
Shape
Coverage 9⁄10
Dick Hess
1–ε
Andrew Bayly 		9⁄10
Dick Hess
24⁄25
Livio Zucca 		14⁄15 .960+
Maurizio Morandi
Enlarge 		.949+
Maurizio Morandi
Enlarge 		44⁄45
Joe DeVincentis 		.970+
Maurizio Morandi
Enlarge
Shape
Coverage 1 9⁄10
Dick Hess
1 .933+
Maurizio Morandi
Enlarge 		.967+
Maurizio Morandi
Enlarge 		.971+
Maurizio Morandi
Enlarge 		.993+
Livio Zucca
Enlarge 		44⁄45
Joe DeVincentis 		1
Joe DeVincentis
Shape
Coverage 1 15⁄16
Dick Hess 		1 1
Dick Hess 		.970+
Livio Zucca
Enlarge 		.966+
Maurizio Morandi
Enlarge 		.968+
Maurizio Morandi
Enlarge 		44⁄45
Joe DeVincentis 		1
Joe DeVincentis
Shape
Coverage 1 .925+
Maurizio Morandi
Enlarge 		.944+
Maurizio Morandi
Enlarge 		1
Dick Hess 		.957+
Maurizio Morandi
Enlarge 		.980+
Maurizio Morandi
Enlarge 		.979+
Maurizio Morandi
Enlarge 		44⁄45
Joe DeVincentis 		.984+
Maurizio Morandi
Enlarge
Shape
Coverage 1 9⁄10
Dick Hess 		9⁄10
Dick Hess 		.908+
Gavin Theobald
Enlarge 		14⁄15 .960+
Maurizio Morandi
Enlarge 		.950+
Bryce Herdt
Enlarge 		44⁄45
Joe DeVincentis 		24⁄25
Bryce Herdt
Shape
Coverage 1 .869+
Maurizio Morandi
Enlarge 		1 1–ε
Livio Zucca 		14⁄15
Joe DeVincentis 		1 153⁄160
Joe DeVincentis 		.969+
Maurizio Morandi
Enlarge 		1
Joe DeVincentis
Shape
Coverage .864+
Dick Hess
Enlarge 		9⁄10
Dick Hess 		1 24⁄25
Dick Hess 		.960+
Livio Zucca
Enlarge 		.958+
Maurizio Morandi
Enlarge 		.971+
Maurizio Morandi
Enlarge 		.989+
Maurizio Morandi
Enlarge 		1
Joe DeVincentis
Shape
Coverage 1 15⁄16
Dick Hess 		1 1
Dick Hess 		.970+
Livio Zucca
Enlarge 		.966+
Maurizio Morandi
Enlarge 		.968+
Maurizio Morandi
Enlarge 		44⁄45
Joe DeVincentis 		1
Joe DeVincentis

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

Hexominoes In 3 Parts
1–ε
(Andrew Bayly)
(Andrew Bayly)
(Andrew Bayly)
(Andrew Bayly)
.935+
(Maurizio Morandi) 		(Enlarge)
11⁄12
(Maurizio Morandi)
.878+
(Gavin Theobald) 		(Enlarge)
7⁄8
(Maurizio Morandi)
(Maurizio Morandi)
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)
.855+
(Maurizio Morandi) 		(Enlarge)
5⁄6

Hexominoes In 5 Parts
1-ε
(Maurizio Morandi)
.9725+
(Maurizio Morandi) 		(Enlarge)
35⁄36
.968+
(Maurizio Morandi) 		(Enlarge)
.963+
(Maurizio Morandi) 		(Enlarge)
.937+
(Maurizio Morandi) 		(Enlarge)
15⁄16
(Gavin Theobald)

(Maurizio Morandi)
14⁄15
(Maurizio Morandi) 		(Enlarge)
.932+
(Maurizio Morandi) 		(Enlarge)
.931+
(Maurizio Morandi) 		(Enlarge)
25⁄27
(Gavin Theobald)
(Gavin Theobald)
(Maurizio Morandi)
(Maurizio Morandi)
(Maurizio Morandi)
.916+
(Maurizio Morandi) 		(Enlarge)
.910+
(Maurizio Morandi) 		(Enlarge)
65⁄72
(Gavin Theobald)

Triangle in 5 Parts
5(21√3-34)/12 = .988+
(Maurizio Morandi) 		(Enlarge)

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

Triangle in 10 Parts
5(48√3 - 35)/242 = .994+
(Maurizio Morandi) 		(Enlarge)

Triamond in 5 Parts
5(21√3 - 34)/12 = .988+
(Maurizio Morandi) 		(Enlarge)

