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
CopiesSmallestArea
41
3 4
2 5
1 5
0 7

2 Unique
CopiesSmallestArea
41
3 6
2 5
1 (and 19 others) 9
0 7

2 Rotations
CopiesSmallestArea
41
3 4
2 4
1 5
0 7

2 Unique Rotations
CopiesSmallestArea
41
3 7
2 5
1
8
0 7

2 Translations
CopiesSmallestArea
41
3 3
2 4
1 5
0 7

2 Unique Translations
CopiesSmallestArea
41
3 3
2 5
1 7
0 7



3
CopiesSmallestArea
9 1
8 4
7
5
6 5
5 6
4
7
3 7
2 7
1 9
0
10

3 Unique
CopiesSmallestArea
9 1
8 4
7 6
6 7
5? ?
4? ?
3 7
2 (and 17 others)11
1 11
0
10

3 Rotations
CopiesSmallestArea
9 1
8 4
7 4
6 5
5 6
4 6
3 7
2 7
1 9
0
10

3 Unique Rotations
CopiesSmallestArea
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
CopiesSmallestArea
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
CopiesSmallestArea
9 1
8 8
7 3
6 5
5 6
4 7
3 7
2 7
1 10
0

10



4 Translations
CopiesSmallestArea
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
CopiesSmallestArea
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
CopiesSmallestArea
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
CopiesSmallestArea
16 1
15? ?
14? ?
13? ?
12? ?
11? ?
10? ?
9? ?
8? ?
7? ?
6??
5??
49
3? ?
2? ?
1? ?
0? ?



5 Translations
CopiesSmallestArea
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
CopiesSmallestArea
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
CopiesSmallestArea
41
34
26
17
010

3
CopiesSmallestArea
91
84
75
66
57
48
310
210
113
015

4
CopiesSmallestArea
161
154
146
135
126
117
106
9 (and 12 others)10
810
712
6 (and 18 others)14
5 (and 24 others)14
413
314
216
116
021


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.