Problem of the Month (October 2009)

This month we examine two problems involving tiling the plane with polyominoes.

1.

Pick polyominoes B (shown in Blue) and Y (shown in Yellow). If we tile the plane with copies of B and Y, what is the largest density that copies of B can have in the plane if the copies don't touch, even at the corners? (In other words, how sparsely can we pack the plane with copies of Y so that the holes don't touch and are congruent to B?)

2.

Pick polyominoes B (shown in Blue), Y (shown in Yellow), and R (shown in Red). Can we tile the plane with B, Y, and R so that no tile touches another of the same color? (In other words, is there a 3-color tiling of the plane with these tiles?)

ANSWERS

Contributors this month include George Sicherman, Jeremy Galvagni, Andrew Bayly, and Joshua Taylor.

1.

Here are the best known results:

Blue Triomino, Yellow Triomino

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

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

Blue Tetromino, Yellow Triomino

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

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

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

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

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

Blue Triomino, Yellow Tetromino

3/8 (GS)
3/11 (GS)
1/3 (GS)
1/3 (GS)
3/11 (GS)

1/3 (GS)
3/14 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

Blue Tetromino, Yellow Tetromino

2/5
1/4
2/5
1/3
1/3

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

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

1/3
0
2/5 (GS)
1/3
1/3

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

Blue Pentomino, Yellow Pentomino

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

1/3 (GS)
3/8 (GS)
2/5 (GS)
3/8 (GS)
1/3 (GS)
1/4 (GS)

1/4 (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)
3/8 (GS)
1/3 (GS)
1/4 (GS)

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

1/4 (GS)
1/3 (GS)
5/14 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)

1/4 (GS)
1/3 (GS)
1/3 (GS)
3/8 (GS)
1/3 (GS)
1/3 (GS)

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

1/3 (GS)
3/8 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/6 (GS)

1/5 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/3 (GS)
1/4 (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)
1/4 (GS)
1/3 (GS)
1/6 (GS)
0 (GS)
1/4 (GS)

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

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

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

5/12 (GS)
3/8 (GS)
5/13 (GS)
1/3 (GS)
1/3 (GS)
3/8 (GS)

3/8 (GS)
5/16 (GS)
3/10 (GS)
2/7 (GS)
1/4 (GS)
1/3 (GS)

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

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

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

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

1/3 (GS)
1/3 (GS)
1/6 (GS)
1/3 (GS)
1/4 (GS)
1/4 (GS)

1/3 (GS)
1/3 (GS)
5/13 (GS)
2/7 (GS)
1/4 (GS)
1/3 (GS)

Blue Trihex, Yellow Trihex

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

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

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

2.

Here are the known results.

Blue Triomino, Yellow Triomino, Red Triomino

(GS)
(GS)
(GS)
(GS)

Blue Tetromino, Yellow Tetromino, Red Tetromino
none
(JG)
(JT)

(GS)

(JG) 		none
(GS)
(GS)
none
(GS) 		?

(GS)
(GS)


(JT)
(GS)
(GS)
?
none (JG)

none (JG)
(GS)
?
(GS)
(GS)

Jeremy Galvagni noted that if the polyominoes didn't have to be on the grid, that there is an additional solution:

Blue Pentomino, Yellow Pentomino, Red Pentomino

(GS) 		? ?
(GS)
(GS) 		?

(GS)
(GS)
(GS)
(GS)
(GS)
(GS)

(GS)
(GS)
(GS) 		?
(GS)
(GS)

(GS)
(GS)
(GS)
(GS)
(GS)
(GS)

(GS)
(GS)
(GS)
(GS)
(GS)
(JT)
?
(GS)
(GS) 		?
(GS)
(GS)
?
(GS)
(GS)
(GS)
(GS)
(GS)
?
(GS)
(GS) 		?
(GS) 		?

(GS)
(GS)
(GS)
(GS)
(GS) 		none (JT)

(GS)
(GS)
(GS) 		?
(GS) 		?
?
(GS)
(GS)
(GS)
(GS)
(GS)

(GS)
(GS)
(GS)
(GS)
(GS) 		?

? ?
(GS) 		? ? ?
? ?
(GS) 		? ? ?

(GS)
(GS)
(GS) 		? ? ?

(GS) 		?
(GS)
(AB) 		? ?

(GS)
(GS)
(GS)
(GS) 		? ?

(GS)
(GS)
(GS)
(GS) 		? ?

(GS)
(GS)
(GS)
(GS) 		?
(GS)

(GS)
(GS) 		? ? ? ?
? ?
(GS)
(AB) 		? ?

(GS) 		? ?
(GS) 		?
(GS)
?
(GS) 		? ? none
(GS)

(GS) 		?
(GS) 		? ?
(GS)


?
?
(GS)
(GS)
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
(GS)

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

(GS)
(GS)
?
(GS)
(GS)

(GS)
?
?
?
(GS)

?
(GS)
(GS)
(GS)
(GS)

?
(GS)
(GS)
(GS)
(GS)

?
(GS)
(GS)
(GS)
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

(GS)
(GS)
(GS)
(GS)
?

?
?
(GS)
(GS)
(GS)

(GS)
?
(GS)
(GS)
(GS)

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
(GS)
?
?

(GS)
?
(GS)
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
?

?
?
?
?
(GS)

?
?
(GS)
?
?

?
?
(GS)
?
?

?
?
?
?
?

?
?
?
(GS)
?

?
?
?
?
(GS)

?
(GS)
?
?
?

?
?
?
(GS)
?

?
?
?
(GS)
?

?
?
?
?
(GS)

?
?
?
?
?

?
none (JT)
?
?
?

George Sicherman sent solutions for all the trihexes as well.

Blue Trihex, Yellow Trihex, Red Trihex

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/26/09.