Problem of the Month (March 2016)

In October 2009, our problem was to find the sparsest packing of the plane by a polyomino Y so that all the holes were congruent to a polyomino B, and the holes did not touch, even at the corners. This month we relax the condition that the holes can't touch at corners. So what is the densest packing of holes B so that the rest of the plane can be tiled by copies of Y?

ANSWERS

Improvements were received by Bryce Herdt, George Sicherman, Jeremy Galvagni, and Maurizio Morandi. The best known solutions are shown below. The holes are color-coded by their density. Can you improve any of the packings shown? Can you extend the results to larger polyominoes?

Monomino Holes
1
1/2 		2
1/3 		3
1/3 (BH)
1/4

4
1/3
1/3
3/11
1/5
1/5

5
3/8
1/3
1/3
2/7
2/7
2/7

2/7
3/13
3/13
3/13
3/13
1/6

6
1/3
1/3
1/3
1/3
5/17
5/17
5/17

5/17
5/17 (GS)
5/17 (GS)
5/17 (GS)
5/17 (GS)
5/17 (GS)
5/17 (GS)

3/11 (GS)
1/4
1/4
1/4
1/4
1/4
1/4

1/4
1/4
1/4
1/4
1/4
1/4
1/4

1/5
1/5
1/5
1/5
1/5
1/7
1/7

7
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)

3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)

3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)
3/10 (GS)

3/10 (GS)
3/10 (GS)
5/19 (BH)
5/19 (BH)
5/19 (BH)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)

5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)

5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)

5/19 (GS)
5/19 (GS)
5/19 (GS)
5/19 (GS)
1/4 (GS)
9/37 (GS)
9/37 (GS)
9/37 (GS)
2/9 (GS)

2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)

2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)

2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)

2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
2/9 (GS)
3/17 (GS)
3/17 (GS)

3/17 (GS)
3/17 (GS)
3/17 (GS)
3/17 (GS)
3/17 (GS)
3/17 (GS)
5/33 (GS)
1/8 (GS)
1/8 (GS)

Domino Holes
1
1/2 		2
1/2 		3
2/5
2/5

4
3/7
1/3
1/3
1/3
1/3

5
4/9
3/8
3/8
3/8
3/8
3/8

3/8 (GS)
1/3 (GS)
2/7
2/7
2/7
2/7 (GS)

6
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)

2/5 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
5/17 (GS)
2/7 (GS)

1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)

Straight Triomino Holes
1
1/2 		2
3/7 		3
1/2
2/5

4
3/7
3/7
3/7
1/3 (GS)
3/11

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)
3/13 (GS)
none (GS)

6
3/7 (GS)
2/5 (GS)
2/5 (GS)
3/8 (GS)
3/8 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
3/11 (GS)
1/4 (GS)
1/5 (GS)
1/5 (GS)

Bent Triomino Holes
1
18/35 (GS) 		2
1/2 		3
1/2
2/5 (GS)

4
3/7
3/7
3/7 (GS)
1/3
1/3 (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)

6
3/7 (GS)
3/7 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/4 (GS)
1/4 (GS)

I Tetromino Holes
1
1/2 (GS) 		2
1/2 (GS) 		3
2/5 (GS)
2/5 (GS)

4
1/2 (GS)
3/7 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)

2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
none (GS)

O Tetromino Holes
1
1/2 (GS) 		2
1/2 (GS) 		3
2/5 (GS)
2/5 (GS)

4
1/2 (GS)
1/2 (GS)
1/3 (GS)
4/11 (GS)
4/11 (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)

4/9 (GS)
4/9 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)
2/7 (GS)

T Tetromino Holes
1
4/7 (GS) 		2
4/9 (GS) 		3
4/9 (GS)
2/5 (GS)

4
1/2 (GS)
2/5 (GS)
1/3 (GS)
4/9 (GS)
none (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
8/23 (GS)

8/23 (GS)
8/23 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
none (GS)

S Tetromino Holes
1
4/7 (GS) 		2
1/2 (GS) 		3
2/5 (GS)
2/5 (GS)

4
1/2 (GS)
3/7 (GS)
2/5 (GS)
1/3 (GS)
1/3 (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)

4/9 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)

L Tetromino Holes
1
8/15 (GS) 		2
1/2 (GS) 		3
8/17 (GS)
4/9 (GS)

4
1/2 (GS)
2/5 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)
4/9 (GS)

4/9 (GS)
4/9 (GS)
3/8 (GS)
8/23 (GS)
2/7 (GS)
none (GS)

