Problem of the Month (July 2008)

Given several copies of a polyomino P, and a polyomino frame F, define pack(P,F) to be the maximum number of copies of P that can be packed without overlap inside F. Given a polyomino P, what is the smallest frame F, so that for some positive integer n, P is the only polyomino with pack(P,F)=n ? In other words, how small can the frame be so that giving the maximum number of copies of a polyomino that fit uniquely determines the polyomino? Are the frames shown below the smallest ones? What about larger polyominoes?


ANSWERS

Here are the smallest known solutions:

Small Polyominoes

(George
Sicherman)

(George
Sicherman)

(George
Sicherman)

Pentominoes

George Sicherman found these polyiamond and polyhex solutions:

Polyiamonds

Polyhexes


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/1/08.