Strong Surround Numbers for Polycairos

A polycairo is a plane figure formed by joining isosceles pentagons in the Cairo Grid. The strong surround number of a polycairo is the fewest number of copies of the polycairo that can surround it strongly; that is, including its corners. The polycairos must conform to the Cairo grid.

Strong surround numbers for polyominoes were proposed by Jaime Poniachik in Issue 8 of Puzzle Fun. He asked for the smallest polyominoes with a given strong surround number. In Issue 10, Rodolfo Kurchan extended the problem to polyiamonds, polyhexes, and polyaboloes. He also investigated the smallest polyforms that cannot surround themselves, and the smallest holeless such polyforms. However, his results were not complete.

Here I show minimal strong surrounds for small polycairos, the smallest polycairos with given surround numbers, and the smallest polycairos with no strong surrounds.

See also Strong Surround Numbers for Polyaboloes.

Minimal Strong Surrounds

An exclamation point (!) indicates that the solution is unique for the minimum number of tiles.





Minimal Polycairos with Given Strong Surround Numbers

3 Copies, 9 Cells

4 Copies, 3 Cells

5 Copies, 3 Cells

6 Copies, 2 Cells

7 Copies, 1 Cell

8 Copies, 2 Cells

9 Copies, 6 Cells

10 Copies, 8 Cells

11 Copies, 8 Cells

12 Copies, 7 Cells

Minimal Polycairos that Cannot Surround Themselves Strongly

Last revised 2023-06-30.

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Col. George Sicherman [ HOME | MAIL ]