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?)

1.

Here are the best known results:

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

	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)

	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)

	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)

	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)

	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.

(GS)

(GS)

(GS)

(GS)

		none	(JG)	(JT)
			(GS)
	(JG)	none		(GS)	(GS)
		none	(GS)		none (PH)
			(GS)	(GS)

(JT)	(GS)	(GS)	none (PH)	none (JG)
none (JG)	(GS)	none (PH)	(GS)	(GS)

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

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

	none (GS)	(PH)	(GS)	none (GS)	none (GS)	none (GS)
	(GS)	(PH)	(GS)	none (GS)	none (GS)	(PH)
	(GS)	(GS)	(GS)	(PH)	none (GS)	none (GS)
	(GS)	(PH)	(GS)	(AB)	none (GS)	(PH)
	(GS)	(GS)	(GS)	(GS)	none (GS)	none (GS)
	(GS)	(GS)	(GS)	(GS)	none (GS)	(PH)
	(GS)	(GS)	(GS)	(GS)	none (GS)	(GS)
	(GS)	(GS)	none (GS)	none (GS)	none (GS)	none (GS)
	none (GS)	none (GS)	(GS)	(AB)	none (PH)	(PH)
	(GS)	none (GS)	none (GS)	(GS)	none (GS)	(GS)
	none (GS)	(GS)	none (GS)	none (GS)	none	(GS)
	(GS)	none (GS)	(GS)	(PH)	none (PH)	(GS)

?	(GS)	(GS)	(GS)	none (GS)
none (GS)	(PH)	(PH)	none (PH)	none (PH)
none (GS)	none (PH)	none (PH)	(PH)	(PH)
none (PH)	none (PH)	none (PH)	none (PH)	(GS)
none (PH)	(PH)	none (PH)	(PH)	none (PH)
none (PH)	none (PH)	none (PH)	(PH)	(PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
(GS)	(GS)	?	(GS)	(GS)
(GS)	(PH)	none (PH)	(PH)	(GS)
(PH)	(GS)	(GS)	(GS)	(GS)
none (PH)	(GS)	(GS)	(GS)	(GS)
?	(GS)	(GS)	(GS)	(PH)
none (PH)	(PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
(PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	(PH)	none (PH)	(PH)	none (PH)
(GS)	(GS)	(GS)	(GS)	none (PH)
none (PH)	none (PH)	(GS)	(GS)	(GS)
(GS)	none (PH)	(GS)	(GS)	(GS)
(GS)	(PH)	none (PH)	(PH)	none (PH)
none (PH)	(PH)	none (PH)	none (PH)	none (PH)
(PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	(GS)	(GS)	(GS)	(GS)
(GS)	none (PH)	(GS)	?	(PH)
none (PH)	none (PH)	none (PH)	(PH)	(PH)
(PH)	none (PH)	none (PH)	(PH)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (PH)	(PH)	none (PH)	(GS)
none (PH)	none (PH)	(GS)	none (PH)	none (PH)
(PH)	(PH)	(GS)	(PH)	(GS)
(PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	(PH)	(PH)	(GS)	none (PH)
(PH)	none (PH)	none (PH)	none (PH)	(GS)
(PH)	(GS)	none (PH)	(PH)	none (PH)
none (PH)	none (PH)	none (PH)	(GS)	none (PH)
none (PH)	none (PH)	none (PH)	(GS)	none (PH)
none (PH)	none (PH)	none (PH)	none (PH)	(GS)
none (PH)	none (PH)	none (PH)	none (PH)	none (PH)
none (PH)	none (JT)	none (PH)	none (PH)	none (PH)

George Sicherman sent solutions for all the trihexes as well.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/16/10.