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

2

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

3

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	?	?
4	?	?
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	?	?

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	?	?
0	?	?

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	?	?
0	?	?

4 Unique Rotations

Copies	Smallest	Area
16		1
15	?	?
14	?	?
13	?	?
12	?	?
11	?	?
10	?	?
9	?	?
8	?	?
7	?	?
6	?	?
5	?	?
4		9
3	?	?
2	?	?
1	?	?
0	?	?

5 Translations

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

5 Unique Translations

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

George Sicherman tackled polyiamonds:

2

Copies	Smallest	Area
4		1
3		4
2		6
1		7
0		10

3

Copies	Smallest	Area
9		1
8		4
7		5
6		6
5		7
4		8
3		10
2		10
1		13
0		15

4

Copies	Smallest	Area
16		1
15		4
14		6
13		5
12		6
11		7
10		6
9	(and 12 others)	10
8		10
7		12
6	(and 18 others)	14
5	(and 24 others)	14
4		13
3		14
2		16
1		16
0		21

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/17/08.