Problem of the Month (April 2006)

This month we investigate how many size-1 copies of a polyomino can fit inside a size-n copy. In particular, for n≥2 and 0≤k≤n², what are the smallest area polyominoes P so that exactly k size-1 copies of P will fit inside a size-n copy of P? There are separate cases as to whether we can use all rigid motions of P, only rotations of P, or only translations of P, and whether the packing is unique.

ANSWERS

Copies	Smallest	Area
4		1
3		4
2		5
1		5
0		7

2 Unique

Copies	Smallest	Area
4		1
3		6
2		5
1	(and 19 others)	9
0		7

2 Rotations

Copies	Smallest	Area
4		1
3		4
2		4
1		5
0		7

2 Unique Rotations

Copies	Smallest	Area
4		1
3		7
2		5
1		8
0		7

2 Translations

Copies	Smallest	Area
4		1
3		3
2		4
1		5
0		7

2 Unique Translations

Copies	Smallest	Area
4		1
3		3
2		5
1		7
0		7

Copies	Smallest	Area
9		1
8		4
7		5
6		5
5		6
4		7
3		7
2		7
1		9
0		10

3 Unique

Copies	Smallest	Area
9		1
8		4
7		6
6		7
5	(George Sicherman)	8
4	(George Sicherman)	8
3		7
2	(and 17 others)	11
1		11
0		10

3 Rotations

Copies	Smallest	Area
9		1
8		4
7		4
6		5
5		6
4		6
3		7
2		7
1		9
0		10

3 Unique Rotations

Copies	Smallest	Area
9		1
8		5
7		4
6		6
5		7
4		7
3		7
2	(and 44 others)	11
1		10
0		10

3 Translations

Copies	Smallest	Area
9		1
8		8
7		3
6		4
5		5
4	(and 10 others)	5
3		5
2		7
1		8
0		10

3 Unique Translations

Copies	Smallest	Area
9		1
8		8
7		3
6		5
5		6
4		7
3		7
2		7
1		10
0		10

4 Translations

Copies	Smallest	Area
16		1
15	?	?
14		7
13		6
12		3
11		4
10		5
9		5
8		5
7		6
6		7
5		7
4		8
3	(and 10 others)	9
2		9
1		10
0	(and 17 others) (George Sicherman)	13

4 Unique Translations

Copies	Smallest	Area
16		1
15	?	?
14		7
13		6
12		5
11		7
10		6
9		8
8		7
7		7
6		8
5		9
4		9
3		9
2	(and 21 others)	11
1	(George Sicherman)	13
0	(and 17 others) (George Sicherman)	13

4 Rotations

Copies	Smallest	Area
16		1
15		5
14		5
13		4
12		5
11		6
10		5
9		7
8		7
7		8
6	(and 9 others)	9
5	(and 9 others)	9
4	(and 10 others)	9
3		9
2		9
1	(George Sicherman)	12
0	(George Sicherman)	13

4 Unique Rotations

Copies	Smallest	Area
16		1
15	?	?
14	?	?
13	?	?
12	(George Sicherman)	8
11	(George Sicherman)	8
10	(George Sicherman)	8
9	(George Sicherman)	9
8	(George Sicherman)	10
7	(George Sicherman)	10
6	(George Sicherman)	10
5	(George Sicherman)	11
4		9
3	(George Sicherman)	10
2	(and 61 others) (George Sicherman)	14
1	(George Sicherman)	13
0	(George Sicherman)	13

5 Translations

Copies	Smallest	Area
25		1
24	?	?
23	?	?
22		4
21	?	?
20		5
19		3
18		4
17		5
16		5
15	(George Sicherman)	6
14		5
13	(George Sicherman)	6
12	(George Sicherman)	7
11	(George Sicherman)	7
10	(George Sicherman)	8
9	(George Sicherman)	8
8	(and 13 others) (George Sicherman)	9
7	(George Sicherman)	9
6	(George Sicherman)	9
5	(George Sicherman)	10
4	(and 20 others) (George Sicherman)	11
3	(and 14 others) (George Sicherman)	11
2	(George Sicherman)	11
1	(George Sicherman)	13
0	?	?