George Sicherman thought we should also consider polyiamonds:

Triangle Holes
1
1/2 (GS) 		2
1/3 (GS) 		3
5/14 (GS) 		4
1/3 (GS)
1/3 (GS)
3/11 (GS)

5
2/7 (GS)
2/7 (GS)
1/4 (GS)
11/56 (GS)

6
4/13 (GS)
4/13 (GS)
10/37 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)

1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)

Diamond Holes
1
5/9 (GS) 		2
1/2 (GS) 		3
2/5 (GS) 		4
3/8 (GS)
1/3 (GS)
1/3 (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
2/7 (GS)

6
2/5 (GS)
2/5 (GS)
5/14 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/4 (GS)
1/4 (GS)

Triamond Holes
1
2/3 (GS) 		2
1/2 (GS) 		3
1/2 (GS) 		4
3/7 (GS)
3/7 (GS)
1/3 (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

6
3/7 (GS)
2/5 (GS)
2/5 (GS)
3/8 (GS)
4/11 (GS)
1/3 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
none (GS)

I Tetriamond Holes
1
4/5 (GS) 		2
1/2 (GS) 		3
8/17 (GS) 		4
1/2 (GS)
1/2 (GS)
3/8 (GS)

5
4/9 (GS)
4/9 (GS)
4/9 (GS)
2/7 (GS)

6
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)

2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
1/3 (GS)
none (GS)

C Tetriamond Holes
1
4/7 (GS) 		2
1/2 (GS) 		3
2/5 (GS) 		4
1/2 (GS)
2/5 (GS)
none (GS)

5
4/9 (GS)
4/9 (GS)
1/3 (GS)
2/7 (GS)

6
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
1/3 (GS)

4/13 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (MM)

A Tetriamond Holes
1
2/3 (GS) 		2
1/2 (GS) 		3
2/5 (GS) 		4
1/2 (GS)
1/2 (GS)
1/4 (GS)

5
2/5 (GS)
3/8 (GS)
2/7 (GS)
2/7 (GS)

6
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
2/5 (GS)
4/13 (GS)

4/13 (GS)
4/13 (MM)
2/11 (GS)
4/25 (GS)
none (GS)
none (GS)

George Sicherman thought we should also consider polyhexes:

Monohex Holes
1
1/3 (GS) 		2
1/3 (GS) 		3
1/3 (GS)
1/4 (GS)
1/4 (GS)

4
1/3 (GS)
1/3 (GS)
1/3 (GS)
3/11 (GS)
3/11 (GS)
3/13 (GS)
1/5 (GS)

5
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
2/7 (GS)

2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
3/11 (GS)

3/11 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
1/4 (GS)
3/13 (GS)

3/13 (GS)
3/13 (GS)
3/13 (GS)
11/51 (GS)
1/6 (GS)

Dihex Holes
1
2/5 (GS) 		2
2/5 (GS) 		3
2/5 (GS)
2/5 (GS)
7/22 (GS)

4
2/5 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
5/13 (GS)

5
2/5 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
14/39 (GS)
8/23 (GS)

1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
2/7 (GS)

2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)
2/7 (GS)

D Trihex Holes
1
4/9 (GS) 		2
4/9 (GS) 		3
3/7 (GS)
2/5 (GS)
1/3 (GS)

4
3/7 (GS)
3/7 (GS)
3/7 (GS)
3/7 (GS)
3/7 (GS)
1/3 (GS)
1/3 (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
1/3 (GS)

2/7 (GS)
2/7 (GS)
2/7 (GS)
12/47 (GS)
3/13 (GS)

C Trihex Holes
1
3/7 (GS) 		2
3/7 (GS) 		3
3/7 (GS)
1/3 (GS)
1/3 (GS)

4
3/8 (GS)
3/8 (GS)
5/13 (GS)
9/25 (GS)
1/3 (GS)
3/10 (GS)
none (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
12/37 (GS)
9/29 (GS)

9/29 (GS)
3/10 (GS)
2/7 (GS)
2/7 (GS)
12/47 (GS)

I Trihex Holes
1
3/7 (GS) 		2
3/7 (GS) 		3
2/5 (GS)
3/8 (GS)
1/3 (GS)

4
3/7 (GS)
3/7 (GS)
3/7 (GS)
3/8 (GS)
9/25 (GS)
3/10 (GS)
none (GS)

5
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)
3/8 (GS)

9/29 (GS)
2/7 (GS)
3/11 (GS)
3/13 (GS)
3/13 (GS)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/19/16.