# Problem of the Month(September 2008)

A shape that can tile itself with smaller copies, not necessarily all the same size, is called an irreptile. This month we study the ability of two shapes to tile one another. We say 2 different shapes A and B are n-symbiotic if for every 0 ≤ k ≤ n, k smaller copies of A and (n-k) smaller copies of B will tile either A or B. For shapes A and B, the natural question is to find the smallest n>1 for which A and B are n-symbiotic. When this value of n is finite, we call it the symbiosis number of A and B. Find the symbiosis number of some pairs of shapes. Can you find a 4-symbiotic pair? Are there 3-symbiotic or 2-symbiotic pairs?

Here are the smallest known symbiosis numbers for various pairs of polyominoes:

Symbiosis Numbers of Polyominoes

4 (CB) 4 (CB) 6 7 (AB) 6 7 (GS) 8 (BH) 7 (GS) 4 (AS) 6 (AS)
4 8 5 (AS) 4 (GS) 6 (GS) 10 (CB) 4 (AS) 5 (GS) 6 (AB)
4 6 (GS) 7 (GS) 7 (GS) 10 (GS) 6 (GS) 4 (GS) 7 (GS)
9 (JD) 8 (JD) 11 (BH) 13 (GS) 10 (GS) 7 (GS) 6 (AS)
7 (JD) 6 (GS) 10 (GS) 10 (GS) 5 (JD) 8 (AB)
8 (GS) 10 (GS) 7 (GS) 6 (GS) 8 (GS)
10 (GS) 9 (GS) 7 (GS) 8 (GS)
10 (CB) 10 (GS) 13 (GS)
7 (GS) 9 (GS)
4 (AS)

Symbiosis Numbers of Polytans

3 (AB) 8 (GS) 8 (GS) 5 (GS) 8 (GS)
8 (GS) 8 (GS) 5 (GS) 8 (GS)
8 (GS) 10 (GS) 10 (GS)
8 (GS) 14 (GS)
8 (GS)

Symbiosis Numbers of Polydrafters

4 (AB) 5 (AB) 8 (GS) 8 (GS) 4 (BH) 7 (GS) 7 (GS)
8 (GS) 11 (GS) 8 (GS) 7 (GS) 8 (GS) 7 (GS)
19 (GS) ? 8 (GS) 12 (GS) ?
13 (GS) 13 (GS) ? ?
8 (GS) ? ?
7 (GS) 10 (GS)
8 (GS)

CB = Claudio Baiocchi
BH = Bryce Herdt
AS = Anti Sőlg
AB = Andrew Bayly
JD = Joe DeVincentis
JT = Joshua Taylor
GS = George Sicherman

Here are all the other best known n-symbiotic tilings for n ≤ 6:

Andrew Bayly found the following symbiotic tilings:

Joe DeVincentis found this 2-symbiotic tiling:

George Sicherman found these symbiotic tilings:

Bryce Herdt found an infinite family of symbiotic tilings of trapezoids and triangles that works for all even n≥6:

Andrew Bayly conjectured that an n-symbiotic pair exists for every n>1, though he was only able to prove this for even n. Then Joe DeVincentis provided the argument for odd n.

Bryce Herdt also proved that any 2-symbiotic pair is also n-symbiotic for all n≥3.

Jeremy Galvagni wonders whether every positive integer n>1 is the symbiosis number for some pair of shapes. The smallest n which is in doubt is n=10.

Joshua Taylor also played around with symbiosis numbers of "sliced polyominoes":

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/26/18.