Triamond in 7 Parts
7(32√3 - 55)/3 = .993+
(Maurizio Morandi) 		(Enlarge)

Triamond in 8 Parts
1
(Maurizio Morandi)

Pentiamonds in 2 Parts
14⁄15
(George Sicherman)
.903+
(Maurizio Morandi) 		(Enlarge)

Pentiamonds in 3 Parts
1
(Maurizio Morandi)
.942+
(Maurizio Morandi) 		(Enlarge)
14⁄15
(Maurizio Morandi)
9⁄10
(George Sicherman)

Pentiamonds in 4 Parts
1
(George Sicherman)
.951+
(Joe DeVincentis) 		(Enlarge)
19⁄20
(Joe DeVincentis)
.945+
(Maurizio Morandi) 		(Enlarge)

Pentiamonds in 6 Parts
1
(George Sicherman)
.976+
(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)
.935+
(Maurizio Morandi) 		(Enlarge)

Pentiamonds in 8 Parts
1
(George Sicherman)
.989+
(Maurizio Morandi) 		(Enlarge)
.971+
(Maurizio Morandi) 		(Enlarge)
.963+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 2 Parts
1–ε
(George Sicherman)

Hexiamonds in 3 Parts
1–ε
(George Sicherman)
(Andrew Bayly)
.894+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 4 Parts
.932+
(Maurizio Morandi) 		(Enlarge)
.928+
(Maurizio Morandi) 		(Enlarge)
.906+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 5 Parts
185⁄192
(Károly Hajba) 		(Enlarge)
.950+
(Maurizio Morandi) 		(Enlarge)
.948+
(Maurizio Morandi) 		(Enlarge)
.940+
(Maurizio Morandi) 		(Enlarge)
14⁄15
(Maurizio Morandi)
.932+
(Károly Hajba) 		(Enlarge)
.927+
(Maurizio Morandi) 		(Enlarge)
25⁄27
(George Sicherman)
.923+
(Maurizio Morandi) 		(Enlarge)

Hexiamonds in 7 Parts
35⁄36 (Enlarge)
.972+ (Enlarge)
.965+ (Enlarge)
.950+ (Enlarge) (Enlarge)
.949+ (Enlarge)
.947+ (Enlarge)
.941+ (Enlarge)
14⁄15
(most 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)
41⁄42
20⁄21
.951+ (Enlarge)
.948+ (Enlarge) (Enlarge)
.933+ (Enlarge)
13⁄14
.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)
.907+ (Enlarge) (Enlarge)
19⁄21
.899+ (Enlarge)
44⁄49 (Enlarge) (Enlarge)
.864+ (Enlarge)
6⁄7
(George Sicherman)
(most by Maurizio Morandi)

Heptiamonds in 5 Parts
1
20⁄21
(George Sicherman)
355⁄378 (Enlarge)
.937+ (Enlarge)
.932+ (Enlarge)
13⁄14 (Enlarge)
.926+ (Enlarge)
25⁄27 (Enlarge)
.923+ (Enlarge)
205⁄224 (Enlarge)
32⁄35 (Enlarge)
115⁄126 (Enlarge) (Enlarge) (Enlarge)
25⁄28
(George Sicherman)
(most by Maurizio Morandi)

Heptiamonds in 6 Parts
1
(George Sicherman)
41⁄42
(Joe DeVincentis)
(Joe DeVincentis)
.978+
(Maurizio Morandi) 		(Enlarge)
.978+
(Maurizio Morandi) 		(Enlarge)
.967+
(Joe DeVincentis) 		(Enlarge)
.961+ (Enlarge)
.957+ (Enlarge) (Enlarge)
.956+ (Enlarge) (Enlarge)
.953+ (Enlarge) (Enlarge)
.952+ (Enlarge)
.950+ (Enlarge)
.945+ (Enlarge)
.945+ (Enlarge) (Enlarge) (Enlarge)
33⁄35
.942+ (Enlarge)
.937+ (Enlarge)
.926+ (Enlarge)
(most by Maurizio Morandi)

Heptiamonds in 8 Parts
1
(George Sicherman)
55⁄56 (Enlarge) (Enlarge)
48⁄49
335⁄343 (Enlarge)
.975+ (Enlarge)
334⁄343 (Enlarge)
.960+ (Enlarge)
121⁄126 (Enlarge) (Enlarge) (Enlarge)
.957+ (Enlarge)
328⁄343 (Enlarge) (Enlarge) (Enlarge)
(Enlarge) (Enlarge)
.955+ (Enlarge)
20⁄21
72⁄77 (Enlarge) (Enlarge)
13⁄14
(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 3/24/17.