Problem of the Month (April 2016)

Imagine you are standing on an ice floe in the shape of a polyomino, floating in the ocean. Each minute, the polar wind blows you North, South, East or West one square until eventually you are forced off the ice floe into the ocean and drown. You would naturally be interested in the average time that you will live. We are interested in shapes of polyominoes where all of your expected survival times are integer numbers of minutes. We call polyominoes with this property polyomino floes, and show your expected survival time on each square. Notice that each number is one fourth of the sum of one more than the adjacent numbers. Can you show that there are arbitrarily large polyomino floes? Can you show that there are polyomino floes with arbitrarily many holes, or arbitrarily large holes?

Is there a polyomino floe with a square that contains the number 5? Are there polyomino floes containing arbitrarily large integers? What is the smallest size of an integer floe that contains a square labeled n?

What about this problem in 1 dimension or 3 dimensions? What about other lattices, like the triangular or hexagonal lattice? What about other face-transitive tessellations involving regular polygons?

ANSWERS

The best known results are shown below.

Polyominoes

Smallest Polyomino Floe Containing n

n=1

n=2, 4

n=3

n=5

(Kang Jin Cho)

n=6

n=7, 8
n=9
n=10, 12, 14, 16
(George Sicherman)

Infinite Families of Polyomino Floes Containing Sets of Values

Set	Samples		Areas
{1}			4, even n ≥ 8
{1, 2, 3}	(Bryce Herdt)	(Jon Palin)	20, 40+10n, 52+6n, 84+6n, 116+6n, so all even n ≥ 106
{1, 2, 3, 4}	(George Sicherman)		13, 22, 31, even n ≥ 32, odd n ≥ 41
{2, 3, 4}			76+18m+40n+46p+56q, so all even n ≥ 202
{1, 2, 3, 4, 5, 6}	(Kang Jin Cho)		68, 204 + 68n
{1, 2, 3, 4, 6}	(Kang Jin Cho)		244 + 6m + 190n, 260 + 6m + 190n, 418 + 6m so all even n ≥ 250 (Bryce Herdt)
{1, 2, 3, 4, 5, 6, 7}	(George Sicherman)		156n (Bryce Herdt)

Polyiamonds

Smallest Polyiamond Floe Containing n

n=1, 2

n=3

n=4

(George Sicherman)

n=5

n=6

n=7

n=8, 11

n=9, 12
(George Sicherman)
n=10
(George Sicherman)
n=13
(George Sicherman)
n=14, 17, 20
(George Sicherman)
n=15, 18
(George Sicherman)

n=16, 25, 26
(George Sicherman)
n=19, 21, 24, 27, 29, 31, 34
(George Sicherman)
n=22
? n=23
? n=30
(George Sicherman)

Infinite Families of Polyiamond Floes Containing Sets of Values

Set	Samples		Areas
{2}			6, even n ≥ 12
{1, 2, 3, 4, 5}	(George Sicherman)		33 + 12n

Some other small polyiamond floes are shown below. Are there some with arbitrarily many holes?

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

(George Sicherman)

Polyhexes

Smallest Polyhex Floe Containing n

n=1

n=2, 3

n=4
?

Infinite Families of Polyhex Floes Containing Sets of Values

Set	Samples		Areas
{1}		(Bryce Herdt)	24, 32, 34, 36, even n ≥ 40
{1, 2, 3}		(George Sicherman)	multiples of 10

These last pictures show that there are polyhex floes with arbitrarily large holes and arbitrarily many holes.

Triangular/Hexagonal Grid

Smallest Polyform Floe Containing n

n=1, 2

n=3

n=4, 5

n=6

Infinite Families of Polyform Floes Containing Sets of Values

Set	Samples		Areas
{1, 2}			7 + 3n, 15 + 3n, 20 + 3n, so all n ≥ 18
{2}			18, 24 + 3n (Bryce Herdt)
{1, 2, 3}			10, 30, ?

Square/Octagonal Grid

Smallest Polyform Floe Containing n

n=1

n=2
?

Infinite Families of Polyform Floes Containing Sets of Values

Set	Samples		Areas
{1}			20 + 8m + 10n, 34 + 8m + 10n, so all even n ≥ 34

The smallest floe known on the triangular/dodecagonal lattice has 96 cells, shown below.

(Bryce Herdt)

Bryce Herdt considered and completely classified the 1-dimensional floes. All lines of cells are floes, such as 11, 232, 3553, 47874, and their values are quadratic in position.

Bryce Herdt also showed that if we omit one face of a platonic solid in 3-dimensions, the remaining faces form a 2-dimensional floe. A tetrahedron has faces with 2's, a cube has 4's and a 5, an octahedron has 6's, 8's, and a 9, a dodecahedron has 10's, 13's, and a 14, and an icosahedron has 18's, 26's, 31's, 33's, and a 34. He also gave many other subsets of platonic solid faces that served as floes.

Jon Palin considered cubes in 3-dimensions, with 6 possible directions of movement. He showed that any loop of 1's in 2-dimensions has a height 2 analog of cubes. The same argument shows there are arbitrarily large polycube floes in any dimension.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/16.